Content-Type: multipart/mixed; boundary="-------------0307170315994" This is a multi-part message in MIME format. ---------------0307170315994 Content-Type: text/plain; name="03-336.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-336.keywords" Nonlinear wave equation; Periodic solutions; Lindstedt series method; Tree formalism; Perturbation theory; Counterterms; Renormalization Group; Diophantine conditions. < ---------------0307170315994 Content-Type: application/postscript; name="gm-pde2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="gm-pde2.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.78 Copyright 1998 Radical Eye Software (www.radicaleye.com) %%Title: gm-pde2.dvi %%Pages: 42 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: dvips -o gm-pde2.ps gm-pde2 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.07.17:0947 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict 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b(is)f(analytic)g(in)h Fr(")p Fu(.)189 5031 y(The)38 b(ab)r(o)n(v)n(e)e(result)h(is)h(not)f(new,)j (as)d(it)h(w)n(as)f(pro)n(v)n(ed)f(b)n(y)h(Kuksin)g([13],)i(P\177)-42 b(osc)n(hel)36 b([15])h(and)118 5138 y(W)-7 b(a)n(yne)22 b([16])g(b)n(y)g(sho)n(wing)f(that)h(the)h(nonlinear)e(w)n(a)n(v)n(e)g (equation)h(can)g(b)r(e)g(written)h(as)f(an)g(in\014nite-)118 5244 y(dimensional)31 b(Hamiltonian)h(system)g(and)g(extending)f(in)h (a)g(suitable)g(w)n(a)n(y)e(KAM)i(tec)n(hniques;)1633 5443 y(3)p eop %%Page: 4 4 4 3 bop 118 555 a Fu(the)29 b(existence)g(of)g(a)f(p)r(erio)r(dic)h(or) f(quasi-p)r(erio)r(dic)g(solution)g(means)h(the)g(existence)f(of)h(a)g (\014nite-)118 662 y(dimensional)g(in)n(v)-5 b(arian)n(t)29 b(torus.)43 b(Indeed)30 b(m)n(uc)n(h)g(more)f(general)g(results)g (exist)h(ab)r(out)g(the)g(exis-)118 768 y(tence)h(of)g(p)r(erio)r(dic)g (or)g(quasi-p)r(erio)r(dic)f(solutions)g(of)h(nonlinear)f(PDEs,)h (based)g(on)g(the)g(ab)r(o)n(v)n(e)118 874 y(KAM)d(metho)r(ds)g(or)f 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Fr(!)s(;)14 b(")p Fu(\))p Fr(;)g(")p Fu(\))27 b(is)h(not)g(analytic)f(in)h Fr(")p Fu(,)f(it)h(turns)g(out)g(that)g(the)118 1618 y(function)k(^)-47 b Fr(u)490 1630 y Fp(n;m)613 1618 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))26 b(\(that)g(is)g(the)h (function)f(obtained)g(b)n(y)g(k)n(eeping)f(\014xed)h(the)g(parameters) 31 b(~)-49 b Fr(!)s Fu(,)118 1725 y(without)22 b(expliciting)g(the)g (dep)r(endence)g(of)28 b(~)-48 b Fr(!)24 b Fu(on)d Fr(")p Fu(\))h(is)f(analytic)g(in)h Fr(")p Fu(,)h(pro)n(vided)e(that)28 b(~)-49 b Fr(!)25 b Fu(satis\014es)118 1831 y(the)34 b(conditions)f(\(1.15\))g(and)h Fr(")f Fu(is)g(small)h(enough.)54 b(By)33 b(in)n(v)n(erting)g(\(1.13\))g(one)g(writes)40 b(~)-49 b Fr(!)37 b Fu(as)32 b(a)118 1937 y(function)27 b(of)f Fr(!)j Fu(and)d Fr(")d Fo(2)g(E)7 b Fu(.)37 b(The)26 b(main)h(adv)-5 b(an)n(tage)24 b(of)i(suc)n(h)g(an)g(approac)n(h)f(is)h (that)g(it)h(is)f(rather)118 2044 y(constructiv)n(e,)36 b(in)f(the)g(sense)f(that)h(w)n(e)g(obtain)f(an)h(explicit)g(represen)n (tation)e(of)i(the)g(p)r(erio)r(dic)118 2150 y(solution)25 b(on)f(the)i(nonlinear)e(w)n(a)n(v)n(e)f(equation.)36 b(As)25 b(a)g(new)g(result,)g(it)g(turns)g(out)g(the)h(analyticit)n(y) 118 2256 y(of)37 b(^)-48 b Fr(u)264 2268 y Fp(n;m)388 2256 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))31 b(as)g(a)f (function)i(of)g Fr(")p Fu(,)g(at)f(\014xed)38 b(~)-49 b Fr(!)s Fu(.)48 b(Our)31 b(analysis)e(of)j(the)f(Lindstedt)h(series)f (for)123 2363 y(^)-47 b Fr(u)166 2375 y Fp(n;m)289 2363 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))25 b(is)f(based)g(on)g (Renormalization)f(Group)h(metho)r(ds)h(similar)f(to)g(the)h(ones)f (applied)g(to)118 2469 y(the)32 b(problem)f(of)g(the)h(con)n(v)n (ergence)d(of)i(the)h(series)e(of)i(classical)e(KAM)h(tori)g(b)n(y)g (Galla)n(v)n(otti)f([5])118 2575 y(\(see)h(also)f(Eliasson)f([4]\).)46 b(In)31 b(the)g(latter)g(problem)f(one)h(has)f(to)h(exploit)f(a)h(n)n (um)n(b)r(er)f(of)h(partial)118 2681 y(cancellations)i(in)i(the)f(p)r (erturbativ)n(e)g(expansion)f(whic)n(h)h(at)g(the)h(end)f(ensure)g(the) h(analiticit)n(y)118 2788 y(of)28 b(the)g(series,)f(while)g(in)h(the)g (case)f(of)h(Lindstedt)g(series)f(for)g(PDE)g(suc)n(h)h(cancellations)e (are)h(ab-)118 2894 y(sen)n(t.)38 b(On)28 b(the)g(con)n(trary)e(here)i (one)g(has)f(to)h(p)r(erform)g(a)f(suitable)h(resummation)f(of)h(the)h (formal)118 3000 y(p)r(erturbativ)n(e)24 b(expansion,)g(whic)n(h,)h(at) g(the)g(end,)g(implies)g(non-analyticit)n(y)e(of)i(the)f(solution;)i (for)118 3107 y(similar)35 b(results)g(along)g(the)h(same)f(direction)g (see)h(also)f([6])g(and)h([7].)61 b(Our)35 b(metho)r(d)h(is)g(based)118 3213 y(on)28 b(tec)n(hniques)g(whic)n(h)g(w)n(ere)f(dev)n(elop)r(ed)h (for)f(the)i(pro)r(of)e(of)i(the)f(con)n(v)n(ergence)e(of)i(p)r (erturbativ)n(e)118 3319 y(series)f(for)g(quan)n(tities)g(of)h(in)n (terest)f(in)h(quan)n(tum)f(\014eld)h(theory)f(or)g(statistical)g(mec)n (hanics.)189 3429 y(Finally)40 b(one)f(could)g(try)h(to)f(study)h(b)n (y)g(the)g(same)f(metho)r(ds)h(the)g(case)f(of)g(quasi-p)r(erio)r(dic) 118 3535 y(solutions)26 b(with)i(Diric)n(hlet)f(b)r(oundary)f (conditions,)h(or)f(the)h(case)f(of)h(b)r(oth)g(p)r(erio)r(dic)g(and)g (quasi-)118 3641 y(p)r(erio)r(dic)35 b(solutions)g(with)h(p)r(erio)r (dic)f(b)r(oundary)f(conditions)h(\(in)h(whic)n(h)f(the)h(eigen)n(v)-5 b(alues)34 b(are)118 3748 y(degenerate\),)27 b(and)g(w)n(e)g(hop)r(e)h (to)g(come)f(bac)n(k)f(on)i(that)g(in)g(the)f(near)g(future.)1119 3996 y Fw(2.)50 b(Recursiv)m(e)36 b(relations)118 4176 y Fu(By)27 b(\(1.7\),)h(w)n(e)f(can)g(rewrite)g(\(1.14\))g(as)663 4391 y(^)-47 b Fr(u)706 4403 y Fp(n;m)843 4324 y Fq(\002)877 4391 y Fo(\000)7 b Fu(~)-49 b Fr(!)997 4357 y Fn(2)994 4412 y(1)1034 4391 y Fr(n)1084 4357 y Fn(2)1139 4391 y Fu(+)25 b(~)-49 b Fr(!)1277 4357 y Fn(2)1274 4412 y Fp(m)1337 4324 y Fq(\003)1395 4391 y Fu(=)22 b Fr(\027)1528 4357 y Fn(\()p Fp(a)p Fn(\))1523 4412 y Fp(m)1626 4391 y Fu(^)-47 b 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584 y Fp(m)861 567 y Fj(0)930 559 y Fo(\025)i Fu(0)d(for)g(all)h Fr(m)1450 529 y Fv(0)1513 559 y Fo(\025)h Fu(2)f(and)f Fr(c)j Fu(=)g Fr(a;)14 b(b)p Fu(.)66 b(W)-7 b(e)38 b(set)g(also)f Fo(j)p Fr(k)p 2748 572 46 4 v 3 w Fo(j)j Fu(=)f Fr(k)3007 516 y Fn(\()p Fp(a)p Fn(\))3004 581 y(1)3124 559 y Fu(+)118 679 y Fr(k)164 635 y Fn(\()p Fp(b)p Fn(\))161 701 y(1)272 679 y Fu(+)23 b Fr(k)406 635 y Fn(\()p Fp(a)p Fn(\))403 701 y(2)521 679 y Fu(+)g Fr(k)655 635 y Fn(\()p Fp(b)p Fn(\))652 701 y(2)763 679 y Fu(+)g Fr(:)14 b(:)g(:)p Fu(;)38 b(of)d(course)f(w)n(e)g(are)g(using)g(the)h(symmetry)g(prop)r (ert)n(y)e(in)i(\(1.13\))f(to)118 785 y(restrict)28 b(the)g(dep)r (endence)h(only)f(on)g(the)h(p)r(ositiv)n(e)f(lab)r(els)g Fr(m)2053 755 y Fv(0)2076 785 y Fu(.)39 b(W)-7 b(e)29 b(shall)f(refer)g(to)g(\(2.3\))g(as)g(the)118 891 y Fs(Lindste)l(dt)37 b(series)f Fu(of)e(the)i(p)r(erio)r(dic)f(solution,)h(for)e(the)i (manifest)f(analogy)e(with)j(the)f(case)f(of)118 997 y(\014nite-dimensional)27 b(Hamiltonian)h(systems.)118 1175 y Fs(R)l(emark)33 b(4.)47 b Fu(W)-7 b(e)31 b(shall)f(see)g(that)h (for)f(\014xed)h Fr(k)j Fu(the)d(v)n(ector)e Fr(k)p 2015 1188 V 34 w Fu(can)h(ha)n(v)n(e)g(only)g(a)g(\014nite)h(n)n(um)n(b)r (er)118 1281 y(\()p Fo(\024)d Fr(k)s Fu(\))i(of)h(comp)r(onen)n(ts)f (di\013eren)n(t)g(from)g(zero,)g(whic)n(h)h(giv)n(es)e(sense)h(to)g (the)h(sum)g(app)r(earing)e(in)118 1387 y(\(2.2\).)189 1576 y(F)-7 b(or)27 b Fo(j)p Fr(m)p Fo(j)c Fr(>)g Fu(1)k(one)g(has)32 b(^)-47 b Fr(u)985 1531 y Fn(\(1)p Fp(;)p Fn(0)p 1064 1544 34 4 v -1 w(\))985 1586 y Fp(n;m)1150 1576 y Fu(=)22 b(0,)28 b(while)f(for)g Fr(m)c Fu(=)g Fo(\006)p Fu(1)k(one)g(has)1357 1784 y(^)-47 b Fr(u)1400 1739 y Fn(\(1)p Fp(;)p Fn(0)p 1479 1752 V -1 w(\))1400 1794 y Fp(n;m)1565 1784 y Fu(=)22 b Fr(\016)1689 1796 y Fp(n;)p Fv(\006)p Fn(1)1849 1728 y Fr(m)p 1849 1765 73 4 v 1850 1841 a Fu(4)p Fr(i)1932 1784 y Fu(;)1063 b(\(2)p Fr(:)p Fu(3\))118 1994 y(this)36 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2190 y Fn(0)2170 2178 y Fo(\024)k(j)p Fr(x)p Fo(j)h(\024)g Fu(2)2569 2147 y Fv(\000)p Fp(h)p Fn(+1)2747 2178 y Fr(C)2806 2190 y Fn(0)2844 2178 y Fu(,)f(while)d(if)118 2284 y Fr(\037)170 2296 y Fv(\000)p Fn(1)259 2284 y Fu(\()p Fr(x)p Fu(\))24 b Fo(6)p Fu(=)f(0)k(one)g(has)g Fo(j)p Fr(x)p Fo(j)d(\025)f Fr(C)1115 2296 y Fn(0)1152 2284 y Fu(.)189 2391 y(W)-7 b(e)36 b(write)g(the)g(propagator)d(as)i(sum)h (of)g(propagators)c(on)k(single)f(scales)g(in)h(the)g(follo)n(wing)118 2497 y(w)n(a)n(y:)649 2661 y Fr(g)s Fu(\()7 b(~)-49 b Fr(!)776 2673 y Fn(1)813 2661 y Fr(n;)14 b(m)p Fu(\))23 b(=)1170 2557 y Fv(1)1143 2582 y Fq(X)1116 2760 y Fp(h)p Fn(=)p Fv(\000)p Fn(1)1314 2604 y Fr(\037)1366 2616 y Fp(h)1409 2604 y Fu(\()p Fo(j)7 b Fu(~)-49 b Fr(!)1516 2616 y Fn(1)1553 2604 y Fr(n)p Fo(j)19 b(\000)24 b Fu(~)-48 b Fr(!)1780 2616 y Fp(m)1843 2604 y Fu(\))p 1314 2642 561 4 v 1365 2718 a Fo(\000)7 b Fu(~)-49 b Fr(!)1485 2689 y Fn(2)1482 2740 y(1)1521 2718 y Fr(n)1571 2694 y Fn(2)1627 2718 y Fu(+)24 b(~)-48 b Fr(!)1765 2694 y Fn(2)1762 2738 y Fp(m)1908 2661 y Fu(=)2050 2557 y Fv(1)2023 2582 y Fq(X)1996 2760 y Fp(h)p Fn(=)p Fv(\000)p Fn(1)2184 2661 y Fr(g)2227 2626 y Fn(\()p Fp(h)p Fn(\))2322 2661 y Fu(\()7 b(~)-49 b Fr(!)2406 2673 y Fn(1)2443 2661 y Fr(n;)14 b(m)p Fu(\))p Fr(:)360 b Fu(\(4)p Fr(:)p Fu(2\))118 2918 y(Note)28 b(that)g(w)n(e)f(can)g(b)r(ound)h Fo(j)p Fr(g)1095 2888 y Fn(\()p Fp(h)p Fn(\))1190 2918 y Fu(\()7 b(~)-49 b Fr(!)1274 2930 y Fn(1)1311 2918 y Fr(n;)14 b(m)p Fu(\))p Fo(j)23 b(\024)f Fu(2)1678 2888 y Fv(\000)p Fp(h)p Fn(+1)1857 2918 y Fr(C)1916 2930 y Fn(0)1954 2918 y Fu(.)189 3025 y(This)27 b(means)g(that)h(w)n(e)f(can)g(attac)n(h)g(a) g(scale)g(lab)r(el)g Fr(h)c Fo(\025)g(\000)p Fu(1)j(to)i(eac)n(h)e (line)i Fr(`)22 b Fo(2)i Fr(L)p Fu(\()p Fr(\022)r Fu(\),)k(whic)n(h)f (is)118 3143 y(the)k(scale)e(of)i(the)f(propagator)e(whic)n(h)i(is)g (asso)r(ciated)f(to)h Fr(`)p Fu(.)45 b(W)-7 b(e)31 b(can)f(denote)g (with)h(\002)2895 3097 y Fn(\()p Fp(k)q(;k)p 2977 3110 37 4 v 1 w Fn(\))2895 3152 y Fp(n;m)3074 3143 y Fu(the)118 3249 y(set)j(of)f(trees)g(whic)n(h)g(di\013er)h(from)f(the)h(previous)e (ones)h(simply)g(b)r(ecause)g(the)h(lines)f(carry)f(also)118 3355 y(the)27 b(scale)e(lab)r(els.)36 b(The)27 b(set)f(\002)1085 3310 y Fn(\()p Fp(k)q(;k)p 1167 3323 V 2 w Fn(\))1085 3365 y Fp(n;m)1260 3355 y Fu(is)g(de\014ned)h(according)d(to)j(the)f (rules)g(\(1\))p Fo(\004)p Fu(\(4\))h(of)f(Section)g(3,)118 3461 y(b)n(y)h(c)n(hanging)g(item)h(\(2\))g(in)n(to)118 3568 y(\(2'\))g(T)-7 b(o)27 b(eac)n(h)g(line)h Fr(`)22 b Fo(2)i Fr(L)p Fu(\()p Fr(\022)r Fu(\))k(one)f(asso)r(ciates)f(a)h (scale)g(lab)r(el)830 3819 y Fr(h)878 3831 y Fp(`)933 3819 y Fu(=)1021 3702 y Fq(\032)1097 3769 y Fu(0)p Fr(;)14 b Fu(1)p Fr(;)g Fu(2)p Fr(;)g Fu(3)p Fr(;)g(:)g(:)g(:)d(;)83 b Fu(if)29 b(\()p Fr(n)1786 3781 y Fp(`)1818 3769 y Fr(;)14 b(m)1928 3781 y Fp(`)1959 3769 y Fu(\))24 b Fo(6)p Fu(=)e(\()p Fo(\006)p Fu(1)p Fr(;)14 b Fo(\006)p Fu(1\),)1097 3869 y Fo(\000)p Fu(1)p Fr(;)400 b Fu(if)29 b(\()p Fr(n)1786 3881 y Fp(`)1818 3869 y Fr(;)14 b(m)1928 3881 y Fp(`)1959 3869 y Fu(\))24 b(=)e(\()p Fo(\006)p Fu(1)p Fr(;)14 b Fo(\006)p Fu(1\),)3018 3819 y(\(4)p Fr(:)p Fu(3\))118 4065 y(and)28 b(the)g Fs(pr)l(op)l(agator)245 4386 y Fr(g)288 4343 y Fn(\()p Fp(h)353 4352 y Fh(`)381 4343 y Fn(\))285 4411 y Fp(`)434 4386 y Fo(\021)23 b Fr(g)565 4352 y Fn(\()p Fp(n)632 4361 y Fh(`)660 4352 y Fn(\))691 4386 y Fu(\()7 b(~)-49 b Fr(!)775 4398 y Fn(1)812 4386 y Fr(n)862 4398 y Fp(`)894 4386 y Fr(;)14 b(m)1004 4398 y Fp(`)1035 4386 y Fu(\))24 b(=)1178 4191 y Fq(8)1178 4266 y(>)1178 4291 y(<)1178 4440 y(>)1178 4465 y(:)1276 4259 y Fr(\037)1328 4271 y Fp(h)1367 4280 y Fh(`)1400 4259 y Fu(\()p Fo(j)7 b Fu(~)-49 b Fr(!)1507 4271 y Fn(1)1544 4259 y Fr(n)1594 4271 y Fp(`)1626 4259 y Fo(j)18 b(\000)25 b Fu(~)-49 b Fr(!)1802 4271 y Fp(m)1861 4280 y Fh(`)1894 4259 y Fu(\))p 1276 4296 651 4 v 1357 4372 a Fo(\000)7 b Fu(~)-49 b Fr(!)1477 4343 y Fn(2)1474 4394 y(1)1513 4372 y Fr(n)1563 4343 y Fn(2)1563 4397 y Fp(`)1619 4372 y Fu(+)24 b(~)-48 b Fr(!)1757 4348 y Fn(2)1754 4392 y Fp(m)1813 4401 y Fh(`)1936 4315 y Fr(;)83 b Fu(if)28 b(\()p Fr(n)2200 4327 y Fp(`)2233 4315 y Fr(;)14 b(m)2343 4327 y Fp(`)2374 4315 y Fu(\))24 b Fo(6)p Fu(=)e(\()p Fo(\006)p Fu(1)p Fr(;)14 b Fo(\006)p Fu(1\),)1266 4514 y(1)p Fr(;)711 b Fu(if)28 b(\()p Fr(n)2200 4526 y Fp(`)2233 4514 y Fr(;)14 b(m)2343 4526 y Fp(`)2374 4514 y Fu(\))24 b(=)e(\()p Fo(\006)p Fu(1)p Fr(;)14 b Fo(\006)p Fu(1\),)3018 4386 y(\(4)p Fr(:)p Fu(4\))118 4712 y(with)28 b Fs(momentum)f Fu(\()p Fr(n)820 4724 y Fp(`)852 4712 y Fr(;)14 b(m)962 4724 y Fp(`)994 4712 y Fu(\).)189 4819 y(Lo)r(oking)28 b(at)g(the)h(scale)f(lab)r(els)h(w)n(e)f(iden)n(tify)i(the)f(connected) g(cluster)f Fr(T)40 b Fu(of)29 b(no)r(des)f(whic)n(h)h(are)118 4925 y(link)n(ed)f(b)n(y)g(a)f(con)n(tin)n(uous)g(path)h(of)g(lines)g (with)h(the)f(same)g(scale)f(lab)r(el)h Fr(h)2433 4937 y Fp(T)2513 4925 y Fu(or)g(a)f(lo)n(w)n(er)g(one)g(and)118 5031 y(whic)n(h)h(are)f(maximal;)h(w)n(e)g(shall)f(sa)n(y)g(that)i(the) f(cluster)g(has)g(scale)f Fr(h)2308 5043 y Fp(T)2360 5031 y Fu(.)38 b(W)-7 b(e)29 b(shall)f(denote)g(with)118 5138 y Fr(V)19 b Fu(\()p Fr(T)12 b Fu(\))24 b(and)g Fr(E)5 b Fu(\()p Fr(T)12 b Fu(\))24 b(the)g(set)h(of)f(and)g(the)g(set)h(of)f (end-p)r(oin)n(ts,)g(resp)r(ectiv)n(ely)-7 b(,)24 b(whic)n(h)h(are)e (con)n(tained)118 5244 y(inside)28 b(the)g(cluster)f Fr(T)12 b Fu(,)27 b(and)g(with)h Fr(L)p Fu(\()p Fr(T)12 b Fu(\))27 b(the)h(set)g(of)f(lines)h(connecting)f(them.)1633 5443 y(8)p eop %%Page: 9 9 9 8 bop 189 555 a Fu(Therefore)29 b(an)h(inclusion)f(relation)h(is)f (established)h(b)r(et)n(w)n(een)g(clusters,)g(in)h(suc)n(h)e(a)h(w)n(a) n(y)f(that)118 662 y(the)j(innermost)f(clusters)f(are)h(the)g(clusters) g(with)h(lo)n(w)n(est)e(scale,)h(and)g(so)g(on.)48 b(The)31 b(v)-5 b(alue)31 b(of)g(a)118 768 y(tree)c(can)h(b)r(e)g(written)f (then)i(as)755 939 y(V)-7 b(al\()p Fr(\022)r Fu(\))24 b(=)1091 847 y Fq(\020)1203 860 y(Y)1155 1042 y Fp(`)p Fv(2)p Fp(L)p Fn(\()p Fp(\022)r Fn(\))1372 939 y Fr(g)1415 896 y Fn(\()p Fp(h)1480 905 y Fh(`)1508 896 y Fn(\))1412 964 y Fp(`)1538 847 y Fq(\021\020)1708 860 y(Y)1651 1042 y Fp(v)r Fv(2)p Fp(V)15 b Fn(\()p Fp(\022)r Fn(\))1884 939 y Fr(\021)1925 951 y Fp(v)1965 847 y Fq(\021)o(\020)2133 860 y(Y)2078 1042 y Fp(v)r Fv(2)p Fp(E)s Fn(\()p Fp(\022)r Fn(\))2319 883 y Fr("m)2431 895 y Fp(v)p 2319 920 152 4 v 2359 996 a Fu(4)p Fr(i)2480 847 y Fq(\021)2529 939 y Fr(;)466 b Fu(\(4)p Fr(:)p Fu(5\))118 1200 y(so)27 b(that)h(\(3.1\))f(is)h(replaced)e(with)1243 1383 y(^)-47 b Fr(u)1286 1338 y Fn(\()p Fp(k)q(;k)p 1368 1351 37 4 v 1 w Fn(\))1286 1392 y Fp(n;m)1457 1383 y Fu(=)1616 1304 y Fq(X)1545 1510 y Fp(\022)r Fv(2)p Fn(\002)1675 1473 y Fi(\()p Fh(k)q(;k)p 1748 1486 33 4 v 1 w Fi(\))1675 1521 y Fh(n;m)1821 1383 y Fu(V)-7 b(al)o(\()p Fr(\022)r Fu(\))p Fr(;)950 b Fu(\(4)p Fr(:)p Fu(6\))118 1668 y(with)28 b(the)g(new)g(de\014nition)734 1839 y(V)-7 b(al\()p Fr(\022)r Fu(\))24 b(=)1071 1747 y Fq(\020)1183 1760 y(Y)1134 1942 y Fp(`)p Fv(2)p Fp(L)p Fn(\()p Fp(\022)r Fn(\))1351 1839 y Fr(g)1394 1796 y Fn(\()p Fp(h)1459 1805 y Fh(`)1487 1796 y Fn(\))1391 1864 y Fp(`)1518 1747 y Fq(\021)o(\020)1687 1760 y(Y)1631 1942 y Fp(v)r Fv(2)p Fp(V)14 b Fn(\()p Fp(\022)r Fn(\))1863 1839 y Fr(\021)1904 1851 y Fp(v)1944 1747 y Fq(\021\020)2113 1760 y(Y)2057 1942 y Fp(v)r Fv(2)p Fp(E)s Fn(\()p Fp(\022)r Fn(\))2298 1783 y Fr("m)2410 1795 y Fp(n)2451 1803 y Fh(v)p 2298 1820 193 4 v 2359 1896 a Fu(4)p Fr(i)2500 1747 y Fq(\021)2550 1839 y Fr(:)445 b Fu(\(4)p Fr(:)p Fu(7\))118 2100 y(for)27 b(the)h(tree)f(v)-5 b(alue)28 b(V)-7 b(al\()p Fr(\022)r Fu(\).)189 2207 y(Eac)n(h)32 b(cluster)h Fr(T)44 b Fu(has)33 b(an)f(arbitrary)g(n)n(um)n(b)r(er)g (of)i(lines)f(en)n(tering)f(it)i(\(incoming)f(lines\),)h(but)118 2313 y(only)21 b(one)h(or)e(zero)h(line)h(coming)f(from)g(it)h (\(outcoming)f(line\);)j(w)n(e)e(shall)f(denote)g(the)h(latter)g (\(when)118 2419 y(it)29 b(exists\))g(with)g Fr(`)690 2389 y Fn(1)690 2442 y Fp(T)742 2419 y Fu(.)40 b(W)-7 b(e)29 b(shall)f(call)h Fs(external)h(lines)f Fu(of)g(the)g(cluster)f Fr(T)40 b Fu(the)29 b(lines)g(whic)n(h)f(either)118 2537 y(en)n(ter)22 b(or)g(come)h(out)f(from)h Fr(T)12 b Fu(,)23 b(and)f(w)n(e)h(shall)f(denote)h(b)n(y)f Fr(h)1953 2494 y Fn(\()p Fp(e)p Fn(\))1953 2561 y Fp(T)2063 2537 y Fu(the)i(maxim)n (um)e(among)g(the)h(scales)118 2643 y(of)k(the)g(external)f(lines)h(of) g Fr(T)12 b Fu(.)36 b(De\014ne)27 b Fr(K)6 b Fu(\()p Fr(T)12 b Fu(\))26 b(as)g(the)i(n)n(um)n(b)r(er)e(of)h(end-p)r(oin)n (ts)g(con)n(tained)f(inside)118 2750 y Fr(T)12 b Fu(,)27 b(so)g(that)h Fr(K)6 b Fu(\()p Fr(T)12 b Fu(\))22 b(=)g Fo(j)p Fr(E)5 b Fu(\()p Fr(T)12 b Fu(\))p Fo(j)p Fu(.)189 2856 y(The)28 b(clusters)f(with)h(only)f(one)g(incoming)g(line)h Fr(`)1731 2826 y Fn(2)1731 2879 y Fp(T)1811 2856 y Fu(suc)n(h)f(that)h (one)f(has)1138 3027 y Fr(n)1188 3045 y Fp(`)1216 3025 y Fi(1)1216 3067 y Fh(T)1288 3027 y Fu(=)c Fr(n)1426 3045 y Fp(`)1454 3025 y Fi(2)1454 3067 y Fh(T)1531 3027 y Fu(and)28 b Fr(m)1766 3045 y Fp(`)1794 3025 y Fi(1)1794 3067 y Fh(T)1867 3027 y Fu(=)22 b Fo(\006)p Fr(m)2092 3045 y Fp(`)2120 3025 y Fi(2)2120 3067 y Fh(T)3018 3027 y Fu(\(4)p Fr(:)p Fu(8\))118 3198 y(will)35 b(b)r(e)g(called)g Fs(self-ener)l(gy)i(gr)l(aphs)f Fu(or)e Fs(r)l(esonanc)l(es)h Fu(\(the)g(\014rst)g(nome)f(is)h(usual)f(in)h(quan)n(tum)118 3304 y(\014eld)g(theory)-7 b(,)35 b(the)g(second)f(one)g(w)n(as)g(in)n (tro)r(duced)g(b)n(y)g(Eliasson)f(in)i(his)f(basic)g(pap)r(er)g([4]\).) 58 b(In)118 3411 y(suc)n(h)30 b(a)g(case)g(the)h(line)f Fr(`)899 3380 y Fn(1)899 3433 y Fp(T)982 3411 y Fu(will)g(b)r(e)h (called)f(a)g Fs(r)l(esonant)i(line.)47 b Fu(If)31 b(\()p Fr(n;)14 b(m)p Fu(\))30 b(is)h(the)f(momen)n(tum)h(of)118 3517 y(the)d(line)h Fr(`)454 3487 y Fn(2)454 3540 y Fp(T)533 3517 y Fu(w)n(e)f(shall)f(call)h(\()p Fr(n;)14 b(m)p Fu(\))28 b(the)g(momen)n(tum)g(of)g(the)h(self-energy)d(graph;)h(then)h (for)g(an)n(y)118 3623 y(line)g Fr(`)23 b Fo(2)g Fr(L)p Fu(\()p Fr(T)12 b Fu(\))27 b(one)g(can)h(write)f(\()p Fr(n)1219 3635 y Fp(`)1251 3623 y Fr(;)14 b(m)1361 3635 y Fp(`)1393 3623 y Fu(\))24 b(=)e(\()p Fr(n)1618 3593 y Fn(0)1618 3647 y Fp(`)1656 3623 y Fr(;)14 b(m)1766 3593 y Fn(0)1766 3647 y Fp(`)1803 3623 y Fu(\))k(+)h Fr(\021)1978 3635 y Fp(`)2010 3623 y Fu(\()p Fr(n;)14 b(m)p Fu(\),)28 b(where)f Fr(\021)2566 3635 y Fp(`)2621 3623 y Fu(=)c(1)k(if)h(the)g(line)g Fr(`)118 3729 y Fu(is)g(along)e (the)i(path)g(connecting)f(the)h(external)f(lines)g(of)h Fr(T)38 b Fu(and)28 b Fr(\021)2213 3741 y Fp(`)2268 3729 y Fu(=)23 b(0)k(otherwise.)189 3836 y(The)h Fs(value)g Fu(of)f(the)h(self-energy)f(graph)f Fr(T)39 b Fu(is)27 b(de\014ned)h(as)686 4020 y Fo(V)744 3986 y Fp(h)737 4041 y(T)789 4020 y Fu(\()7 b(~)-49 b Fr(!)873 4032 y Fn(1)910 4020 y Fr(n;)14 b(m)p Fu(\))23 b(=)1213 3928 y Fq(\020)1284 3941 y(Y)1276 4120 y Fp(`)p Fv(2)p Fp(T)1411 4020 y Fr(g)1454 3977 y Fn(\()p Fp(h)1519 3986 y Fh(`)1547 3977 y Fn(\))1451 4045 y Fp(`)1577 3928 y Fq(\021\020)1754 3941 y(Y)1690 4123 y Fp(v)r Fv(2)p Fp(V)15 b Fn(\()p Fp(T)9 b Fn(\))1938 4020 y Fr(\021)1979 4032 y Fp(v)2019 3928 y Fq(\021)o(\020)2194 3941 y(Y)2132 4123 y Fp(v)r Fv(2)p Fp(E)s Fn(\()p Fp(T)g Fn(\))2387 3964 y Fr("m)2499 3976 y Fp(v)p 2387 4001 152 4 v 2428 4077 a Fu(4)p Fr(i)2548 3928 y Fq(\021)2598 4020 y Fr(:)397 b Fu(\(4)p Fr(:)p Fu(9\))118 4306 y(where)22 b Fr(h)h Fu(=)g Fr(h)560 4263 y Fn(\()p Fp(e)p Fn(\))560 4330 y Fp(T)669 4306 y Fu(is)g(the)f(maxim)n (um)h(b)r(et)n(w)n(een)f(the)h(scales)e(of)h(the)h(t)n(w)n(o)f (external)f(lines)i(of)f Fr(T)34 b Fu(\(they)118 4412 y(can)27 b(di\013er)h(at)f(most)h(b)n(y)f(a)g(unit\),)i(and)e(one)h (has)632 4593 y Fr(n)p Fu(\()p Fr(T)12 b Fu(\))23 b Fo(\021)974 4514 y Fq(X)918 4696 y Fp(v)r Fv(2)p Fp(E)s Fn(\()p Fp(T)9 b Fn(\))1163 4593 y Fr(n)1213 4605 y Fp(v)1276 4593 y Fu(=)22 b(0)p Fr(;)180 b(m)p Fu(\()p Fr(T)12 b Fu(\))22 b Fo(\021)1972 4514 y Fq(X)1916 4696 y Fp(v)r Fv(2)p Fp(E)s Fn(\()p Fp(T)9 b Fn(\))2162 4593 y Fr(m)2235 4605 y Fp(v)2297 4593 y Fo(2)24 b(f)p Fu(0)p Fr(;)14 b Fu(2)p Fr(m)p Fo(g)p Fr(;)300 b Fu(\(4)p Fr(:)p Fu(10\))118 4854 y(b)n(y)34 b(de\014nition)h(of)f(self-energy)f(graph;)j(one)e(has) g Fr(c)g Fu(=)f Fr(a)i Fu(when)f Fr(m)p Fu(\()p Fr(T)12 b Fu(\))34 b(=)f(0)h(and)g Fr(c)g Fu(=)g Fr(b)g Fu(when)118 4961 y Fr(m)p Fu(\()p Fr(T)12 b Fu(\))23 b(=)f(2)p Fr(m)p Fu(.)118 5138 y Fs(R)l(emark)34 b(7.)52 b Fu(Note)32 b(that,)i(if)e Fo(j)p Fr(m)p Fo(j)f Fu(=)f(1)i(and)g Fo(j)p Fr(n)p Fo(j)f(6)p Fu(=)f(1)i(the)h(scale)e(of)h(the)h(external)e (lines)h(is)g(suc)n(h)118 5244 y(that)27 b Fr(h)c Fo(\024)f Fr(h)503 5256 y Fn(0)541 5244 y Fu(,)k(where)h Fr(h)878 5256 y Fn(0)941 5244 y Fu(is)g(a)f(suitable)g(constan)n(t)g(\(see)h (Remark)e(6\):)36 b(one)27 b(can)f(tak)n(e)g Fr(h)2874 5256 y Fn(0)2934 5244 y Fu(=)c(0)27 b(for)1633 5443 y(9)p eop %%Page: 10 10 10 9 bop 118 555 a Fr(C)177 567 y Fn(0)245 555 y Fu(small)29 b(enough.)43 b(In)30 b(particular)f(this)h(implies)g(that)g(no)f (self-energy)g(graph)f(with)j Fr(m)26 b Fu(=)h(1)i(is)118 662 y(p)r(ossible;)e(w)n(e)h(shall)f(rep)r(eatedly)g(use)g(suc)n(h)h(a) f(prop)r(ert)n(y)f(in)i(the)g(follo)n(wing.)189 839 y(Giv)n(en)h(a)f (tree)h Fr(\022)r Fu(,)h(w)n(e)e(shall)h(denote)g(b)n(y)g Fr(N)1535 851 y Fp(h)1577 839 y Fu(\()p Fr(\022)r Fu(\))h(the)g(n)n(um) n(b)r(er)f(of)f(lines)h(with)h(scale)e Fr(h)p Fu(,)i(and)e(b)n(y)118 945 y Fr(C)177 957 y Fp(h)220 945 y Fu(\()p Fr(\022)r Fu(\))37 b(the)f(n)n(um)n(b)r(er)g(of)f(clusters)g(with)i(scale)e Fr(h)p Fu(.)61 b(W)-7 b(e)36 b(will)g(get)f(immediately)h(the)g(follo)n (wing)118 1063 y(b)r(ound,)28 b(for)f Fr(\022)f Fo(2)d Fu(\002)732 1018 y Fn(\()p Fp(k)q(;k)p 814 1031 37 4 v 2 w Fn(\))732 1073 y Fp(n;m)908 1063 y Fu(and)28 b(assuming)e Fo(j)p Fr(\027)1499 1020 y Fn(\()p Fp(c)p Fn(\))1494 1073 y Fp(m)1586 1063 y Fo(j)d(\024)f Fr(C)6 b Fo(j)p Fr(")p Fo(j)p Fu(,)999 1310 y Fo(j)p Fu(V)-7 b(al\()p Fr(\022)r Fu(\))p Fo(j)24 b(\024)f(j)p Fr(")p Fo(j)1467 1275 y Fp(k)q Fn(+)p Fv(j)p Fp(k)p 1574 1288 V 2 w Fv(j)1634 1310 y Fr(D)1705 1275 y Fp(k)q Fn(+)p Fv(j)p Fp(k)p 1812 1288 V 3 w Fv(j)1916 1206 y(1)1896 1231 y Fq(Y)1887 1410 y Fp(h)p Fn(=0)2024 1310 y Fu(2)2066 1275 y Fp(hN)2158 1284 y Fh(h)2196 1275 y Fn(\()p Fp(\022)r Fn(\))2285 1310 y Fr(;)669 b Fu(\(4)p Fr(:)p Fu(11\))118 1595 y(if)32 b Fr(D)i Fu(is)d(a)g(suitable)h(constan)n(t.)47 b(Note)32 b(that)g(one)f(has)g Fo(j)p Fr(n)1923 1607 y Fp(`)1955 1595 y Fo(j)p Fr(;)14 b Fo(j)p Fr(m)2111 1607 y Fp(`)2142 1595 y Fo(j)30 b(\024)f Fr(k)34 b Fu(for)d(all)h(trees)f Fr(\022)g Fo(2)f Fu(\002)3040 1550 y Fn(\()p Fp(k)q(;k)p 3122 1563 V 2 w Fn(\))3040 1605 y Fp(n;m)118 1702 y Fu(and)e(for)f(all) g(lines)g Fr(`)c Fo(2)h Fr(L)p Fu(\()p Fr(\022)r Fu(\).)37 b(The)28 b(follo)n(wing)e(result)h(is)h(pro)n(v)n(ed)e(in)i(App)r (endix)g(A1.)118 1890 y Fk(Lemma)i(1.)36 b Fs(F)-6 b(or)30 b(any)g(tr)l(e)l(e)f Fr(\022)c Fo(2)f Fu(\002)1250 1845 y Fn(\()p Fp(k)q(;k)p 1332 1858 V 1 w Fn(\))1250 1900 y Fp(n;m)1428 1890 y Fs(and)31 b(for)f(al)t(l)h Fr(h)23 b Fo(\025)f Fu(0)30 b Fs(one)g(has)746 2077 y Fr(N)813 2089 y Fp(h)856 2077 y Fu(\()p Fr(\022)r Fu(\))24 b Fo(\024)e Fu(4)p Fr(K)6 b Fu(\()p Fr(\022)r Fu(\)2)1338 2043 y Fn(\(2)p Fv(\000)p Fp(h)p Fn(\))p Fp(=\034)1607 2077 y Fo(\000)18 b Fr(C)1749 2089 y Fp(h)1793 2077 y Fu(\()p Fr(\022)r Fu(\))h(+)f Fr(S)2051 2089 y Fp(h)2094 2077 y Fu(\()p Fr(\022)r Fu(\))h(+)f Fr(M)2391 2043 y Fp(\027)2382 2097 y(h)2432 2077 y Fu(\()p Fr(\022)r Fu(\))p Fr(;)417 b Fu(\(4)p Fr(:)p Fu(12\))118 2263 y Fs(wher)l(e)35 b Fr(K)6 b Fu(\()p Fr(\022)r Fu(\))30 b(=)664 2201 y Fq(P)752 2288 y Fp(v)r Fv(2)p Fp(E)s Fn(\()p Fp(\022)r Fn(\))987 2263 y Fo(j)p Fr(n)1060 2275 y Fp(v)1100 2263 y Fo(j)p Fs(,)35 b(while)g Fr(S)1450 2275 y Fp(h)1493 2263 y Fu(\()p Fr(\022)r Fu(\))g Fs(is)f(the)g(numb)l(er)f(of)h(self-ener)l(gy)h(gr)l (aphs)g Fr(T)45 b Fs(in)34 b Fr(\022)118 2398 y Fs(with)j Fr(h)353 2355 y Fn(\()p Fp(e)p Fn(\))353 2422 y Fp(T)476 2398 y Fu(=)e Fr(h)h Fs(and)h Fr(M)918 2368 y Fp(\027)909 2421 y(h)959 2398 y Fu(\()p Fr(\022)r Fu(\))g Fs(is)g(the)f(numb)l(er)g (of)h Fr(\027)5 b Fs(-vertic)l(es)37 b(in)g Fr(\022)i Fs(such)d(that)h(the)f(maximum)118 2504 y(sc)l(ale)31 b(of)f(the)g(two)g(external)f(lines)i(is)f Fr(h)p Fs(.)189 2681 y Fu(Let)f(us)g(consider)f(a)h(tree)g(with)h(no)f(self-energy)e (graphs)h(and)h(with)h(no)f Fr(\027)5 b Fu(-v)n(ertices.)40 b(Then)29 b(b)n(y)118 2788 y(\(4.10\))e(and)g(\(4.12\))g(w)n(e)g(get) 856 3034 y Fo(j)p Fu(V)-7 b(al\()p Fr(\022)r Fu(\))p Fo(j)24 b(\024)f(j)p Fr(")p Fo(j)1324 3000 y Fp(k)1364 3034 y Fr(D)1435 3000 y Fp(k)1518 2930 y Fv(1)1499 2955 y Fq(Y)1490 3134 y Fp(h)p Fn(=0)1627 3034 y Fu(2)1669 3000 y Fn(4)p Fp(hk)q Fn(2)1810 2975 y Fj(\000)p Fi(\()p Fh(h)p Fj(\000)p Fi(2\))p Fh(=\034)2103 3034 y Fo(\024)2210 3013 y Fu(~)2190 3034 y Fr(D)2261 3000 y Fp(k)2302 3034 y Fo(j)p Fr(")p Fo(j)2387 3000 y Fp(k)2428 3034 y Fr(;)526 b Fu(\(4)p Fr(:)p Fu(13\))118 3307 y(for)24 b(a)g(suitable)h(constan)n (t)966 3286 y(~)947 3307 y Fr(D)r Fu(,)g(and)g(w)n(e)f(ha)n(v)n(e)f (used)i(that)g(one)f(has)g Fr(k)i Fu(=)c Fr(K)6 b Fu(\()p Fr(\022)r Fu(\);)26 b(see)f(\(3.6\).)35 b(On)25 b(the)118 3414 y(other)30 b(hand)h(the)g(b)r(ound)h(for)e(a)g(generic)g(tree)h Fs(with)g Fu(self-energy)e(graphs)h(is)h(quite)g(bad;)h(this)f(is)118 3520 y(not)24 b(just)g(a)f(tec)n(hnical)h(problem,)g(as)f(one)g(can)h (easily)e(iden)n(tify)j(trees)e(whose)g(v)-5 b(alue)23 b(is)h(essen)n(tially)118 3626 y(giv)n(en)33 b(b)n(y)g Fr(C)527 3596 y Fp(k)568 3626 y Fr(k)s Fu(!)637 3596 y Fp(\013)684 3626 y Fr(")723 3596 y Fp(k)764 3626 y Fu(,)i(for)e(some)g(p)r(ositiv)n(e)g(constan)n(ts)g Fr(\013)h Fu(and)f Fr(C)6 b Fu(.)55 b(W)-7 b(e)34 b(ha)n(v)n(e)e(then)i(to)g (de\014ne)f(a)118 3732 y(di\013eren)n(t)28 b(expansion,)e(as)h(it)h (will)g(b)r(e)g(sho)n(wn)f(in)h(the)g(follo)n(wing)e(section.)893 3980 y Fw(5.)50 b(The)38 b(renormalized)e(expansion)118 4158 y Fu(W)-7 b(e)25 b(ha)n(v)n(e)f(seen)h(that)g(the)h(expansion)d (en)n(visaged)h(in)h(the)g(previous)f(sections)h(can)f(b)r(e)i(written) f(as)118 4264 y(sum)e(o)n(v)n(er)d(trees)i(and)h(some)f(of)g(suc)n(h)g (trees)g(\(the)i(trees)d(con)n(taining)h(self-energy)f(graphs\))h (cannot)118 4370 y(b)r(e)i(b)r(ounded)g(b)n(y)f Fr(C)738 4340 y Fp(k)780 4370 y Fo(j)p Fr(")p Fo(j)865 4340 y Fp(k)905 4370 y Fu(,)i(as)e(they)h(are)e(of)i(order)e(of)i Fr(C)1828 4340 y Fp(k)1869 4370 y Fo(j)p Fr(")p Fo(j)1954 4340 y Fp(k)1994 4370 y Fr(k)s Fu(!)2063 4340 y Fp(\013)2111 4370 y Fu(.)35 b(In)24 b(this)g(section)f(w)n(e)g(will)h(set)g(up)118 4477 y(a)h(di\013eren)n(t)h(expansion,)f(whic)n(h,)h(b)n(y)g(c)n(ho)r (osing)e(in)i(a)f(prop)r(er)g(w)n(a)n(y)f(the)j(parameters)d Fr(\027)2829 4489 y Fp(m)2892 4477 y Fu(,)i Fr(m)d(>)f Fu(1,)118 4583 y(can)27 b(b)r(e)h(written)g(as)f(sum)h(o)n(v)n(er)e (trees)h(in)g(whic)n(h)h(all)f(of)h(them)g(can)f(b)r(e)h(b)r(ounded)g (b)n(y)g Fr(C)2870 4553 y Fp(k)2911 4583 y Fo(j)p Fr(")p Fo(j)2996 4553 y Fp(k)3036 4583 y Fu(.)189 4689 y(W)-7 b(e)21 b(in)n(tro)r(duce)f(a)g Fs(lo)l(c)l(alization)25 b(op)l(er)l(ator)c Fu(acting)f(on)g(the)h(self-energy)e(graphs)g(in)h (the)h(follo)n(wing)118 4795 y(w)n(a)n(y)-7 b(,)27 b(for)g Fr(m)c(>)f Fu(1,)1154 4902 y Fo(LV)1269 4867 y Fp(h)1262 4922 y(T)1314 4902 y Fu(\()7 b(~)-49 b Fr(!)1398 4914 y Fn(1)1435 4902 y Fr(n;)14 b(m)p Fu(\))23 b(=)g Fo(V)1796 4867 y Fp(h)1789 4922 y(T)1841 4902 y Fu(\()7 b(~)-49 b Fr(!)1925 4914 y Fp(m)1988 4902 y Fr(;)14 b(m)p Fu(\))p Fr(;)865 b Fu(\(5)p Fr(:)p Fu(1\))118 5058 y(and)28 b(w)n(e)f(de\014ne) h(the)g Fs(r)l(e)l(gularization)i(op)l(er)l(ator)f Fo(R)24 b Fu(=)e Fa(1)-27 b Fu(1)18 b Fo(\000)g(L)28 b Fu(as)888 5244 y Fo(RV)1016 5210 y Fp(h)1009 5265 y(T)1062 5244 y Fu(\()7 b(~)-49 b Fr(!)1146 5256 y Fn(1)1183 5244 y Fr(n;)14 b(m)p Fu(\))23 b(=)g Fo(V)1544 5210 y Fp(h)1537 5265 y(T)1589 5244 y Fu(\()7 b(~)-49 b Fr(!)1673 5256 y Fn(1)1710 5244 y Fr(n;)14 b(m)p Fu(\))k Fo(\000)g(V)2061 5210 y Fp(h)2054 5265 y(T)2107 5244 y Fu(\()7 b(~)-49 b Fr(!)2191 5256 y Fp(m)2254 5244 y Fr(;)14 b(m)p Fu(\))p Fr(:)599 b Fu(\(5)p Fr(:)p Fu(2\))1612 5443 y(10)p eop %%Page: 11 11 11 10 bop 118 555 a Fu(W)-7 b(e)28 b(shall)f(de\014ne,)h(for)f Fr(m)c(>)g Fu(1)k(and)g Fr(h)c Fo(\025)g Fu(0,)k(the)h Fs(running)h(c)l(oupling)i(c)l(onstants)1006 721 y Fu(2)1048 687 y Fv(\000)p Fp(h)1143 721 y Fr(\027)1189 678 y Fn(\()p Fp(c)p Fn(\))1184 747 y Fp(h;m)1329 721 y Fu(=)22 b Fr(\027)1462 687 y Fn(\()p Fp(c)p Fn(\))1457 742 y Fp(m)1567 721 y Fu(+)1701 643 y Fq(X)1650 843 y Fp(T)9 b Fv(2T)1795 812 y Fi(\()p Fh(c)p Fi(\))1781 865 y Fh()g Fu(1\))k(the)h Fo(R)g Fu(op)r(eration)f(is)g(applied;)118 2271 y(\(6\))36 b(there)f(are)f(no)r(des)h Fr(v)k Fu(with)c Fr(s)1181 2283 y Fp(v)1257 2271 y Fu(=)g(1)g(suc)n(h)g(that,)i(if)f(the)g(momen)n (ta)f(of)g(the)h(external)e(lines)118 2377 y(are,)k(resp)r(ectiv)n(ely) -7 b(,)38 b(\()p Fr(n;)14 b(m)p Fu(\))37 b(and)f(\()p Fr(n;)14 b Fo(\006)p Fr(m)p Fu(\),)39 b(with)e Fr(m)h(>)f Fu(1,)h(and)f Fr(h)f Fu(is)h(the)f(maximal)g(scale)g(of)118 2493 y(the)d(external)f(lines)g(\(they)i(can)e(di\013er)h(at)f(most)g (b)n(y)h(1\),)h(then)f(a)f(factor)g(2)2507 2463 y Fv(\000)p Fp(h)2601 2493 y Fr(\027)2647 2450 y Fn(\()p Fp(c)p Fn(\))2642 2518 y Fp(h;m)2764 2493 y Fu(,)i Fr(c)d Fu(=)g Fr(a;)14 b(b)32 b Fu(is)118 2599 y(asso)r(ciated)26 b(to)i Fr(v)j Fu(\(where)c Fr(a)h Fu(corresp)r(onds)d(to)j(the)g(sign)f(+)g(and)h Fr(b)f Fu(to)g(the)h(sign)f Fo(\000)p Fu(\).)189 2705 y(W)-7 b(e)35 b(call)g Fs(r)l(e)l(gularize)l(d)i(self-ener)l(gy)g(gr)l (aphs)f Fu(the)g(self-energy)d(graphs)h(on)g(whic)n(h)h Fo(R)g Fu(applies,)118 2812 y(and)26 b(w)n(e)g(denote)g(with)g Fo(S)6 b Fu(\()p Fr(\022)r Fu(\))28 b(the)e(set)g(of)h(regularized)d (self-energy)g(graphs)h(con)n(tained)h(in)g Fr(\022)r Fu(.)37 b(W)-7 b(e)118 2918 y(still)28 b(call)f Fr(\027)5 b Fu(-v)n(ertices)27 b(the)h(no)r(des)f Fr(v)k Fu(with)d Fr(s)1482 2930 y Fp(v)1544 2918 y Fu(=)23 b(1.)189 3024 y(In)28 b(the)h(follo)n(wing)e(it)i(will)f(b)r(e)g(useful)h(to)f (de\014ne)g(also)g(the)g Fs(r)l(enormalize)l(d)k(self-ener)l(gy)f(gr)l (aphs)p Fu(,)118 3130 y(whic)n(h)26 b(are)e(de\014ned)i(as)f(the)g (self-energy)f(graphs)h(except)g(that)h(the)g(same)e(items)i(\(1\))g (and)f(\(2\))h(as)118 3237 y(for)d(the)h(renormalized)f(trees)g(apply)g (to)h(the)g(self-energy)e(graphs)h(and)g(no)r(des)h(con)n(tained)f (inside.)189 3343 y(The)38 b(crucial)g(p)r(oin)n(t)g(is)g(no)n(w)f (that,)42 b(if)c Fo(j)p Fr(\027)1525 3300 y Fn(\()p Fp(c)p Fn(\))1520 3368 y Fp(h;m)1642 3343 y Fo(j)j(\024)f Fr(C)6 b Fo(j)p Fr(")p Fo(j)38 b Fu(for)g(some)g(constan)n(t)f Fr(C)6 b Fu(,)41 b(then)e(eac)n(h)118 3449 y(renormalied)33 b(tree)h Fr(\022)i Fu(admit)f(a)e(go)r(o)r(d)h(b)r(ound.)57 b(First)34 b(of)g(all)g(note)g(that)h(a)e(b)r(ound)i(lik)n(e)f (\(4.12\))118 3556 y(holds)d(also)f(for)h(a)g(renormalized)f(tree)h Fr(\022)r Fu(,)i(namely)e(one)g(has)g(the)g(follo)n(wing)g(result)g (\(pro)n(v)n(ed)f(in)118 3662 y(App)r(endix)f(A2\).)118 3851 y Fk(Lemma)h(2.)36 b Fs(F)-6 b(or)30 b(any)g(tr)l(e)l(e)f Fr(\022)c Fo(2)f Fu(\002)1250 3805 y Fn(\()p Fp(k)q(;k)p 1332 3818 V 1 w Fn(\))p Fv(R)1250 3860 y Fp(n;m)1485 3851 y Fs(one)30 b(has)746 4017 y Fr(N)813 4029 y Fp(h)856 4017 y Fu(\()p Fr(\022)r Fu(\))24 b Fo(\024)e Fu(4)p Fr(K)6 b Fu(\()p Fr(\022)r Fu(\)2)1338 3982 y Fn(\(2)p Fv(\000)p Fp(h)p Fn(\))p Fp(=\034)1607 4017 y Fo(\000)18 b Fr(C)1749 4029 y Fp(h)1793 4017 y Fu(\()p Fr(\022)r Fu(\))h(+)f Fr(S)2051 4029 y Fp(h)2094 4017 y Fu(\()p Fr(\022)r Fu(\))h(+)f Fr(M)2391 3982 y Fp(\027)2382 4037 y(h)2432 4017 y Fu(\()p Fr(\022)r Fu(\))p Fr(;)458 b Fu(\(5)p Fr(:)p Fu(5\))118 4183 y Fs(wher)l(e)31 b(the)e(notations)h (ar)l(e)g(as)h(in)e(lemma)i(1.)189 4360 y Fu(Therefore,)37 b(b)n(y)e(using)h(lemma)f(2,)j(w)n(e)d(obtain)h(the)g(follo)n(wing)e (result)i(\(whic)n(h)g(is)g(pro)n(v)n(ed)e(in)118 4466 y(App)r(endix)40 b(A3\),)i(whic)n(h,)g(under)d(the)h(assumption)f(that) g(the)h(running)e(coupling)h(constan)n(ts)118 4573 y(remain)28 b(b)r(ounded)h(of)f(order)f Fr(")p Fu(,)h(allo)n(ws)f(us)i(to)f(pro)n (v)n(e)f(the)h(con)n(v)n(ergence)e(of)j(the)f(Linstedt)h(series)118 4679 y(as)e(a)g(function)h(of)g Fr(")f Fu(and)h(the)g(parameters)e Fr(\027)1556 4636 y Fn(\()p Fp(c)p Fn(\))1551 4704 y Fp(m)1610 4687 y Fj(0)1632 4704 y Fp(;h)1691 4687 y Fj(0)1717 4679 y Fu(,)i(with)g Fr(c)23 b Fu(=)g Fr(a;)14 b(b)27 b Fu(and)g Fo(j)p Fr(m)2505 4649 y Fv(0)2528 4679 y Fo(j)d Fr(>)e Fu(1.)118 4856 y Fk(Lemma)31 b(3.)39 b Fs(Assume)30 b(that)g(for)i(al)t(l)f Fo(j)p Fr(m)p Fo(j)25 b Fr(>)f Fu(1)30 b Fs(and)i(al)t(l)f Fr(h)25 b Fo(\025)f Fu(0)30 b Fs(ther)l(e)h(exist)f(a)h(c)l(onstant)f Fr(C)37 b Fs(such)118 4972 y(that)c(one)g(has)g Fo(j)p Fr(\027)671 4929 y Fn(\()p Fp(c)p Fn(\))666 4997 y Fp(h;m)788 4972 y Fo(j)28 b(\024)g Fr(C)6 b Fo(j)p Fr(")p Fo(j)p Fs(,)34 b(with)f Fr(c)28 b Fu(=)g Fr(a;)14 b(b)p Fs(.)47 b(Then)33 b(ther)l(e)g(exists)f Fr(")2368 4984 y Fn(0)2433 4972 y Fr(>)c Fu(0)k Fs(such)h(that)f(for)i (al)t(l)118 5078 y Fo(j)p Fr(")p Fo(j)23 b Fr(<)g(")353 5090 y Fn(0)419 5078 y Fs(and)31 b(for)f(al)t(l)h Fu(\()p Fr(n;)14 b(m)p Fu(\))23 b Fo(6)p Fu(=)g(\()p Fo(\006)p Fu(1)p Fr(;)14 b Fo(\006)p Fu(1\))28 b Fs(one)i(has)1054 5244 y Fo(j)5 b Fu(^)-47 b Fr(u)1125 5256 y Fp(n;m)1248 5244 y Fo(j)23 b(\024)g Fr(D)1451 5256 y Fn(0)1488 5244 y Fr(D)1559 5210 y Fv(j)p Fp(n)p Fv(j)p Fn(+)p Fv(j)p Fp(m)p Fv(j)1793 5244 y Fo(j)p Fr(")p Fo(j)1878 5210 y Fn(\()p Fv(j)p Fp(n)p Fv(j)p Fn(+)p Fv(j)p Fp(m)p Fv(j)p Fn(\))p Fp(=)p Fn(2)2230 5244 y Fr(;)765 b Fu(\(5)p Fr(:)p Fu(6\))1612 5443 y(11)p eop %%Page: 12 12 12 11 bop 118 555 a Fs(wher)l(e)31 b Fr(D)g Fs(and)f Fr(D)683 567 y Fn(0)750 555 y Fs(ar)l(e)g(p)l(ositive)h(c)l(onstants.) 38 b(Mor)l(e)l(over)f Fu(^)-48 b Fr(u)2002 567 y Fp(n;m)2155 555 y Fs(dep)l(end)31 b(analytic)l(al)t(ly)h(on)e Fr(")f Fs(and)118 674 y(on)h(al)t(l)h(the)f(p)l(ar)l(ameters)g Fr(\027)964 631 y Fn(\()p Fp(c)p Fn(\))959 699 y Fp(m)1018 683 y Fj(0)1040 699 y Fp(;h)1099 683 y Fj(0)1126 674 y Fs(,)g(with)g Fr(c)23 b Fu(=)g Fr(a;)14 b(b)29 b Fs(and)h Fo(j)p Fr(m)1911 644 y Fv(0)1934 674 y Fo(j)23 b Fr(>)g Fu(1)p Fs(.)118 853 y(R)l(emark)k(8.)37 b Fu(One)25 b(should)g(compare) f(the)h(ab)r(o)n(v)n(e)f(p)r(o)n(w)n(er)f(series)h(expansion)g(with)i (the)f(analogous)118 960 y(Lindstedt)30 b(series)f(for)g(in)n(v)-5 b(arian)n(t)29 b(KAM)h(tori)f(in)h(quasi-in)n(tegrable)d(Hamiltonian)j (system)f(\(see)118 1075 y([4],)h([5])f(and)h([8]\).)43 b(The)30 b(main)f(di\013erence)h(is)f(that)h(in)g(the)g(latter)g(case)e (the)i(analogous)e(of)h Fr(\027)3072 1032 y Fn(\()p Fp(c)p Fn(\))3067 1100 y Fp(h;m)118 1182 y Fu(are)21 b(exactly)g(v)-5 b(anishing,)23 b(as)e(a)h(consequence)f(of)g(p)r(eculiar)h (cancellations.)34 b(Hence)22 b(one)f(can)h(pro)n(v)n(e)118 1288 y(analyticit)n(y)28 b(in)h Fr(")f Fu(for)g(the)h(p)r(o)n(w)n(er)e (series.)39 b(In)28 b(the)h(presen)n(t)f(case)g Fr(\027)2231 1245 y Fn(\()p Fp(c)p Fn(\))2226 1313 y Fp(h;m)2376 1288 y Fu(are)f(not)i(v)-5 b(anishing,)28 b(and)118 1419 y(w)n(e)j(ha)n(v)n (e)g(to)g(c)n(ho)r(ose)g(carefully)g(the)h(coun)n(teterms)f Fr(\027)1815 1431 y Fp(m)1910 1419 y Fu(so)g(that)h Fo(j)p Fr(\027)2269 1376 y Fn(\()p Fp(c)p Fn(\))2264 1444 y Fp(h;m)2385 1419 y Fo(j)e(\024)g Fr(C)6 b Fo(j)p Fr(")p Fo(j)p Fu(;)33 b(for)f(a)f(similar)118 1526 y(approac)n(h)26 b(,)i(see)f(for)g(instance)g([10])g(or)g([11].)189 1714 y(The)h(quan)n(tities)f Fr(\027)789 1671 y Fn(\()p Fp(c)p Fn(\))784 1739 y Fp(h;m)906 1714 y Fu(,)g(for)g Fr(h)19 b Fu(+)f(1)k Fo(\025)h Fu(0)k(and)h Fo(j)p Fr(m)p Fo(j)23 b Fr(>)f Fu(1,)27 b(v)n(erify)g(the)h(recursiv)n(e)e(relations)965 1956 y Fr(\027)1011 1913 y Fn(\()p Fp(c)p Fn(\))1006 1981 y Fp(h)p Fn(+1)p Fp(;m)1235 1956 y Fu(=)c(2)p Fr(\027)1410 1913 y Fn(\()p Fp(c)p Fn(\))1405 1981 y Fp(h;m)1545 1956 y Fu(+)c Fr(\014)1679 1913 y Fn(\()p Fp(c)p Fn(\))1675 1981 y Fp(h;m)1797 1956 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(";)g Fo(f)p Fr(\027)2085 1913 y Fn(\()p Fp(c)2141 1887 y Fj(0)2162 1913 y Fn(\))2080 1981 y Fp(h)2119 1964 y Fj(0)2140 1981 y Fp(;m)2219 1964 y Fj(0)2245 1956 y Fo(g)p Fu(\))p Fr(;)676 b Fu(\(5)p Fr(:)p Fu(7\))118 2193 y(where,)30 b(b)n(y)g(de\014ning)h Fo(T)885 2149 y Fn(\()p Fp(c)p Fn(\))864 2218 y Fp(h)1001 2193 y Fu(as)e(the)i(set)f(of)g(self-energy) f(graphs)g(in)h Fo(T)2331 2149 y Fn(\()p Fp(c)p Fn(\))2309 2218 y Fp()f Fu(1)30 b Fs(and)i(al)t(l)f Fr(h)25 b Fo(\025)f Fu(0)30 b Fs(ther)l(e)h(exist)f(a)h(c)l(onstant)f Fr(C)37 b Fs(such)118 3197 y(that)29 b(one)f(has)h Fo(j)p Fr(\027)658 3154 y Fn(\()p Fp(c)p Fn(\))653 3222 y Fp(h;m)775 3197 y Fo(j)23 b(\024)g Fr(C)6 b Fo(j)p Fr(")p Fo(j)p Fs(,)29 b(with)g Fr(c)23 b Fu(=)g Fr(a;)14 b(b)p Fs(.)38 b(Then)29 b(for)g(al)t(l)g Fr(h)23 b Fo(\025)g Fu(0)28 b Fs(and)h(for)g(al)t(l)h Fr(T)k Fo(2)23 b(T)2948 3154 y Fn(\()p Fp(c)p Fn(\))2926 3222 y Fp(h)3062 3197 y Fs(one)118 3303 y(has)798 3410 y Fo(jV)879 3375 y Fp(h)p Fn(+1)872 3435 y Fp(T)1006 3410 y Fu(\()7 b(~)-49 b Fr(!)1090 3422 y Fp(m)1153 3410 y Fr(;)14 b(m)p Fu(\))p Fo(j)23 b(\024)g Fr(B)1492 3422 y Fn(0)1529 3410 y Fr(B)1596 3376 y Fp(K)t Fn(\()p Fp(T)9 b Fn(\))1760 3410 y Fo(j)p Fr(")p Fo(j)1845 3376 y Fn(2)1878 3351 y Fi(\()p Fh(h)p Fj(\000)p Fi(1\))p Fh(=\034)2099 3376 y Fp(=)p Fn(2)2170 3410 y Fo(j)p Fr(")p Fo(j)2255 3376 y Fp(K)t Fn(\()p Fp(T)g Fn(\))p Fp(=)p Fn(2)2486 3410 y Fr(;)468 b Fu(\(5)p Fr(:)p Fu(12\))118 3576 y Fs(wher)l(e)31 b Fr(K)6 b Fu(\()p Fr(T)12 b Fu(\))22 b(=)664 3513 y Fq(P)752 3601 y Fp(v)r Fv(2)p Fp(V)14 b Fn(\()p Fp(T)9 b Fn(\))1004 3576 y Fo(j)p Fr(n)1077 3588 y Fp(v)1116 3576 y Fo(j)p Fs(,)30 b(for)h(some)f(p)l(ositive)h(c)l (onstants)e Fr(B)2266 3588 y Fn(0)2333 3576 y Fs(and)h Fr(B)t Fs(.)189 3786 y Fu(Note)c(that)h(in)g(eac)n(h)e(con)n(tribution) h(to)g Fr(\014)1471 3743 y Fn(\()p Fp(c)p Fn(\))1467 3811 y Fp(h;m)1615 3786 y Fu(con)n(taining)f(a)h Fr(\027)2131 3743 y Fn(\()p Fp(c)2187 3718 y Fj(0)2210 3743 y Fn(\))2126 3811 y Fp(h)2165 3794 y Fj(0)2187 3811 y Fp(;m)2270 3786 y Fu(,)h(there)f(are)f(at)i(least)f Fr(M)e Fo(\000)15 b Fu(1)118 3892 y(end-p)r(oin)n(ts;)23 b(in)f(fact)f(the)h(self-energy) d(graphs)h(\(with)i(an)f(incoming)g(external)f(line)h(carrying)f(mo-) 118 3998 y(men)n(tum)25 b(\()p Fr(n;)14 b(m)p Fu(\)\))25 b(with)f(only)g Fr(\027)5 b Fu(-v)n(ertices)23 b(are)g(suc)n(h)h(that)h (all)e(the)i(propagators)c(ha)n(v)n(e)i(argumen)n(t)118 4105 y(giv)n(en)32 b(b)n(y)39 b(~)-48 b Fr(!)513 4117 y Fn(1)549 4105 y Fr(n)22 b Fo(\000)28 b Fu(~)-48 b Fr(!)760 4117 y Fp(m)822 4105 y Fu(;)36 b(hence)d(when)g(the)g Fo(L)g Fu(op)r(eration)f(is)g(applied)h(\(consisting)f(in)h(replacing) 125 4211 y(~)-49 b Fr(!)170 4223 y Fn(1)207 4211 y Fr(n)26 b Fu(with)33 b(~)-49 b Fr(!)522 4223 y Fp(m)585 4211 y Fu(\))26 b(they)g(are)f(v)-5 b(anishing)26 b(b)n(y)g(the)g(supp)r (ort)g(prop)r(erties)f(of)g(the)i(propagators.)33 b(This,)118 4317 y(together)27 b(with)h(the)g(inequalit)n(y)f Fr(M)32 b Fo(\025)22 b Fu(3,)28 b(implies)f(that)h(w)n(e)f(can)h(b)r(ound)g(in) g(\(A2.2\))1171 4447 y Fq(\014)1171 4497 y(\014)1171 4546 y(\014)1199 4542 y Fr(\014)1250 4499 y Fn(\()p Fp(c)p Fn(\))1246 4567 y Fp(h;m)1368 4447 y Fq(\014)1368 4497 y(\014)1368 4546 y(\014)1418 4542 y Fo(\024)23 b Fr(C)6 b Fo(j)p Fr(")p Fo(j)1656 4508 y Fp(M)g Fv(\000)p Fn(1)1838 4542 y Fo(\024)22 b Fr(C)6 b Fo(j)p Fr(")p Fo(j)2075 4508 y Fn(2)2113 4542 y Fr(;)841 b Fu(\(5)p Fr(:)p Fu(13\))118 4763 y(whic)n(h)32 b(follo)n(ws)f(from)g(the)h(v)n(ery)f(de\014nition)h (for)f(the)i(con)n(tributions)e(arising)f(from)h(self-energy)118 4869 y(graphs)22 b(con)n(taining)h(only)g(one)h(no)r(de)f(\(hence)h(on) g(scale)f Fr(h)f Fu(=)h Fo(\000)p Fu(1\),)h(and)g(from)f(the)h(b)r (ound)g(\(A3.2\))118 4975 y(for)31 b(the)i(con)n(tribution)e(con)n (taining)g(at)g(least)h(t)n(w)n(o)f(no)r(des)h(\(so)f(that)h(one)g (line)g(can)f(b)r(e)h(on)g(scale)118 5082 y Fr(h)23 b Fo(\025)g Fu(0\).)1612 5443 y(13)p eop %%Page: 14 14 14 13 bop 820 555 a Fw(6.)50 b(The)37 b(c)m(hoice)g(of)g(the)h(coun)m (terterms)118 733 y Fu(In)29 b(this)f(section)g(w)n(e)g(sho)n(w)g(that) h(it)f(is)h(p)r(ossible)f(to)g(c)n(ho)r(ose)f Fr(\027)2067 703 y Fn(\()p Fp(c)p Fn(\))2178 733 y Fu(=)d Fo(f)p Fr(\027)2355 690 y Fn(\()p Fp(c)p Fn(\))2350 742 y Fp(m)2440 733 y Fo(g)2482 748 y Fv(j)p Fp(m)p Fv(j\025)p Fn(1)2698 733 y Fu(suc)n(h)k(that,)h(for)118 857 y(a)c(suitable)g(p)r(ositiv)n(e)g (constan)n(t)g Fr(C)6 b Fu(,)26 b(one)f(has)g Fo(j)p Fr(\027)1609 814 y Fn(\()p Fp(c)p Fn(\))1604 882 y Fp(h;m)1725 857 y Fo(j)e(\024)g Fr(C)6 b Fo(j)p Fr(")p Fo(j)26 b Fu(for)e(all)h Fr(h)e Fo(\025)g Fu(0)i(and)g(for)g(all)g Fo(j)p Fr(m)p Fo(j)e Fr(>)f Fu(1.)189 963 y(F)-7 b(or)27 b(an)n(y)g(sequence)g Fr(a)c Fo(\021)f(f)p Fr(a)1079 975 y Fp(m)1142 963 y Fo(g)1184 978 y Fv(j)p Fp(m)p Fv(j\025)p Fn(1)1399 963 y Fu(w)n(e)27 b(in)n(tro)r(duce)g(the)h(norm)1313 1129 y Fo(k)o Fr(a)p Fo(k)1440 1154 y Fv(1)1533 1129 y Fu(=)52 b(sup)1621 1203 y Fv(j)p Fp(m)p Fv(j\025)p Fn(1)1818 1129 y Fo(j)p Fr(a)1885 1141 y Fp(m)1948 1129 y Fo(j)p Fr(;)1024 b Fu(\(6)p Fr(:)p Fu(1\))118 1351 y(and)28 b(the)g(seminorm)1331 1457 y Fo(j)p Fr(a)p Fo(j)1421 1482 y Fv(1)1515 1457 y Fu(=)52 b(sup)1602 1531 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)1800 1457 y Fo(j)p Fr(a)1867 1469 y Fp(m)1930 1457 y Fo(j)p Fr(;)1042 b Fu(\(6)p Fr(:)p Fu(2\))118 1660 y(whic)n(h)28 b(will)f(b)r(e)h(useful)g(in)g(order)f (to)g(b)r(ound)h(the)g(coun)n(terterms.)189 1766 y(W)-7 b(e)28 b(pro)n(v)n(e)e(the)i(follo)n(wing)e(lemma)i(in)g(App)r(endix)g (A7.)118 1943 y Fk(Lemma)34 b(7.)47 b Fs(Ther)l(e)34 b(exists)e Fr(")1094 1955 y Fn(0)1160 1943 y Fr(>)d Fu(0)j Fs(such)h(that)g(for)h Fo(j)p Fr(")p Fo(j)29 b Fr(<)g(")2076 1955 y Fn(0)2146 1943 y Fs(ther)l(e)k(is)g(a)h(family)g(of)g(intervals) 118 2064 y Fr(I)161 2021 y Fn(\()188 2006 y(\026)187 2021 y Fp(h)p Fn(\))154 2074 y Fp(c;m)267 2064 y Fs(,)325 2042 y Fu(\026)324 2064 y Fr(h)26 b Fo(\025)g Fu(0)p Fs(,)32 b Fo(j)p Fr(m)p Fo(j)26 b Fr(>)g Fu(1)p Fs(,)32 b Fr(c)26 b Fu(=)f Fr(a;)14 b(b)p Fs(,)32 b(such)g(that)f Fr(I)1654 2021 y Fn(\()1681 2006 y(\026)1680 2021 y Fp(h)p Fn(+1\))1647 2074 y Fp(c;m)1859 2064 y Fo(\032)26 b Fr(I)1993 2021 y Fn(\()2020 2006 y(\026)2019 2021 y Fp(h)p Fn(\))1986 2074 y Fp(c;m)2099 2064 y Fs(,)32 b Fo(j)p Fr(I)2223 2019 y Fn(\026)2222 2034 y Fp(h)2215 2085 y(c;m)2328 2064 y Fo(j)26 b(\024)g Fu(2)p Fo(j)p Fr(")p Fo(j)p Fu(\()2627 1996 y Fo(p)p 2696 1996 42 4 v 68 x Fu(2)o(\))2769 2034 y Fv(\000)p Fn(\()2848 2019 y(\026)2847 2034 y Fp(h)p Fn(+1\))3032 2064 y Fs(and,)118 2196 y(if)31 b Fr(\027)245 2153 y Fn(\()p Fp(c)p Fn(\))240 2206 y Fp(m)354 2196 y Fo(2)23 b Fr(I)475 2153 y Fn(\()502 2138 y(\026)501 2153 y Fp(h)p Fn(\))468 2206 y Fp(c;m)581 2196 y Fs(,)30 b(then)1105 2312 y Fo(j)p Fr(\027)1174 2269 y Fn(\()p Fp(c)p Fn(\))1169 2337 y Fp(h)1260 2312 y Fo(j)1283 2324 y Fv(1)1376 2312 y Fo(\024)23 b Fr(C)6 b Fo(j)p Fr(")p Fo(j)p Fr(;)1822 2290 y Fu(\026)1821 2312 y Fr(h)23 b Fo(\025)f Fr(h)h Fo(\025)g Fu(0)p Fr(;)815 b Fu(\(6)p Fr(:)p Fu(3\))118 2473 y Fs(for)28 b(some)f(p)l(ositive)i(c)l(onstant)d Fr(C)6 b Fs(.)38 b(Final)t(ly)28 b(one)g(has)f Fr(\027)1838 2429 y Fn(\()p Fp(c)p Fn(\))1833 2498 y Fp(h;)p Fv(\000)p Fp(m)2030 2473 y Fu(=)c Fr(\027)2164 2429 y Fn(\()p Fp(c)p Fn(\))2159 2498 y Fp(h;m)2280 2473 y Fs(,)28 b Fr(c)23 b Fu(=)g Fr(a;)14 b(b)p Fs(,)27 b(for)h(al)t(l)2896 2451 y Fu(\026)2895 2473 y Fr(h)23 b Fo(\025)g Fr(h)f Fo(\025)118 2579 y Fu(0)29 b Fs(and)i(for)f(al)t(l)h Fr(m)23 b(>)g Fu(1)p Fs(.)189 2756 y Fu(It)j(will)h(b)r(e)f(useful)h(to)e(really)g (construct)h(the)g Fr(\027)1658 2713 y Fn(\()p Fp(c)p Fn(\))1653 2781 y Fp(h;m)1801 2756 y Fu(b)n(y)g(a)g(con)n(traction)e (metho)r(d.)37 b(By)26 b(iterating)118 2862 y(\(5.7\))h(w)n(e)h (\014nd,)g(for)f Fr(m)c(>)f Fu(1)679 3103 y Fr(\027)725 3060 y Fn(\()p Fp(c)p Fn(\))720 3128 y Fp(h;m)864 3103 y Fu(=)h(2)994 3069 y Fp(h)p Fn(+1)1134 2961 y Fq( )1200 3103 y Fr(\027)1246 3069 y Fn(\()p Fp(c)p Fn(\))1241 3124 y Fp(m)1351 3103 y Fu(+)1458 3000 y Fp(h)p Fv(\000)p Fn(1)1460 3024 y Fq(X)1434 3203 y Fp(k)q Fn(=)p Fv(\000)p Fn(1)1620 3103 y Fu(2)1662 3069 y Fv(\000)p Fp(k)q Fv(\000)p Fn(2)1839 3103 y Fr(\014)1890 3060 y Fn(\()p Fp(c)p Fn(\))1886 3128 y Fp(k)q(;m)2006 3103 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(";)g Fo(f)p Fr(\027)2294 3060 y Fn(\()p Fp(c)2350 3035 y Fj(0)2371 3060 y Fn(\))2289 3128 y Fp(k)2325 3112 y Fj(0)2347 3128 y Fp(;m)2426 3112 y Fj(0)2452 3103 y Fo(g)p Fu(\))2526 2961 y Fq(!)2605 3103 y Fr(:)390 b Fu(\(6)p Fr(:)p Fu(4\))118 3377 y(where)25 b Fr(\014)407 3334 y Fn(\()p Fp(c)p Fn(\))403 3402 y Fp(k)q(;m)523 3377 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(";)g Fo(f)p Fr(\027)811 3334 y Fn(\()p Fp(c)867 3309 y Fj(0)888 3334 y Fn(\))806 3402 y Fp(k)842 3386 y Fj(0)864 3402 y Fp(;m)943 3386 y Fj(0)969 3377 y Fo(g)p Fu(\))26 b(dep)r(ends)g(on)f Fr(\027)1542 3389 y Fp(k)1578 3373 y Fj(0)1601 3389 y Fp(;m)1680 3373 y Fj(0)1732 3377 y Fu(with)h Fr(k)1965 3347 y Fv(0)2012 3377 y Fo(\024)c Fr(k)c Fo(\000)c Fu(1.)36 b(If)26 b(w)n(e)f(put)h Fr(h)d Fu(=)2851 3356 y(\026)2850 3377 y Fr(h)i Fu(in)h(\(6.4\))118 3484 y(w)n(e)h(get)h(the)g(follo)n (wing)e(iden)n(tit)n(y)704 3728 y Fr(\027)750 3694 y Fn(\()p Fp(c)p Fn(\))745 3749 y Fp(m)859 3728 y Fu(=)d Fo(\000)1051 3609 y Fn(\026)1050 3625 y Fp(h)o Fv(\000)p Fn(1)1052 3650 y Fq(X)1026 3828 y Fp(k)q Fn(=)p Fv(\000)p Fn(1)1212 3728 y Fu(2)1254 3694 y Fv(\000)p Fp(k)q Fv(\000)p Fn(2)1431 3728 y Fr(\014)1482 3685 y Fn(\()p Fp(c)p Fn(\))1478 3753 y Fp(k)q(;m)1597 3728 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(";)g Fo(f)p Fr(\027)1885 3685 y Fn(\()p Fp(c)1941 3660 y Fj(0)1963 3685 y Fn(\))1880 3753 y Fp(k)1916 3737 y Fj(0)1938 3753 y Fp(;m)2017 3737 y Fj(0)2044 3728 y Fo(g)p Fu(\))k(+)g(2)2261 3694 y Fv(\000)2314 3679 y Fn(\026)2313 3694 y Fp(h)o Fv(\000)p Fn(1)2440 3728 y Fr(\027)2486 3685 y Fn(\()p Fp(c)p Fn(\))2482 3746 y(\026)2481 3762 y Fp(h;m)3018 3728 y Fu(\(6)p Fr(:)p Fu(5\))118 3980 y(and)28 b(\(6.4\),)f(\(6.5\))g(are)g(equiv)-5 b(alen)n(t,)27 b(for)1383 3958 y(\026)1382 3980 y Fr(h)c(>)g(h)g Fo(\025)f Fu(0,)28 b(to)566 4239 y Fr(\027)612 4196 y Fn(\()p Fp(c)p Fn(\))607 4264 y Fp(h;m)752 4239 y Fu(=)23 b Fo(\000)p Fu(2)947 4205 y Fp(h)p Fn(+1)1087 4072 y Fq(0)1087 4222 y(@)1162 4120 y Fn(\026)1161 4135 y Fp(h)p Fv(\000)p Fn(1)1163 4160 y Fq(X)1159 4339 y Fp(k)q Fn(=)p Fp(h)1300 4239 y Fu(2)1342 4205 y Fv(\000)p Fp(k)q Fv(\000)p Fn(2)1519 4239 y Fr(\014)1570 4196 y Fn(\()p Fp(c)p Fn(\))1566 4264 y Fp(k)q(;m)1685 4239 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(";)g Fo(f)p Fr(\027)1973 4196 y Fn(\()p Fp(c)2029 4171 y Fj(0)2051 4196 y Fn(\))1968 4264 y Fp(k)2004 4247 y Fj(0)2027 4264 y Fp(;m)2106 4247 y Fj(0)2132 4239 y Fo(g)p Fu(\))2206 4072 y Fq(1)2206 4222 y(A)2297 4239 y Fu(+)k(2)2422 4205 y Fp(h)p Fv(\000)2514 4189 y Fn(\026)2513 4205 y Fp(h)2555 4239 y Fr(\027)2601 4196 y Fn(\()p Fp(c)p Fn(\))2597 4257 y(\026)2596 4272 y Fp(h;m)2718 4239 y Fr(:)277 b Fu(\(6)p Fr(:)p Fu(6\))118 4522 y(The)42 b(sequences)e Fo(f)p Fr(\027)781 4479 y Fn(\()p Fp(c)p Fn(\))776 4547 y Fp(h;m)898 4522 y Fo(g)940 4537 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)1127 4522 y Fu(,)1196 4500 y(\026)1195 4522 y Fr(h)46 b(>)g(h)g Fo(\025)g Fu(0,)e(parametrized)d(b)n(y)g Fo(f)p Fr(\027)2449 4479 y Fn(\()p Fp(c)p Fn(\))2445 4540 y(\026)2444 4556 y Fp(h)o(;m)2565 4522 y Fo(g)2607 4537 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)2836 4522 y Fu(suc)n(h)g(that) 118 4662 y Fo(j)p Fr(\027)187 4619 y Fn(\()p Fp(c)p Fn(\))183 4680 y(\026)182 4695 y Fp(h)273 4662 y Fo(j)296 4674 y Fv(1)402 4662 y Fo(\024)35 b Fr(C)6 b Fo(j)p Fr(")p Fo(j)p Fu(,)38 b(can)c(b)r(e)i(obtained)f(as)f(the)i(limit)g(as)e Fr(q)39 b Fo(!)c(1)h Fu(of)f(the)g(sequences)f Fo(f)p Fr(\027)2954 4619 y Fn(\()p Fp(c)p Fn(\)\()p Fp(q)r Fn(\))2949 4687 y Fp(h;m)3124 4662 y Fo(g)p Fu(,)118 4768 y Fr(q)26 b Fo(\025)d Fu(0,)k(de\014ned)h(recursiv)n(ely)e(as)637 4939 y Fr(\027)683 4896 y Fn(\()p Fp(c)p Fn(\)\(0\))678 4964 y Fp(h;m)877 4939 y Fu(=)d(0)p Fr(;)638 5169 y(\027)684 5135 y Fn(\()p Fp(c)p Fn(\)\()p Fp(q)r Fn(\))679 5190 y Fp(m)877 5169 y Fu(=)g Fo(\000)1045 5050 y Fn(\026)1044 5065 y Fp(h)p Fv(\000)p Fn(1)1046 5090 y Fq(X)1044 5269 y Fp(k)q Fn(=)p Fp(h)1183 5169 y Fu(2)1225 5135 y Fv(\000)p Fp(k)q Fv(\000)p Fn(2)1403 5169 y Fr(\014)1454 5126 y Fn(\()p Fp(c)p Fn(\))1450 5194 y Fp(k)q(;m)1569 5169 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(";)g Fo(f)p Fr(\027)1857 5126 y Fn(\()p Fp(c)1913 5101 y Fj(0)1934 5126 y Fn(\)\()p Fp(q)r Fv(\000)p Fn(1\))1852 5194 y Fp(k)1888 5178 y Fj(0)1910 5194 y Fp(;m)1989 5178 y Fj(0)2134 5169 y Fo(g)p Fu(\))k(+)g(2)2351 5135 y Fp(h)p Fv(\000)2443 5120 y Fn(\026)2442 5135 y Fp(h)2484 5169 y Fr(\027)2530 5126 y Fn(\()p Fp(c)p Fn(\))2526 5187 y(\026)2525 5203 y Fp(h;m)2647 5169 y Fr(:)3018 5085 y Fu(\(6)p Fr(:)p Fu(7\))1612 5443 y(14)p eop %%Page: 15 15 15 14 bop 118 559 a Fu(In)31 b(fact,)i(it)e(is)g(easy)f(to)h(sho)n(w)f (inductiv)n(ely)i(that,)g(if)g Fr(")e Fu(is)h(small)g(enough,)h Fo(j)p Fr(\027)2568 516 y Fn(\()p Fp(q)r Fn(\))2563 584 y Fp(h)2656 559 y Fo(j)2679 571 y Fv(1)2779 559 y Fo(\024)c Fr(C)6 b Fo(j)p Fr(")p Fo(j)3022 529 y Fn(2)3060 559 y Fu(,)32 b(so)118 665 y(that)c(\(6.7\))f(is)h(meaningful,)g(and)1091 843 y(max)1060 905 y Fn(0)p Fv(\024)p Fp(h)p Fv(\024)1237 890 y Fn(\026)1236 905 y Fp(h)1289 843 y Fo(j)p Fr(\027)1358 800 y Fn(\()p Fp(q)r Fn(\))1353 868 y Fp(h)1465 843 y Fo(\000)18 b Fr(\027)1594 800 y Fn(\()p Fp(q)r Fv(\000)p Fn(1\))1589 868 y Fp(h)1768 843 y Fo(j)1791 855 y Fv(1)1885 843 y Fo(\024)k Fu(\()p Fr(C)6 b Fo(j)p Fr(")p Fo(j)p Fu(\))2186 808 y Fp(q)2224 843 y Fr(:)771 b Fu(\(6)p Fr(:)p Fu(8\))118 1081 y(F)-7 b(or)34 b Fr(q)j Fu(=)c(1)h(this)h(is)f (true)g(as)f Fr(\027)1117 1038 y Fn(\()p Fp(c)p Fn(\)\(0\))1112 1106 y Fp(h)1323 1081 y Fu(=)g(0;)k(for)d Fr(q)j(>)d Fu(1)g(it)g(follo)n(ws)f(trivially)h(b)n(y)g(the)g(fact)h(that)118 1206 y Fr(\014)169 1162 y Fn(\()p Fp(c)p Fn(\))165 1231 y Fp(k)255 1206 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(";)g Fo(f)p Fr(\027)543 1162 y Fn(\()p Fp(c)599 1137 y Fj(0)620 1162 y Fn(\)\()p Fp(q)r Fv(\000)p Fn(1\))538 1231 y Fp(k)574 1214 y Fj(0)596 1231 y Fp(;m)675 1214 y Fj(0)820 1206 y Fo(g)p Fu(\))24 b Fo(\000)g Fr(\014)1058 1162 y Fn(\()p Fp(c)p Fn(\))1054 1231 y Fp(k)1144 1206 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(";)g Fo(f)p Fr(\027)1432 1162 y Fn(\()p Fp(c)1488 1137 y Fj(0)1510 1162 y Fn(\)\()p Fp(q)r Fv(\000)p Fn(2\))1427 1231 y Fp(k)1463 1214 y Fj(0)1485 1231 y Fp(;m)1564 1214 y Fj(0)1709 1206 y Fo(g)p Fu(\))36 b(can)h(b)r(e)g (written)f(as)g(a)g(sum)h(of)g(terms)f(in)118 1341 y(whic)n(h)f(there)g (are)f(at)h(least)f(one)h Fr(\027)5 b Fu(-v)n(ertex,)36 b(with)f(a)g(di\013erence)g Fr(\027)2255 1298 y Fn(\()p Fp(c)2311 1273 y Fj(0)2333 1298 y Fn(\)\()p Fp(q)r Fv(\000)p Fn(1\))2250 1366 y Fp(h)2289 1350 y Fj(0)2556 1341 y Fo(\000)23 b Fr(\027)2690 1298 y Fn(\()p Fp(c)2746 1273 y Fj(0)2768 1298 y Fn(\)\()p Fp(q)r Fv(\000)p Fn(2\))2685 1366 y Fp(h)2724 1350 y Fj(0)2968 1341 y Fu(,)37 b(with)118 1466 y Fr(h)166 1436 y Fv(0)221 1466 y Fo(\025)31 b Fr(k)s Fu(,)j(in)f(place)f(of)h(the)g(corresp)r(onding)e Fr(\027)1572 1423 y Fn(\()p Fp(c)1628 1398 y Fj(0)1650 1423 y Fn(\))1567 1491 y Fp(h)1606 1474 y Fj(0)1680 1466 y Fu(,)j(and)f(one)f(no)r(de)h (carrying)e(an)h Fr(")p Fu(.)52 b(Then)33 b Fr(\027)3100 1423 y Fn(\()p Fp(q)r Fn(\))3095 1491 y Fp(h)118 1572 y Fu(con)n(v)n(erges)25 b(as)i Fr(q)f Fo(!)d(1)p Fu(,)28 b(for)1023 1550 y(\026)1022 1572 y Fr(h)23 b(<)g(h)g Fo(\024)f Fu(1,)28 b(to)f(a)g(limit)i Fr(\027)1842 1584 y Fp(h)1885 1572 y Fu(,)f(satisfying)f(the)h(b)r(ound)g Fo(j)p Fr(\027)2769 1584 y Fp(h)2812 1572 y Fo(j)2835 1584 y Fv(1)2928 1572 y Fo(\024)23 b Fr(C)6 b Fo(j)p Fr(")p Fo(j)p Fu(.)118 1679 y(Since)28 b(the)g(solution)f(is)g(unique,) h(it)g(m)n(ust)g(coincide)f(with)h(the)g(one)g(implied)g(b)n(y)f(lemma) h(7.)189 1856 y(The)f(ab)r(o)n(v)n(e)f(prescription)g(\014xes)g Fr(\027)1283 1813 y Fn(\()p Fp(c)p Fn(\))1278 1881 y Fp(h;m)1427 1856 y Fu(for)g(all)h Fo(j)p Fr(m)p Fo(j)c Fr(>)g Fu(1.)36 b(The)27 b(v)-5 b(alues)27 b Fr(\027)2462 1813 y Fn(\()p Fp(c)p Fn(\))2457 1878 y Fv(\006)p Fn(1)2575 1856 y Fu(are)f(then)h(\014xed)g(b)n(y)118 1987 y(inserting)h(the)h(v) -5 b(alues)29 b(of)g Fr(\027)996 1944 y Fn(\()p Fp(c)p Fn(\))991 2012 y Fp(h;m)1141 1987 y Fu(for)f(all)h Fo(j)p Fr(m)p Fo(j)c Fr(>)f Fu(1)29 b(in)n(to)f(the)h(expressions)e(of)34 b(^)-47 b Fr(u)2581 1999 y Fp(n;m)2733 1987 y Fu(app)r(earing)28 b(in)118 2093 y(the)g(righ)n(t)f(hand)h(side)f(of)h(\(2.7\).)189 2209 y(Finally)41 b(note)g(that)g(one)g(can)f(c)n(ho)r(ose)g Fr(\027)1530 2166 y Fn(\()p Fp(c)p Fn(\))1525 2219 y Fp(m)1661 2209 y Fu(=)45 b Fr(\027)1817 2166 y Fn(\()p Fp(c)p Fn(\))1812 2230 y Fv(\000)p Fp(m)1968 2209 y Fu(and)c(then)h Fr(\027)2392 2166 y Fn(\()p Fp(c)p Fn(\))2387 2234 y Fp(h;m)2554 2209 y Fu(=)j Fr(\027)2710 2166 y Fn(\()p Fp(c)p Fn(\))2705 2234 y Fp(h;)p Fv(\000)p Fp(m)2919 2209 y Fu(for)c(an)n(y)118 2341 y Fo(j)p Fr(m)p Fo(j)c Fr(>)g Fu(1)e(and)h(an)n(y)789 2319 y(\026)788 2341 y Fr(h)h Fo(\025)f Fr(h)h Fo(\025)g Fu(0;)i(this)d(follo)n(ws)f(from)h (the)g(fact)g(that)h(the)f(function)g Fr(\014)2968 2297 y Fn(\()p Fp(c)p Fn(\))2964 2366 y Fp(k)q(;m)3120 2341 y Fu(in)118 2447 y(\(6.7\))c(is)f(ev)n(en)h(under)f(the)h(exc)n(hange)f Fr(m)f Fo(!)g(\000)p Fr(m)p Fu(;)j(it)f(dep)r(ends)g(on)g Fr(m)g Fu(through)37 b(~)-48 b Fr(!)2765 2459 y Fp(m)2859 2447 y Fu(\(whic)n(h)32 b(is)118 2553 y(an)f(ev)n(en)g(function)h(of)f Fr(m)p Fu(\),)h(through)f(the)g(end-p)r(oin)n(ts)g Fr(v)i Fo(2)c Fr(E)5 b Fu(\()p Fr(\022)r Fu(\))32 b(\(whic)n(h)g(are)e(o)r(dd) i(under)f(the)118 2669 y(exc)n(hange)22 b Fr(m)h Fo(!)g(\000)p Fr(m)p Fu(;)h(but)g(their)f(n)n(um)n(b)r(er)g(m)n(ust)g(b)r(e)h(ev)n (en\))f(and)g(\014nally)g(through)f Fr(\027)2782 2626 y Fn(\()p Fp(q)r Fv(\000)p Fn(1\))2777 2694 y Fp(k)q(;m)2979 2669 y Fu(whic)n(h)118 2775 y(are)27 b(assumed)g(inductiv)n(ely)g(to)h (b)r(e)g(ev)n(en.)525 3023 y Fw(7.)100 b(Construction)36 b(of)i(the)f(p)s(erturb)s(ed)h(frequencies)118 3200 y Fu(In)29 b(the)g(follo)n(wing)f(it)i(will)f(b)r(e)g(con)n(v)n(enien)n (t)f(to)g(set)h Fr(!)f Fu(=)d Fo(f)p Fr(!)1985 3212 y Fp(m)2047 3200 y Fo(g)2089 3215 y Fv(j)p Fp(m)p Fv(j\025)p Fn(1)2305 3200 y Fu(and)35 b(~)-48 b Fr(!)28 b Fu(=)c Fo(f)7 b Fu(~)-49 b Fr(!)2731 3212 y Fp(m)2794 3200 y Fo(g)2836 3215 y Fv(j)p Fp(m)p Fv(j\025)p Fn(1)3023 3200 y Fu(.)40 b(By)118 3307 y(the)26 b(analysis)e(of)i(the)g(previous)e (sections)h(w)n(e)g(ha)n(v)n(e)f(found)i(the)g(coun)n(terterms)f Fo(f)p Fr(\027)2703 3319 y Fp(m)2765 3307 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))p Fo(g)3002 3322 y Fv(j)p Fp(m)p Fv(j\025)p Fn(1)118 3413 y Fu(as)27 b(functions)h(of)g Fr(")f Fu(and)34 b(~)-49 b Fr(!)s Fu(.)37 b(W)-7 b(e)28 b(ha)n(v)n(e)e(no)n(w)h(to)h(in)n(v)n(ert)f(the)h(relations)1278 3581 y(~)-48 b Fr(!)1327 3547 y Fn(2)1324 3601 y Fp(m)1405 3581 y Fu(+)18 b Fr(\027)1529 3593 y Fp(m)1592 3581 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))23 b(=)g Fr(!)1953 3547 y Fn(2)1950 3601 y Fp(m)2012 3581 y Fr(:)983 b Fu(\(7)p Fr(:)p Fu(1\))118 3749 y(This)28 b(is)f(giv)n(en)g(b)n(y)g(the)h(follo) n(wing)f(result.)118 3926 y Fk(Prop)s(osition)k(1.)39 b Fs(F)-6 b(or)30 b(al)t(l)i Fr(\026)1074 3938 y Fn(0)1135 3926 y Fr(>)24 b Fu(0)30 b Fs(ther)l(e)h(exists)f(a)h(ful)t(l)g(me)l (asur)l(e)f(set)g Fo(M)24 b(\032)g Fu([0)p Fr(;)14 b(\026)2769 3938 y Fn(0)2806 3926 y Fu(])30 b Fs(such)h(that)118 4032 y(for)36 b(al)t(l)f Fr(\026)e Fo(2)f(M)i Fs(ther)l(e)h(ar)l(e)g Fr(\030)h(>)c Fu(0)i Fs(and)h(a)g(set)g Fo(E)k(\032)32 b Fu([)p Fo(\000)p Fr(")1971 4044 y Fn(0)2007 4032 y Fr(;)14 b(")2083 4044 y Fn(0)2120 4032 y Fu(])35 b Fs(with)g(c)l (omplement)g(of)h(r)l(elative)118 4138 y(L)l(eb)l(esgue)29 b(me)l(asur)l(e)g(of)h(or)l(der)g Fr(")1135 4099 y Fp(\030)1135 4161 y Fn(0)1201 4138 y Fs(such)f(that)g(for)h(al)t(l)g Fr(")22 b Fo(2)i(E)36 b Fs(ther)l(e)29 b(exists)35 b Fu(~)-48 b Fr(!)26 b Fu(=)j(~)-49 b Fr(!)s Fu(\()p Fr(!)s(;)14 b(")p Fu(\))29 b Fs(solution)118 4245 y(of)i(\(7.1\).)189 4422 y Fu(W)-7 b(e)28 b(shall)g(sho)n(w)f(that)h(there)g(exists)f(a)h (sequence)f(of)h(sets)g Fo(fE)2120 4392 y Fn(\()p Fp(p)p Fn(\))2210 4422 y Fo(g)2252 4392 y Fv(1)2252 4442 y Fp(p)p Fn(=0)2401 4422 y Fu(in)h([)p Fo(\000)p Fr(")2626 4434 y Fn(0)2662 4422 y Fr(;)14 b(")2738 4434 y Fn(0)2775 4422 y Fu(],)28 b(suc)n(h)g(that)118 4528 y Fo(E)169 4498 y Fn(\()p Fp(p)p Fn(+1\))367 4528 y Fo(\032)22 b(E)505 4498 y Fn(\()p Fp(p)p Fn(\))596 4528 y Fu(,)h(and)e(a)g(sequence)g(of)h (functions)g Fo(f)7 b Fu(~)-49 b Fr(!)1736 4498 y Fn(\()p Fp(p)p Fn(\))1825 4528 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fo(g)2062 4498 y Fv(1)2062 4549 y Fp(p)p Fn(=0)2184 4528 y Fu(,)23 b(with)f(eac)n(h)27 b(~)-48 b Fr(!)2649 4498 y Fn(\()p Fp(p)p Fn(\))2762 4528 y Fo(\021)29 b Fu(~)-49 b Fr(!)2904 4498 y Fn(\()p Fp(p)p Fn(\))2994 4528 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))118 4634 y(de\014ned)28 b(for)f Fr(")c Fo(2)g(E)722 4604 y Fn(\()p Fp(p)p Fn(\))813 4634 y Fu(,)28 b(suc)n(h)f(that)h(for)f(all)g Fr(")c Fo(2)g(E)7 b Fu(,)28 b(with)1208 4858 y Fo(E)i Fu(=)1396 4754 y Fv(1)1383 4779 y Fq(\\)1370 4954 y Fp(p)p Fn(=0)1502 4858 y Fo(E)1553 4823 y Fn(\()p Fp(p)p Fn(\))1667 4858 y Fu(=)48 b(lim)1754 4907 y Fp(p)p Fv(!1)1935 4858 y Fo(E)1986 4823 y Fn(\()p Fp(p)p Fn(\))2076 4858 y Fr(;)919 b Fu(\(7)p Fr(:)p Fu(2\))118 5102 y(there)28 b(exists)f(the)h(limit) 1147 5208 y(~)-48 b Fr(!)1196 5174 y Fn(\()p Fv(1)p Fn(\))1318 5208 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))22 b(=)49 b(lim)1623 5258 y Fp(p)p Fv(!1)1810 5208 y Fu(~)-48 b Fr(!)1859 5174 y Fn(\()p Fp(p)p Fn(\))1948 5208 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fr(;)852 b Fu(\(7)p Fr(:)p Fu(3\))1612 5443 y(15)p eop %%Page: 16 16 16 15 bop 118 555 a Fu(and)28 b(it)g(solv)n(es)e(\(7.1\).)189 662 y(T)-7 b(o)27 b(ful\014ll)i(the)f(program)d(ab)r(o)n(v)n(e)h(w)n(e) i(shall)f(follo)n(w)g(an)g(iterativ)n(e)g(sc)n(heme)g(b)n(y)g(setting) 719 821 y(~)-49 b Fr(!)767 787 y Fn(\(0\)2)764 841 y Fp(m)912 821 y Fu(=)23 b Fr(!)1055 787 y Fn(2)1052 841 y Fp(m)1115 821 y Fr(;)719 961 y Fu(~)-49 b Fr(!)767 927 y Fn(\()p Fp(p)p Fn(\)2)764 982 y Fp(m)913 961 y Fo(\021)30 b Fu(~)-49 b Fr(!)1056 927 y Fn(\()p Fp(p)p Fn(\)2)1053 982 y Fp(m)1179 961 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))23 b(=)f Fr(!)1539 927 y Fn(2)1536 982 y Fp(m)1618 961 y Fo(\000)c Fr(\027)1742 973 y Fp(m)1805 961 y Fu(\()7 b(~)-49 b Fr(!)1892 927 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2067 961 y Fr(;)14 b(")p Fu(\))p Fr(;)180 b(p)23 b Fo(\025)f Fu(1)p Fr(;)3018 887 y Fu(\(7)p Fr(:)p Fu(4\))118 1113 y(and)28 b(reducing)e(recursiv)n(ely)g(the)i(set)g(of)f (admissible)h(v)-5 b(alues)27 b(of)g Fr(")p Fu(.)189 1219 y(W)-7 b(e)28 b(imp)r(ose)f(on)h Fr(!)d Fo(\021)30 b Fu(~)-49 b Fr(!)947 1189 y Fn(\(0\))1064 1219 y Fu(the)28 b(Diophan)n(tine)f(conditions)199 1371 y Fo(j)p Fr(!)274 1383 y Fn(1)311 1371 y Fr(n)19 b Fo(\006)f Fr(!)515 1383 y Fp(m)577 1371 y Fo(j)24 b(\025)e Fr(C)770 1383 y Fn(0)808 1371 y Fo(j)p Fr(n)p Fo(j)904 1337 y Fv(\000)p Fp(\034)987 1345 y Fi(0)1189 1371 y Fo(8)p Fr(n)g Fo(2)h Fm(Z)16 b Fo(n)i(f)p Fu(0)p Fo(g)26 b Fu(and)i Fo(8)p Fr(m)22 b Fo(2)h Fm(Z)16 b Fo(n)i(f)p Fu(0)p Fr(;)c Fo(\006)p Fu(1)p Fo(g)p Fr(;)199 1502 y Fo(j)p Fr(!)274 1514 y Fn(1)311 1502 y Fr(n)19 b Fo(\006)f Fr(m)p Fo(j)23 b(\025)f Fr(C)728 1514 y Fn(0)766 1502 y Fo(j)p Fr(n)p Fo(j)862 1468 y Fv(\000)p Fp(\034)945 1476 y Fi(0)1147 1502 y Fo(8)p Fr(n)g Fo(2)h Fm(Z)16 b Fo(n)i(f)p Fu(0)p Fo(g)26 b Fu(and)i Fo(8)p Fr(m)22 b Fo(2)h Fm(Z)16 b Fo(n)i(f)p Fu(0)p Fr(;)c Fo(\006)p Fu(1)p Fo(g)p Fr(;)199 1634 y Fo(j)p Fr(!)274 1646 y Fn(1)311 1634 y Fr(n)19 b Fo(\006)f Fu(\()p Fr(!)547 1646 y Fp(m)606 1629 y Fj(0)651 1634 y Fo(\006)g Fr(!)786 1646 y Fp(m)848 1634 y Fu(\))p Fo(j)24 b(\025)e Fr(C)1073 1646 y Fn(0)1111 1634 y Fo(j)p Fr(n)p Fo(j)1207 1599 y Fv(\000)p Fp(\034)1290 1607 y Fi(0)1492 1634 y Fo(8)p Fr(n)g Fo(2)i Fm(Z)15 b Fo(n)j(f)p Fu(0)p Fo(g)27 b Fu(and)g Fo(8)p Fr(m;)14 b(m)2378 1599 y Fv(0)2423 1634 y Fo(2)24 b Fm(Z)15 b Fo(n)j(f)p Fu(0)p Fr(;)c Fo(\006)p Fu(1)p Fo(g)p Fr(;)3018 1502 y Fu(\(7)p Fr(:)p Fu(5\))118 1785 y(where)29 b Fr(C)419 1797 y Fn(0)486 1785 y Fu(and)g Fr(\034)685 1797 y Fn(0)752 1785 y Fu(are)f(t)n(w)n(o)g(p)r(ositiv)n(e) h(constan)n(ts.)41 b(This)29 b(will)g(imply)h(some)f(restriction)f(on)h (the)118 1892 y(admissible)24 b(v)-5 b(alues)25 b(of)g Fr(\026)p Fu(,)g(as)f(the)h(follo)n(wing)f(result)h(sho)n(ws)e(\(see)i (App)r(endix)h(A8)e(for)h(the)g(pro)r(of)6 b(\).)118 2069 y Fk(Lemma)25 b(8.)35 b Fs(F)-6 b(or)26 b(al)t(l)h Fr(\026)878 2081 y Fn(0)938 2069 y Fr(>)c Fu(0)j Fs(the)g(set)f(of)i (values)g Fr(\026)c Fo(2)g Fu([0)p Fr(;)14 b(\026)1995 2081 y Fn(0)2032 2069 y Fu(])26 b Fs(such)g(that)g(\(7.5\))h(ar)l(e)g (satis\014e)l(d)f(for)118 2175 y(some)k(p)l(ositive)h(c)l(onstant)e Fr(C)1019 2187 y Fn(0)1087 2175 y Fs(is)h(of)g(ful)t(l)h(me)l(asur)l(e) e(pr)l(ovide)l(d)j(that)e(one)g(has)g Fr(\034)2580 2187 y Fn(0)2648 2175 y Fs(lar)l(ge)g(enough.)189 2352 y Fu(The)e(sets)f Fo(E)573 2322 y Fn(\()p Fp(p)p Fn(\))691 2352 y Fu(will)h(b)r(e)g (de\014ned)g(recursiv)n(ely)e(as)583 2511 y Fo(E)634 2477 y Fn(\(0\))746 2511 y Fu(=)d(\()p Fo(\000)p Fr(")970 2523 y Fn(0)1007 2511 y Fr(;)14 b(")1083 2523 y Fn(0)1120 2511 y Fu(\))p Fr(;)582 2669 y Fo(E)633 2635 y Fn(\()p Fp(p)p Fn(\))746 2669 y Fu(=)834 2577 y Fq(n)889 2669 y Fr(")23 b Fo(2)g(E)1080 2635 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1302 2669 y Fu(:)46 b Fo(j)7 b Fu(~)-49 b Fr(!)1449 2626 y Fn(\()p Fp(p)p Fn(\))1446 2691 y(1)1539 2669 y Fr(n)18 b Fo(\006)25 b Fu(~)-49 b Fr(!)1745 2635 y Fn(\()p Fp(p)p Fn(\))1742 2690 y Fp(m)1835 2669 y Fo(j)24 b Fr(>)e(C)2028 2681 y Fn(0)2066 2669 y Fo(j)p Fr(n)p Fo(j)2162 2635 y Fv(\000)p Fp(\034)2255 2669 y Fr(;)1055 2852 y Fo(j)7 b Fu(~)-49 b Fr(!)1133 2809 y Fn(\()p Fp(p)p Fn(\))1130 2874 y(1)1223 2852 y Fr(n)19 b Fo(\006)f Fu(\()7 b(~)-49 b Fr(!)1462 2818 y Fn(\()p Fp(p)p Fn(\))1459 2872 y Fp(m)1570 2852 y Fo(\006)25 b Fu(~)-49 b Fr(!)1708 2809 y Fn(\()p Fp(p)p Fn(\))1705 2876 y Fp(m)1764 2860 y Fj(0)1798 2852 y Fu(\))p Fo(j)24 b Fr(>)e(C)2023 2864 y Fn(0)2061 2852 y Fo(j)p Fr(n)p Fo(j)2157 2818 y Fv(\000)p Fp(\034)2250 2760 y Fq(o)2305 2852 y Fr(;)180 b(p)23 b Fo(\025)g Fu(1)p Fr(;)3018 2690 y Fu(\(7)p Fr(:)p Fu(6\))118 3029 y(for)k Fr(\034)33 b(>)23 b(\034)438 3041 y Fn(0)503 3029 y Fu(to)k(b)r(e)h (\014xed.)189 3135 y(In)g(App)r(endix)g(A9)g(w)n(e)f(pro)n(v)n(e)f(the) i(follo)n(wing)e(result.)118 3312 y Fk(Lemma)k(9.)36 b Fs(F)-6 b(or)30 b(al)t(l)h Fr(p)22 b Fo(\025)h Fu(1)29 b Fs(one)h(has)752 3393 y Fq(\015)752 3443 y(\015)752 3493 y(\015)805 3489 y Fu(~)-48 b Fr(!)854 3455 y Fn(\()p Fp(p)p Fn(\))943 3489 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))19 b Fo(\000)24 b Fu(~)-48 b Fr(!)1295 3455 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1470 3489 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))1665 3393 y Fq(\015)1665 3443 y(\015)1665 3493 y(\015)1711 3547 y Fv(1)1804 3489 y Fo(\024)23 b Fr(C)6 b(")1996 3449 y Fp(p)1996 3511 y Fn(0)2204 3489 y Fo(8)p Fr(")22 b Fo(2)h(E)2441 3455 y Fn(\()p Fp(p)p Fn(\))2532 3489 y Fr(;)463 b Fu(\(7)p Fr(:)p Fu(7\))118 3670 y Fs(for)31 b(some)f(c)l(onstant)f Fr(C)6 b Fs(.)189 3847 y Fu(This)33 b(implies)h(that)f(there)h(exist)f(a)g(sequence)g Fo(f)7 b Fu(~)-49 b Fr(!)1800 3817 y Fn(\()p Fp(p)p Fn(\))1889 3847 y Fo(g)1931 3817 y Fv(1)1931 3868 y Fp(p)p Fn(=0)2086 3847 y Fu(con)n(v)n(erging)31 b(to)40 b(~)-49 b Fr(!)2667 3817 y Fn(\()p Fv(1)p Fn(\))2823 3847 y Fu(for)32 b Fr(")h Fo(2)g(E)7 b Fu(.)118 3953 y(W)-7 b(e)28 b(ha)n(v)n(e)f(no)n(w)f(to)i (sho)n(w)f(that)h(the)g(set)f Fo(E)35 b Fu(has)27 b(p)r(ositiv)n(e)g (\(large\))g(measure.)189 4060 y(It)h(is)f(con)n(v)n(enien)n(t)g(to)g (in)n(tro)r(duce)h(a)f(set)g(of)h(v)-5 b(ariables)26 b Fr(\026)p Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))28 b(suc)n(h)f(that)1274 4223 y(~)-49 b Fr(!)1319 4235 y Fp(m)1400 4223 y Fu(+)18 b Fr(\026)1533 4235 y Fp(m)1597 4223 y Fu(\()7 b(~)-49 b Fr(!)r(;)14 b(")p Fu(\))23 b(=)g Fr(!)1954 4235 y Fp(m)2017 4223 y Fu(;)978 b(\(7)p Fr(:)p Fu(8\))118 4386 y(the)28 b(v)-5 b(ariables)26 b Fr(\026)p Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))28 b(and)f(the)h(coun)n (terterms)f(are)f(trivially)h(related)g(b)n(y)1000 4549 y Fr(\027)1041 4561 y Fp(m)1104 4549 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))23 b(=)g Fr(\026)1460 4515 y Fn(2)1460 4570 y Fp(m)1523 4549 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))k(+)g(2)7 b(~)-49 b Fr(!)1913 4561 y Fp(m)1976 4549 y Fr(\026)2026 4561 y Fp(m)2089 4549 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))p Fr(:)711 b Fu(\(7)p Fr(:)p Fu(9\))189 4712 y(One)28 b(can)g(write)35 b(~)-49 b Fr(!)786 4682 y Fn(\()p Fp(p)p Fn(\))901 4712 y Fu(=)24 b Fr(!)d Fo(\000)e Fr(\026)p Fu(\()7 b(~)-49 b Fr(!)1284 4682 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1459 4712 y Fr(;)14 b(")p Fu(\),)29 b(according)d(to)i(\(7.4\),)h(so)f(that) g(the)h(Diophan)n(tine)118 4819 y(conditions)e(in)h(\(1.15\))f(can)g(b) r(e)h(written)g(as)346 4982 y Fo(j)p Fr(!)421 4994 y Fn(1)458 4982 y Fr(n)18 b Fo(\000)g Fr(\026)659 4994 y Fn(1)697 4982 y Fu(\()7 b(~)-49 b Fr(!)784 4947 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))959 4982 y Fr(;)14 b(")p Fu(\))p Fr(n)k Fo(\000)g Fr(!)1270 4994 y Fp(m)1351 4982 y Fu(+)g Fr(\026)1484 4994 y Fp(m)1547 4982 y Fu(\()7 b(~)-49 b Fr(!)1634 4947 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1809 4982 y Fr(;)14 b(")p Fu(\))p Fo(j)23 b Fr(>)g(C)2110 4994 y Fn(0)2148 4982 y Fo(j)p Fr(n)p Fo(j)2244 4947 y Fv(\000)p Fp(\034)2337 4982 y Fr(;)617 b Fu(\(7)p Fr(:)p Fu(10\))346 5113 y Fo(j)p Fr(!)421 5125 y Fn(1)458 5113 y Fr(n)18 b Fo(\000)g Fr(\026)659 5125 y Fn(1)697 5113 y Fu(\()7 b(~)-49 b Fr(!)784 5079 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))959 5113 y Fr(;)14 b(")p Fu(\))p Fr(n)696 5244 y Fo(\007)19 b Fu(\(\()p Fr(!)896 5256 y Fp(m)977 5244 y Fo(\006)f Fr(!)1112 5256 y Fp(m)1171 5240 y Fj(0)1197 5244 y Fu(\))h(+)f(\()p Fr(\026)1413 5256 y Fp(m)1477 5244 y Fu(\()7 b(~)-49 b Fr(!)1564 5210 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1739 5244 y Fr(;)14 b(")p Fu(\))k(+)g Fr(\026)1998 5256 y Fp(m)2057 5240 y Fj(0)2084 5244 y Fu(\()7 b(~)-49 b Fr(!)2171 5210 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2346 5244 y Fr(;)14 b(")p Fu(\)\)\))p Fo(j)24 b Fr(>)e(C)2711 5256 y Fn(0)2749 5244 y Fo(j)p Fr(n)p Fo(j)2845 5210 y Fv(\000)p Fp(\034)2938 5244 y Fr(:)1612 5443 y Fu(16)p eop %%Page: 17 17 17 16 bop 118 555 a Fu(Supp)r(ose)33 b(that)h(for)e Fr(")g Fo(2)h(E)977 525 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1185 555 y Fu(the)h(functions)f Fr(\027)5 b Fu(\()i(~)-49 b Fr(!)1830 525 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2006 555 y Fr(;)14 b(")p Fu(\))33 b(are)f(w)n(ell)h(de\014ned;)j(then)e (de\014ne)118 671 y Fo(I)169 641 y Fn(\()p Fp(p)p Fn(\))283 671 y Fu(=)22 b Fo(I)421 628 y Fn(\()p Fp(p)p Fn(\))415 693 y(1)530 671 y Fo([)d(I)655 628 y Fn(\()p Fp(p)p Fn(\))649 693 y(2)764 671 y Fo([)g(I)889 628 y Fn(\()p Fp(p)p Fn(\))883 693 y(3)980 671 y Fu(,)27 b(where)h Fo(I)1322 628 y Fn(\()p Fp(p)p Fn(\))1316 693 y(1)1440 671 y Fu(is)f(the)h(set)g(of)f(v)-5 b(alues)28 b Fr(")f Fu(v)n(erifying)g(the)h(conditions)628 778 y Fq(\014)628 828 y(\014)628 878 y(\014)656 874 y Fr(!)708 886 y Fn(1)745 874 y Fr(n)18 b Fo(\000)g Fr(\026)946 886 y Fn(1)983 874 y Fu(\()7 b(~)-49 b Fr(!)1070 839 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1245 874 y Fr(;)14 b(")p Fu(\))p Fr(n)19 b Fo(\000)f Fr(!)1557 886 y Fp(m)1638 874 y Fu(+)g Fr(\026)1771 886 y Fp(m)1834 874 y Fu(\()7 b(~)-49 b Fr(!)1921 839 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2096 874 y Fr(;)14 b(")p Fu(\))2204 778 y Fq(\014)2204 828 y(\014)2204 878 y(\014)2255 874 y Fo(\024)22 b Fr(C)2401 886 y Fn(0)2453 874 y Fo(j)p Fr(n)p Fo(j)2549 832 y Fv(\000)p Fp(\034)2656 874 y Fr(;)298 b Fu(\(7)p Fr(:)p Fu(11\))118 1099 y Fo(I)169 1056 y Fn(\()p Fp(p)p Fn(\))163 1121 y(2)287 1099 y Fu(is)28 b(the)g(set)f(of)h(v)-5 b(alues)27 b Fr(")h Fu(v)n(erifying)e(the)i(conditions)258 1206 y Fq(\014)258 1256 y(\014)258 1306 y(\014)286 1301 y Fr(!)338 1313 y Fn(1)375 1301 y Fr(n)18 b Fo(\000)g Fr(\026)576 1313 y Fn(1)613 1301 y Fu(\()7 b(~)-49 b Fr(!)700 1267 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))875 1301 y Fr(;)14 b(")p Fu(\))p Fr(n)600 1484 y Fo(\006)p Fu(\(\()p Fr(!)781 1496 y Fp(m)862 1484 y Fo(\000)k Fr(!)997 1496 y Fp(m)1056 1480 y Fj(0)1083 1484 y Fu(\))g Fo(\007)g Fu(\()p Fr(\026)1298 1496 y Fp(m)1362 1484 y Fu(\()7 b(~)-49 b Fr(!)1449 1450 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1624 1484 y Fr(;)14 b(")p Fu(\))k Fo(\000)g Fr(\026)1883 1496 y Fp(m)1942 1480 y Fj(0)1969 1484 y Fu(\()7 b(~)-49 b Fr(!)2056 1450 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2231 1484 y Fr(;)14 b(")p Fu(\)\)\))2403 1389 y Fq(\014)2403 1438 y(\014)2403 1488 y(\014)2454 1484 y Fo(\024)23 b Fr(C)2601 1496 y Fn(0)2638 1484 y Fo(j)p Fr(n)p Fo(j)2734 1450 y Fv(\000)p Fp(\034)2828 1484 y Fr(;)2977 1393 y Fu(\(7)p Fr(:)p Fu(12\))118 1709 y(and)28 b Fo(I)331 1666 y Fn(\()p Fp(p)p Fn(\))325 1732 y(3)449 1709 y Fu(is)f(the)h(set)g(of)f(v)-5 b(alues)28 b Fr(")f Fu(v)n(erifying)g(the)h(conditions)234 1816 y Fq(\014)234 1866 y(\014)234 1916 y(\014)261 1912 y Fr(!)313 1924 y Fn(1)350 1912 y Fr(n)19 b Fo(\000)f Fr(\026)552 1924 y Fn(1)589 1912 y Fu(\()7 b(~)-49 b Fr(!)676 1878 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))851 1912 y Fr(;)14 b(")p Fu(\))p Fr(n)576 2095 y Fo(\006)655 2002 y Fq(\020)704 2095 y Fu(\()p Fr(!)788 2107 y Fp(m)869 2095 y Fu(+)k Fr(!)1004 2107 y Fp(m)1063 2090 y Fj(0)1089 2095 y Fu(\))h Fo(\007)f Fu(\()p Fr(\026)1305 2107 y Fp(m)1369 2095 y Fu(\()7 b(~)-49 b Fr(!)1456 2060 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1631 2095 y Fr(;)14 b(")p Fu(\))k(+)g Fr(\026)1890 2107 y Fp(m)1949 2090 y Fj(0)1976 2095 y Fu(\()7 b(~)-49 b Fr(!)2063 2060 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2238 2095 y Fr(;)14 b(")p Fu(\)\))2378 2002 y Fq(\021)2428 1999 y(\014)2428 2049 y(\014)2428 2099 y(\014)2478 2095 y Fo(\024)23 b Fr(C)2625 2107 y Fn(0)2663 2095 y Fo(j)p Fr(n)p Fo(j)2759 2060 y Fv(\000)p Fp(\034)2852 2095 y Fr(:)2977 2003 y Fu(\(7)p Fr(:)p Fu(13\))118 2320 y(F)-7 b(or)33 b(future)h(con)n(v)n(enience)e(w)n(e)h(shall)g(call,)h (for)f Fr(i)f Fu(=)g(1)p Fr(;)14 b Fu(2)p Fr(;)g Fu(3,)34 b(also)e Fo(I)2267 2277 y Fn(\()p Fp(p)p Fn(\))2261 2343 y Fp(i)2357 2320 y Fu(\()p Fr(n)p Fu(\))i(the)g(subsets)f(of)g Fo(I)3098 2277 y Fn(\()p Fp(p)p Fn(\))3092 2343 y Fp(i)118 2426 y Fu(whic)n(h)k(v)n(erify)e(the)i(Diophan)n(tine)g(conditions)f (\(7.11\),)i(\(7.12\))d(and)i(\(7.13\),)h(resp)r(ectiv)n(ely)-7 b(,)38 b(for)118 2533 y(\014xed)24 b Fr(n)p Fu(.)36 b(W)-7 b(e)25 b(w)n(an)n(t)f(to)g(b)r(ound)g(the)h(measure)e(of)i(the)f(set)h Fo(I)1983 2502 y Fn(\()p Fp(p)p Fn(\))2073 2533 y Fu(.)36 b(First)24 b(w)n(e)g(need)h(to)f(kno)n(w)f(a)h(little)118 2639 y(b)r(etter)36 b(the)g(dep)r(endence)h(on)e Fr(")h Fu(of)g(the)g(coun)n(terterms:)52 b(this)36 b(is)g(pro)n(vided)e(b)n(y) i(the)g(follo)n(wing)118 2745 y(results)27 b(\(to)h(b)r(e)g(pro)n(v)n (ed)e(in)i(App)r(endix)g(A10)f(and)h(in)g(App)r(endix)g(A11,)f(resp)r (ectiv)n(ely\).)118 2922 y Fk(Lemma)j(10.)36 b Fs(F)-6 b(or)30 b(al)t(l)g Fr(p)23 b Fo(\025)g Fu(1)29 b Fs(and)h(for)h(al)t(l) g Fr(")22 b Fo(2)i(E)1717 2892 y Fn(\()p Fp(p)p Fn(\))1837 2922 y Fs(one)30 b(has)916 3097 y Fr(\027)957 3109 y Fn(1)994 3097 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))23 b(=)g Fr(\013)1353 3109 y Fp(M)1427 3097 y Fr(")1466 3063 y Fp(M)6 b Fv(\000)p Fn(1)1643 3097 y Fu(+)18 b Fr(O)r Fu(\()p Fr(")1862 3063 y Fp(M)1936 3097 y Fu(\))p Fr(;)890 3228 y(\027)931 3240 y Fp(m)994 3228 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))23 b(=)g Fr(\014)1347 3240 y Fp(M)1421 3228 y Fr(")1460 3194 y Fp(M)6 b Fv(\000)p Fn(1)1636 3228 y Fu(+)18 b Fr(O)r Fu(\()p Fr(")1855 3194 y Fp(M)1930 3228 y Fu(\))p Fr(;)184 b(m)23 b Fo(\025)f Fu(2)p Fr(;)2977 3161 y Fu(\(7)p Fr(:)p Fu(14\))118 3412 y Fs(for)40 b(suitable)g(p)l(ositive)g(c)l(onstants)e Fr(\013)1306 3424 y Fp(M)1419 3412 y Fs(e)h Fr(\014)1543 3424 y Fp(M)1656 3412 y Fs(such)g(that)g Fo(j)p Fr(n\013)2159 3424 y Fp(M)2258 3412 y Fo(\000)25 b Fr(\014)2395 3424 y Fp(M)2468 3412 y Fo(j)40 b Fr(>)g Fu(1)p Fr(=)p Fu(4)2762 3382 y Fp(M)6 b Fv(\000)p Fn(1)2958 3412 y Fs(for)40 b(al)t(l)118 3518 y Fr(n)23 b Fo(2)g Fm(Z)n Fs(.)118 3696 y Fk(Lemma)30 b(11.)36 b Fs(F)-6 b(or)30 b(al)t(l)g Fr(p)23 b Fo(\025)g Fu(1)29 b Fs(and)h(for)h(al)t(l)g Fr(")22 b Fo(2)i(E)1717 3665 y Fn(\()p Fp(p)p Fn(\))1837 3696 y Fs(one)30 b(has)861 3796 y Fq(\014)861 3846 y(\014)861 3896 y(\014)889 3892 y Fr(@)938 3904 y Fn(~)-38 b Fp(!)975 3920 y Fh(m)1026 3908 y Fj(0)1056 3892 y Fr(\027)1097 3904 y Fp(m)1160 3892 y Fu(\()7 b(~)-49 b Fr(!)1247 3858 y Fn(\()p Fp(p)p Fn(\))1338 3892 y Fr(;)14 b(")p Fu(\))1446 3796 y Fq(\014)1446 3846 y(\014)1446 3896 y(\014)1496 3892 y Fo(\024)23 b Fr(C)d Fo(j)p Fr(")p Fo(j)1748 3850 y Fn(2\()p Fp(M)6 b Fv(\000)p Fn(1\))2005 3892 y Fr(;)184 b(m)23 b Fo(\025)f Fu(2)p Fr(;)861 3979 y Fq(\014)861 4029 y(\014)861 4079 y(\014)889 4075 y Fr(@)933 4087 y Fp(")975 4075 y Fu(~)-49 b Fr(!)1023 4040 y Fn(\()p Fp(p)p Fn(\))1020 4095 y Fp(m)1113 4075 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))1308 3979 y Fq(\014)1308 4029 y(\014)1308 4079 y(\014)1359 4075 y Fo(\024)22 b Fr(C)e Fo(j)p Fr(")p Fo(j)1610 4033 y Fp(M)6 b Fv(\000)p Fn(2)1953 4075 y Fr(m)22 b Fo(\025)h Fu(2)p Fr(;)2977 3983 y Fu(\(7)p Fr(:)p Fu(15\))118 4271 y Fs(wher)l(e)31 b(the)e(derivatives)j(ar)l(e)e (in)g(the)g(sense)g(of)g(Whitney.)189 4448 y Fu(No)n(w)d(w)n(e)g(can)h (b)r(ound)g(the)f(measure)g(of)h(the)g(set)f(w)n(e)g(ha)n(v)n(e)g(to)g (exclude.)189 4554 y(W)-7 b(e)28 b(start)g(with)g(the)h(estimate)f(of)f (the)i(measure)e(of)h(the)g(set)g Fo(I)2179 4511 y Fn(\()p Fp(p)p Fn(\))2173 4577 y(1)2270 4554 y Fu(.)38 b(When)28 b(\(7.11\))g(is)f(satis\014ed)118 4661 y(one)g(m)n(ust)h(ha)n(v)n(e)e (\(b)n(y)i(using)f(also)g(the)h(\014rst)f(of)h(\(7.5\)\))645 4837 y Fr(C)704 4849 y Fn(0)742 4837 y Fo(j)p Fr(n)p Fo(j)838 4802 y Fv(\000)p Fp(\034)921 4810 y Fi(0)980 4837 y Fo(\024)22 b(j)p Fr(!)1142 4849 y Fn(1)1179 4837 y Fr(n)d Fo(\000)f Fr(!)1383 4849 y Fp(m)1446 4837 y Fo(j)980 4977 y(\024)k(j)p Fr(!)1142 4989 y Fn(1)1179 4977 y Fr(n)d Fo(\000)f Fr(\026)1381 4989 y Fn(1)1418 4977 y Fu(\()7 b(~)-49 b Fr(!)1505 4943 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1680 4977 y Fr(;)14 b(")p Fu(\))p Fr(n)k Fo(\000)g Fr(!)1991 4989 y Fp(m)2073 4977 y Fu(+)g Fr(\026)2206 4989 y Fp(m)2269 4977 y Fu(\()7 b(~)-49 b Fr(!)2356 4943 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2531 4977 y Fr(;)14 b(")p Fu(\))p Fo(j)1307 5117 y Fu(+)k Fo(j)p Fr(\026)1463 5129 y Fn(1)1501 5117 y Fu(\()7 b(~)-49 b Fr(!)1588 5083 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1763 5117 y Fr(;)14 b(")p Fu(\))p Fr(n)k Fo(\000)g Fr(\026)2072 5129 y Fp(m)2135 5117 y Fu(\()7 b(~)-49 b Fr(!)2222 5083 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2397 5117 y Fr(;)14 b(")p Fu(\))p Fo(j)980 5248 y(\024)22 b Fr(C)1126 5260 y Fn(0)1164 5248 y Fo(j)p Fr(n)p Fo(j)1260 5214 y Fv(\000)p Fp(\034)1372 5248 y Fu(+)c Fr(C)6 b(")1559 5260 y Fn(0)1596 5248 y Fo(j)p Fr(n)p Fo(j)p Fr(;)2977 5043 y Fu(\(7)p Fr(:)p Fu(16\))1612 5443 y(17)p eop %%Page: 18 18 18 17 bop 118 555 a Fu(whic)n(h)28 b(implies,)g(for)f Fo(j)p Fr(n)p Fo(j)c Fr(>)f Fu(1)27 b(and)h Fr(\034)33 b(>)22 b(\034)1417 567 y Fn(0)1473 555 y Fu(+)c(1,)1126 806 y Fo(j)p Fr(n)p Fo(j)23 b(\025)g(N)1401 818 y Fn(0)1461 806 y Fo(\021)1549 689 y Fq(\022)1663 750 y Fr(C)1722 762 y Fn(0)p 1620 787 183 4 v 1620 863 a Fu(2)p Fr(C)6 b(")1766 875 y Fn(0)1813 689 y Fq(\023)1874 706 y Fn(1)p Fp(=)p Fn(\()p Fp(\034)1998 714 y Fi(0)2030 706 y Fn(+1\))2158 806 y Fr(;)796 b Fu(\(7)p Fr(:)p Fu(17\))118 1028 y(and)28 b(also)195 1213 y Fr(C)254 1225 y Fn(1)292 1213 y Fo(j)p Fr(m)p Fo(j)23 b(\024)g(j)p Fr(!)597 1225 y Fp(m)678 1213 y Fo(\000)18 b Fr(\026)811 1225 y Fp(m)874 1213 y Fu(\()7 b(~)-49 b Fr(!)961 1178 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1136 1213 y Fr(;)14 b(")p Fu(\))p Fo(j)23 b(\024)g(j)p Fu(\()p Fr(!)1485 1225 y Fn(1)1540 1213 y Fo(\000)18 b Fr(\026)1673 1225 y Fn(1)1711 1213 y Fu(\()7 b(~)-49 b Fr(!)1798 1178 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1973 1213 y Fr(;)14 b(")p Fu(\)\))p Fr(n)p Fo(j)k Fu(+)g Fr(C)2346 1225 y Fn(0)2384 1213 y Fo(j)p Fr(n)p Fo(j)2480 1178 y Fv(\000)p Fp(\034)2596 1213 y Fo(\024)23 b Fr(C)2743 1225 y Fn(2)2781 1213 y Fo(j)p Fr(n)p Fo(j)p Fr(;)77 b Fu(\(7)p Fr(:)p Fu(18\))118 1397 y(whic)n(h)28 b(implies)1127 1534 y Fo(j)p Fr(m)p Fo(j)23 b(\024)g(M)1457 1546 y Fn(0)1494 1534 y Fo(j)p Fr(n)p Fo(j)p Fr(;)180 b Fo(M)1893 1546 y Fn(0)1953 1534 y Fu(=)2050 1478 y Fr(C)2109 1490 y Fn(2)p 2050 1515 97 4 v 2050 1591 a Fr(C)2109 1603 y Fn(1)2157 1534 y Fr(:)797 b Fu(\(7)p Fr(:)p Fu(19\))189 1734 y(Let)28 b(us)f(consider)g(the)h(function)g Fr(\026)p Fu(\()7 b(~)-49 b Fr(!)1373 1704 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1548 1734 y Fr(;)14 b(")p Fu(\):)37 b(w)n(e)27 b(can)g(de\014ne)h(a)g(map)f Fr(t)c Fo(!)g Fr(")p Fu(\()p Fr(t)p Fu(\))28 b(suc)n(h)f(that)606 1919 y Fr(f)9 b Fu(\()p Fr(")p Fu(\()p Fr(t)p Fu(\)\))23 b Fo(\021)g Fr(!)1016 1931 y Fn(1)1053 1919 y Fr(n)18 b Fo(\000)g Fr(\026)1254 1931 y Fn(1)1291 1919 y Fu(\()7 b(~)-49 b Fr(!)1378 1884 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1554 1919 y Fr(;)14 b(")p Fu(\()p Fr(t)p Fu(\)\))p Fr(n)k Fo(\000)g Fr(!)1959 1931 y Fp(m)2041 1919 y Fu(+)g Fr(\026)2174 1931 y Fp(m)2237 1919 y Fu(\()7 b(~)-49 b Fr(!)2324 1884 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2499 1919 y Fr(;)14 b(")p Fu(\()p Fr(t)p Fu(\)\))876 2094 y(=)23 b Fr(t)1024 2038 y(C)1083 2050 y Fn(0)p 1004 2075 138 4 v 1004 2151 a Fo(j)p Fr(n)p Fo(j)1100 2127 y Fp(\034)1151 2094 y Fr(;)180 b(t)23 b Fo(2)h Fu([)p Fo(\000)p Fu(1)p Fr(;)14 b Fu(1])p Fr(;)2977 2027 y Fu(\(7)p Fr(:)p Fu(20\))118 2320 y(describ)r(es)27 b(the)h(in)n(terv)-5 b(al)27 b(de\014ned)h(b)n (y)f(\(7.11\);)g(then)h(one)g(has)904 2444 y Fq(Z)950 2632 y Fv(I)992 2602 y Fi(\()p Fh(p)p Fi(\))988 2652 y(1)1089 2557 y Fu(d)p Fr(")23 b Fu(=)1334 2478 y Fq(X)1285 2660 y Fv(j)p Fp(n)p Fv(j\025N)1472 2668 y Fi(0)1628 2478 y Fq(X)1518 2660 y Fv(j)p Fp(m)p Fv(j\024M)1747 2668 y Fi(0)1778 2660 y Fv(j)p Fp(n)p Fv(j)1873 2444 y Fq(Z)1956 2464 y Fn(1)1919 2632 y Fv(\000)p Fn(1)2022 2557 y Fu(d)p Fr(t)2112 2436 y Fq(\014)2112 2486 y(\014)2112 2536 y(\014)2112 2586 y(\014)2149 2501 y Fu(d)p Fr(")p Fu(\()p Fr(t)p Fu(\))p 2149 2538 180 4 v 2201 2614 a(d)p Fr(t)2339 2436 y Fq(\014)2339 2486 y(\014)2339 2536 y(\014)2339 2586 y(\014)2380 2557 y Fr(:)574 b Fu(\(7)p Fr(:)p Fu(21\))118 2831 y(W)-7 b(e)28 b(ha)n(v)n(e)f(from)g(\(7.20\))1295 2918 y(d)p Fr(f)p 1295 2955 96 4 v 1305 3031 a Fu(d)p Fr(t)1423 2974 y Fu(=)1521 2918 y(d)p Fr(f)p 1521 2955 V 1526 3031 a Fu(d)p Fr(")1637 2918 y Fu(d)p Fr(")p 1637 2955 85 4 v 1641 3031 a Fu(d)p Fr(t)1754 2974 y Fu(=)1872 2918 y Fr(C)1931 2930 y Fn(0)p 1852 2955 138 4 v 1852 3031 a Fo(j)p Fr(n)p Fo(j)1948 3007 y Fp(\034)1999 2974 y Fr(;)955 b Fu(\(7)p Fr(:)p Fu(22\))118 3166 y(hence)717 3199 y Fq(Z)763 3387 y Fv(I)805 3357 y Fi(\()p Fh(p)p Fi(\))801 3407 y(1)902 3312 y Fu(d)p Fr(")23 b Fu(=)1147 3233 y Fq(X)1097 3415 y Fv(j)p Fp(n)p Fv(j\025N)1284 3423 y Fi(0)1441 3233 y Fq(X)1330 3415 y Fv(j)p Fp(m)p Fv(j\024M)1559 3423 y Fi(0)1591 3415 y Fv(j)p Fp(n)p Fv(j)1716 3256 y Fr(C)1775 3268 y Fn(0)p 1695 3293 V 1695 3369 a Fo(j)p Fr(n)p Fo(j)1791 3345 y Fp(\034)1857 3199 y Fq(Z)1940 3219 y Fn(1)1903 3387 y Fv(\000)p Fn(1)2006 3312 y Fu(d)p Fr(t)2096 3191 y Fq(\014)2096 3241 y(\014)2096 3291 y(\014)2096 3341 y(\014)2133 3256 y Fu(d)p Fr(f)9 b Fu(\()p 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Fo(\000)18 b Fu(1\))27 b(end-p)r(oin)n(ts)g (and)h(if)g Fo(j)p Fr(m)2913 4515 y Fv(0)2936 4545 y Fo(j)23 b Fr(>)g Fo(j)p Fr(m)p Fo(j)118 4652 y Fu(there)28 b(m)n(ust)f(b)r(e)h(at)g(least)f Fo(j)p Fr(m)1041 4622 y Fv(0)1064 4652 y Fo(j)19 b(\000)f(j)p Fr(m)p Fo(j)27 b Fu(end-p)r(oin)n(ts.)189 4758 y(A)n(t)33 b(the)g(end)g(w)n(e)f(get)g (that)h(the)g(sum)g(in)g(\(7.24\))f(is)g(O\()p Fr(")1979 4728 y Fn(2\()p Fp(M)6 b Fv(\000)p Fn(1\))2223 4758 y Fo(j)p Fr(m)p Fo(j)p Fu(\),)34 b(and)e(using)h(\(7.19\))e(one)118 4864 y(has)405 5024 y Fr(f)9 b Fu(\()p Fr(")p Fu(\()p Fr(t)p Fu(\)\))23 b(=)g Fr(!)815 5036 y Fn(1)852 5024 y Fr(n)18 b Fo(\000)g Fr(!)1055 5036 y Fp(m)1136 5024 y Fo(\000)g Fr(\013)1272 5036 y Fp(M)1346 5024 y Fr(")1385 4990 y Fp(M)6 b Fv(\000)p Fn(1)1543 5024 y Fu(\()p Fr(t)p Fu(\))19 b(+)f Fr(\014)1786 5036 y Fp(M)1860 5024 y Fr(n")1949 4990 y Fp(M)6 b Fv(\000)p Fn(1)2107 5024 y Fu(\()p Fr(t)p Fu(\))19 b(+)f Fr(O)r Fu(\()p Fr(n")2489 4990 y Fp(M)2563 5024 y Fu(\()p Fr(t)p Fu(\)\))675 5200 y(=)23 b Fr(t)823 5143 y(C)882 5155 y Fn(0)p 803 5181 138 4 v 803 5257 a Fo(j)p Fr(n)p Fo(j)899 5233 y Fp(\034)950 5200 y Fr(;)2977 5135 y Fu(\(7)p Fr(:)p Fu(26\))1612 5443 y(18)p eop %%Page: 19 19 19 18 bop 118 555 a Fu(where)27 b Fr(\013)411 567 y Fp(M)513 555 y Fu(and)g Fr(\014)721 567 y Fp(M)822 555 y Fu(are)g(de\014ned)h (in)g(the)g(statemen)n(t)f(of)h(lemma)f(10,)g(hence)1108 662 y Fq(\014)1108 712 y(\014)1108 761 y(\014)1108 811 y(\014)1146 726 y Fr(@)5 b(f)k Fu(\()p Fr(")p Fu(\()p Fr(t)p Fu(\)\))p 1146 763 297 4 v 1203 839 a Fr(@)c(")p Fu(\()p Fr(t)p Fu(\))1452 662 y Fq(\014)1452 712 y(\014)1452 761 y(\014)1452 811 y(\014)1503 782 y Fo(\025)22 b Fr(c)1626 794 y Fp(M)1714 782 y Fo(j)p Fr(n)p Fo(j)14 b(j)p Fr(")p Fu(\()p Fr(t)p Fu(\))p Fo(j)2003 740 y Fp(M)6 b Fv(\000)p Fn(2)2176 782 y Fr(;)778 b Fu(\(7)p Fr(:)p Fu(27\))118 1013 y(for)27 b(a)g(suitable)h(constan)n(t)f Fr(c)995 1025 y Fp(M)1068 1013 y Fu(,)h(and,)g(for)f Fo(j)p Fr(n)p Fo(j)c Fr(>)f Fu(1)27 b(and)h Fr(\034)33 b(>)22 b(\034)2060 1025 y Fn(0)2116 1013 y Fu(+)c(1,)250 1245 y Fr(C)309 1257 y Fn(1)347 1245 y Fo(j)p Fr(n)p Fo(jj)p Fr(")505 1210 y Fp(M)6 b Fv(\000)p Fn(1)663 1245 y Fu(\()p Fr(t)p Fu(\))p Fo(j)24 b(\025)892 1149 y Fq(\014)892 1199 y(\014)892 1249 y(\014)919 1245 y Fr(cn")1044 1210 y Fp(M)6 b Fv(\000)p Fn(1)1203 1245 y Fu(\()p Fr(t)p Fu(\))19 b(+)f Fr(O)r Fu(\()p Fr(n")1585 1210 y Fn(2\()p Fp(M)6 b Fv(\000)p Fn(1\))1829 1245 y Fu(\()p Fr(t)p Fu(\)\))1955 1149 y Fq(\014)1955 1199 y(\014)1955 1249 y(\014)2006 1245 y Fo(\025)2094 1124 y Fq(\014)2094 1174 y(\014)2094 1224 y(\014)2094 1274 y(\014)2122 1245 y Fo(j)p Fr(!)2197 1257 y Fn(1)2234 1245 y Fr(n)18 b Fo(\000)g Fr(!)2437 1257 y Fp(m)2500 1245 y Fo(j)g(\000)2634 1189 y(j)p Fr(tC)2746 1201 y Fn(0)2784 1189 y Fo(j)p 2634 1226 173 4 v 2652 1302 a(j)p Fr(n)p Fo(j)2748 1278 y Fp(\034)2817 1124 y Fq(\014)2817 1174 y(\014)2817 1224 y(\014)2817 1274 y(\014)804 1477 y Fr(>)935 1421 y(C)994 1433 y Fn(0)p 902 1458 164 4 v 902 1534 a Fo(j)p Fr(n)p Fo(j)998 1510 y Fp(\034)1029 1518 y Fi(0)1089 1360 y Fq(\022)1150 1477 y Fu(1)g Fo(\000)1408 1421 y Fu(1)p 1303 1458 253 4 v 1303 1534 a Fo(j)p Fr(n)p Fo(j)1399 1510 y Fp(\034)7 b Fv(\000)p Fp(\034)1520 1518 y Fi(0)1565 1360 y Fq(\023)1649 1477 y Fr(>)1801 1421 y(C)1860 1433 y Fn(0)p 1747 1458 205 4 v 1747 1534 a Fu(2)p Fo(j)p Fr(n)p Fo(j)1885 1510 y Fp(\034)1916 1518 y Fi(0)1961 1477 y Fr(;)2977 1361 y Fu(\(7)p Fr(:)p Fu(28\))118 1704 y(so)27 b(that)h(one)f(has)954 1856 y Fo(j)p Fr(")p Fu(\()p Fr(t)p Fu(\))p Fo(j)d Fr(>)1245 1738 y Fq(\022)1336 1799 y Fr(C)1395 1811 y Fn(0)p 1316 1836 139 4 v 1316 1912 a Fu(2)p Fr(C)1417 1924 y Fn(1)1464 1738 y Fq(\023)1525 1756 y Fn(1)p Fp(=)p Fn(\()p Fp(M)6 b Fv(\000)p Fn(1\))2052 1799 y Fu(1)p 1826 1836 494 4 v 1826 1914 a Fr(n)1876 1890 y Fn(\()p Fp(\034)1933 1898 y Fi(0)1965 1890 y Fn(+1\))p Fp(=)p Fn(\()p Fp(M)g Fv(\000)p Fn(1\))2330 1856 y Fr(;)624 b Fu(\(7)p Fr(:)p Fu(29\))118 2055 y(whic)n(h,)28 b(in)n(tro)r(duced)f(in)n(to)g(\(7.23\),)g(giv)n (es)g(together)f(with)j(\(7.27\),)617 2183 y Fq(Z)664 2372 y Fv(I)706 2341 y Fi(\()p Fh(p)p Fi(\))702 2392 y(1)803 2296 y Fu(d)p Fr(")23 b Fo(\024)f Fu(const.)1274 2217 y Fq(X)1225 2399 y Fv(j)p Fp(n)p Fv(j\025N)1412 2407 y Fi(0)1568 2217 y Fq(X)1458 2399 y Fv(j)p Fp(m)p Fv(j\024M)1687 2407 y Fi(0)1718 2399 y Fv(j)p Fp(n)p Fv(j)1843 2240 y Fr(C)1902 2252 y Fn(0)p 1823 2277 138 4 v 1823 2353 a Fo(j)p Fr(n)p Fo(j)1919 2329 y Fp(\034)1984 2183 y Fq(Z)2067 2204 y Fn(1)2030 2372 y Fv(\000)p Fn(1)2329 2240 y Fu(d)p Fr(t)p 2143 2277 448 4 v 2143 2368 a Fo(j)p Fr(")p Fu(\()p Fr(t)p Fu(\))p Fo(j)2322 2327 y Fp(M)6 b Fv(\000)p Fn(2)2495 2368 y Fo(j)p Fr(n)p Fo(j)911 2585 y(\024)22 b Fu(const.)1274 2506 y Fq(X)1225 2688 y Fv(j)p Fp(n)p Fv(j\025N)1412 2696 y Fi(0)1568 2506 y Fq(X)1458 2688 y Fv(j)p Fp(m)p Fv(j\024M)1687 2696 y Fi(0)1718 2688 y Fv(j)p Fp(n)p Fv(j)1843 2529 y Fr(C)1902 2541 y Fn(0)p 1823 2566 138 4 v 1823 2642 a Fo(j)p Fr(n)p 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1759 3943 V 77 x Fr(C)1818 4032 y Fn(0)1869 3903 y Fq(\022)1951 3964 y Fr(")1990 3976 y Fn(0)p 1940 4001 V 1940 4077 a Fr(C)1999 4089 y Fn(0)2047 3903 y Fq(\023)2108 3920 y Fn(\()p Fp(\034)2172 3895 y Fj(0)2194 3920 y Fv(\000)p Fn(2\))p Fp(=)p Fn(\(1+)p Fp(\034)2480 3928 y Fi(0)2512 3920 y Fn(\))2565 4020 y Fo(\024)h Fu(const.)o Fr(")2904 3980 y Fn(1+)p Fp(\030)3020 3955 y Fj(0)2904 4042 y Fn(0)3047 4020 y Fr(;)118 4246 y Fu(pro)n(vided)k(that)h(one)f (has)944 4552 y Fr(\030)984 4517 y Fv(0)1031 4552 y Fu(=)1129 4496 y Fr(\034)1174 4465 y Fv(0)1216 4496 y Fo(\000)18 b Fr(\034)1335 4508 y Fn(0)1391 4496 y Fo(\000)g Fu(1)p 1129 4533 387 4 v 1214 4609 a Fr(\034)1250 4621 y Fn(0)1306 4609 y Fu(+)g(1)1549 4552 y(=)1646 4462 y Fr(\034)28 b Fo(\000)1803 4406 y Fu(1)p 1803 4443 42 4 v 1803 4519 a(2)1855 4462 y(\(3)p Fr(\034)1965 4474 y Fn(0)2021 4462 y Fu(+)18 b(1\))p 1646 4533 532 4 v 1804 4609 a Fr(\034)1840 4621 y Fn(0)1895 4609 y Fu(+)h(1)2210 4552 y Fr(>)k Fu(0)p Fr(;)614 b Fu(\(7)p Fr(:)p Fu(33\))118 4774 y(whic)n(h)28 b(imp)r(oses)f Fr(\034)33 b(>)22 b Fu(\(3)p Fr(\034)934 4786 y Fn(0)990 4774 y Fu(+)c(1\))p Fr(=)p Fu(2.)189 4889 y(No)n(w)26 b(w)n(e)f(discuss)h(ho)n(w)f(to)h(b)r(ound)h(the)f (measure)f(of)h(the)h(set)f Fo(I)2177 4846 y Fn(\()p Fp(p)p Fn(\))2171 4911 y(2)2268 4889 y Fu(.)36 b(W)-7 b(e)27 b(start)e(b)n(y)h(noting)g(that)118 4996 y(for)h(all)g Fr(p)c Fo(\025)g Fu(0)k(one)g(has)927 5220 y(~)-48 b Fr(!)976 5186 y Fn(\()p Fp(p)p Fn(\))973 5241 y Fp(m)1089 5220 y Fu(=)22 b Fr(!)1228 5232 y Fp(m)1310 5220 y Fu(+)c Fr(\024)g Fu(+)g Fr(\033)1592 5186 y Fn(\()p Fp(p)p Fn(\))1589 5241 y Fp(m)1863 5125 y Fq(\014)1863 5175 y(\014)1863 5225 y(\014)1890 5220 y Fr(\033)1940 5186 y Fn(\()p Fp(p)p Fn(\))1937 5241 y Fp(m)2031 5125 y Fq(\014)2031 5175 y(\014)2031 5225 y(\014)2082 5220 y Fr(<)k(C)2268 5164 y Fo(j)p Fr(")p Fo(j)p 2245 5201 132 4 v 2245 5277 a Fr(m)2318 5253 y Fp(\030)2350 5237 y Fj(0)2977 5220 y Fu(\(7)p Fr(:)p Fu(34\))1612 5443 y(19)p eop %%Page: 20 20 20 19 bop 118 555 a Fu(where)27 b Fr(\030)398 525 y Fv(0)445 555 y Fu(=)22 b(1)27 b(and)h Fr(\024)23 b Fu(=)f Fr(O)r Fu(\()p Fr(")1057 525 y Fn(2\()p Fp(M)6 b Fv(\000)p Fn(1)1276 555 y Fu(\)\))28 b(is)g(a)f(constan)n(t)g(\(in)h Fr(m)p Fu(\).)189 672 y(In)i(fact)f Fr(\027)506 629 y Fn(\()p Fp(p)p Fn(\))501 682 y Fp(m)627 672 y Fu(is)g(giv)n(en)g(b)n(y)g(a)g (sum)h(of)g(v)-5 b(alues)29 b(of)h(renormalized)d(self-energy)i(graphs) f Fr(T)12 b Fu(,)29 b(and,)118 778 y(for)c(eac)n(h)f(v)-5 b(alue,)25 b(the)h(dep)r(endence)f(on)g Fr(m)g Fu(is)g(due)g(to)g(the)g (propagators)d(of)j(lines)g(along)f(the)h(path)118 884 y(connecting)h(the)h(external)f(lines)g(of)h(the)g(self-energy)e(graph) g(\(the)i(self-energy)e(graphs)g(with)i(no)118 991 y(path)32 b(of)g(lines)g(connecting)f(the)i(external)e(lines)g(con)n(tribute)h (to)g(the)g(constan)n(t)f(term)h Fr(\024)p Fu(\).)50 b(The)118 1097 y(propagators)25 b(of)i(suc)n(h)g(lines)h(ha)n(v)n(e)e (the)i(form)1179 1310 y Fr(\037)1231 1322 y Fp(h)1274 1310 y Fu(\()p Fo(j)7 b Fu(~)-49 b Fr(!)1384 1266 y Fn(\()p Fp(p)p Fn(\))1381 1332 y(1)1475 1310 y Fr(n)1525 1322 y Fp(`)1575 1310 y Fu(+)24 b(~)-48 b Fr(!)1713 1266 y Fn(\()p Fp(p)p Fn(\))1710 1319 y Fp(m)1769 1328 y Fh(`)1803 1310 y Fo(j)18 b(\000)25 b Fu(~)-49 b Fr(!)1982 1266 y Fn(\()p Fp(p)p Fn(\))1979 1319 y Fp(m)2072 1310 y Fu(\))p 731 1347 1822 4 v 731 1452 a Fo(\000)810 1360 y Fq(\020)859 1452 y Fo(j)7 b Fu(~)-49 b Fr(!)937 1409 y Fn(\()p Fp(p)p Fn(\))934 1474 y(1)1027 1452 y Fr(n)1077 1464 y Fp(`)1127 1452 y Fu(+)25 b(~)-49 b Fr(!)1265 1409 y Fn(\()p Fp(p)p Fn(\))1262 1462 y Fp(m)1321 1471 y Fh(`)1355 1452 y Fo(j)19 b Fu(+)24 b(~)-48 b Fr(!)1535 1409 y Fn(\()p Fp(p)p Fn(\))1532 1462 y Fp(m)1591 1471 y Fh(`)1625 1360 y Fq(\021)13 b(\020)1738 1452 y Fo(j)7 b Fu(~)-49 b Fr(!)1816 1409 y Fn(\()p Fp(p)p Fn(\))1813 1474 y(1)1906 1452 y Fr(n)1956 1464 y Fp(`)2006 1452 y Fu(+)25 b(~)-49 b Fr(!)2144 1409 y Fn(\()p Fp(p)p Fn(\))2141 1462 y Fp(m)2200 1471 y Fh(`)2234 1452 y Fo(j)18 b(\000)25 b Fu(~)-49 b Fr(!)2413 1409 y Fn(\()p Fp(p)p Fn(\))2410 1462 y Fp(m)2469 1471 y Fh(`)2503 1360 y Fq(\021)2563 1366 y Fr(;)391 b Fu(\(7)p Fr(:)p Fu(35\))118 1682 y(and)32 b(the)h(second)f(factor)g(in)h(the)g(denominator)e(is)h(b)r(ounded)h (prop)r(ortionally)e(to)h(2)2817 1652 y Fv(\000)p Fp(n)2910 1661 y Fh(`)2943 1682 y Fu(,)i(while)118 1788 y(the)39 b(\014rst)f(is)g(b)r(ounded)h(b)n(y)f(a)g(constan)n(t)g(times)g Fr(m)1755 1800 y Fp(`)1787 1788 y Fu(;)44 b(hence)38 b(\(7.34\))g(holds)g(with)h Fr(\030)45 b Fu(=)40 b(1.)69 b(As)118 1894 y Fr(m)191 1906 y Fp(`)246 1894 y Fu(=)23 b Fr(m)407 1864 y Fn(0)407 1918 y Fp(`)462 1894 y Fu(+)18 b Fr(m)28 b Fu(and)f Fr(k)850 1906 y Fp(T)926 1894 y Fr(>)22 b(m)1086 1864 y Fn(0)1086 1918 y Fp(`)1151 1894 y Fu(w)n(e)28 b(get)258 2114 y(~)-49 b Fr(!)306 2071 y Fn(\()p Fp(p)p Fn(\))303 2139 y Fp(m)p Fn(+)p Fp(`)463 2114 y Fo(\000)25 b Fu(~)-49 b Fr(!)601 2080 y Fn(\()p Fp(p)p Fn(\))598 2135 y Fp(m)714 2114 y Fu(=)23 b Fr(!)854 2126 y Fp(m)p Fn(+)p Fp(`)1014 2114 y Fo(\000)18 b Fr(!)1149 2126 y Fp(m)1230 2114 y Fu(+)g Fr(O)r Fu(\()p Fr(")p Fu(\()p Fr(m)h Fu(+)f Fr(`)p Fu(\))1723 2080 y Fv(\000)p Fp(\030)1807 2055 y Fj(0)1834 2114 y Fu(\))g(+)g Fr(O)r Fu(\()p Fr("m)2176 2080 y Fv(\000)p Fp(\030)2260 2055 y Fj(0)2288 2114 y Fu(\))714 2262 y(=)23 b Fr(`)18 b Fu(+)g Fr(O)r Fu(\()p Fr(`m)1143 2228 y Fv(\000)p Fp(\030)1231 2262 y Fu(\))h(+)f Fr(O)r Fu(\()p Fr(")p Fu(\()p Fr(m)h Fu(+)f Fr(`)p Fu(\))1775 2228 y Fv(\000)p Fp(\030)1859 2203 y Fj(0)1886 2262 y Fu(\))h(+)f Fr(O)r Fu(\()p Fr("m)2229 2228 y Fv(\000)p Fp(\030)2313 2203 y Fj(0)2340 2262 y Fu(\))23 b(=)g Fr(`)18 b Fu(+)g Fr(O)r Fu(\()p Fr(`\026)2801 2274 y Fn(0)2839 2262 y Fr(m)2912 2228 y Fv(\000)p Fn(1)3001 2262 y Fu(\))p Fr(;)2977 2361 y Fu(\(7)p Fr(:)p Fu(36\))118 2468 y(where)27 b(1)c(=)f(min)q Fo(f)p Fr(\030)t(;)14 b(\030)808 2438 y Fv(0)831 2468 y Fo(g)22 b Fu(=)h(min)p Fo(f)p Fu(1)p Fr(;)14 b Fu(2)p Fo(g)p Fu(;)26 b(hence)i(w)n(e)f(can)g (b)r(ound)1141 2617 y Fq(\014)1141 2666 y(\014)1141 2716 y(\014)1175 2712 y Fu(~)-48 b Fr(!)1224 2669 y Fn(\()p Fp(p)p Fn(\))1221 2737 y Fp(m)p Fn(+)p Fp(`)1381 2712 y Fo(\000)24 b Fu(~)-48 b Fr(!)1519 2678 y Fn(\()p Fp(p)p Fn(\))1516 2733 y Fp(m)1627 2712 y Fo(\000)18 b Fr(`)1745 2617 y Fq(\014)1745 2666 y(\014)1745 2716 y(\014)1796 2712 y Fo(\024)1893 2656 y Fu(2)p Fr(K)6 b(\026)2062 2668 y Fn(0)2098 2656 y Fr(`)p 1893 2693 240 4 v 1977 2769 a(m)2143 2712 y(;)811 b Fu(\(7)p Fr(:)p Fu(37\))118 2941 y(with)28 b(the)g(same)f(constan)n(t)g Fr(K)33 b Fu(as)27 b(in)h(\(A8.4\).)189 3048 y(The)i(conditions)f(in)i(\(7.12\))e (corresp)r(ond)f(to)i(sev)n(eral)e(p)r(ossibilities.)44 b(If)30 b Fr(n)d(>)f Fu(0,)k Fr(m)2814 3018 y Fv(0)2865 3048 y Fr(>)c(m)h(>)f Fu(0)118 3164 y(and)i Fo(jj)7 b Fu(~)-49 b Fr(!)381 3120 y Fn(\()p Fp(p)p Fn(\))378 3188 y Fp(m)437 3171 y Fj(0)489 3164 y Fo(\000)25 b Fu(~)-49 b Fr(!)627 3120 y Fn(\()p Fp(p)p Fn(\))624 3173 y Fp(m)717 3164 y Fo(j)18 b(\000)25 b Fu(~)-48 b Fr(!)897 3120 y Fn(\()p Fp(p)p Fn(\))894 3186 y(1)986 3164 y Fr(n)p Fo(j)23 b Fr(<)g Fu(1)k(the)h(discussion)f(pro)r(ceeds)g(as)g(follo)n(ws.)189 3270 y(When)h(the)g(conditions)f(\(7.12\))g(are)g(satis\014ed,)g(one)g (has)302 3471 y Fr(C)361 3483 y Fn(0)399 3471 y Fo(j)p Fr(n)p Fo(j)495 3437 y Fv(\000)p Fp(\034)578 3445 y Fi(0)637 3471 y Fo(\024)c(j)p Fr(!)800 3483 y Fn(1)837 3471 y Fr(n)18 b Fo(\000)g Fu(\()p Fr(!)1072 3483 y Fp(m)1131 3467 y Fj(0)1176 3471 y Fo(\000)g Fr(!)1311 3483 y Fp(m)1374 3471 y Fu(\))p Fo(j)637 3620 y(\024)23 b(j)p Fr(!)800 3632 y Fn(1)837 3620 y Fr(n)18 b Fo(\000)g Fr(\026)1038 3632 y Fn(1)1075 3620 y Fu(\()7 b(~)-49 b Fr(!)1162 3586 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1337 3620 y Fr(;)14 b(")p Fu(\))p Fr(n)19 b Fo(\000)f Fu(\()p Fr(!)1681 3632 y Fp(m)1740 3616 y Fj(0)1785 3620 y Fu(+)g Fr(\026)1918 3577 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1918 3645 y Fp(m)1977 3628 y Fj(0)2093 3620 y Fu(\()7 b(~)-49 b Fr(!)2180 3586 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2355 3620 y Fr(;)14 b(")p Fu(\)\))964 3761 y Fo(\000)19 b Fu(\()p Fr(!)1132 3773 y Fp(m)1213 3761 y Fu(+)f Fr(\026)1346 3726 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1346 3781 y Fp(m)1521 3761 y Fu(\()7 b(~)-49 b Fr(!)1608 3726 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1783 3761 y Fr(;)14 b(")p Fu(\)\))p Fo(j)964 3901 y Fu(+)19 b Fo(j)p Fr(\026)1121 3913 y Fn(1)1158 3901 y Fu(\()7 b(~)-49 b Fr(!)1245 3866 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1420 3901 y Fr(;)14 b(")p Fu(\))p Fr(n)k Fo(\000)g Fr(\026)1729 3913 y Fp(m)1788 3897 y Fj(0)1815 3901 y Fu(\()7 b(~)-49 b Fr(!)1902 3866 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2077 3901 y Fr(;)14 b(")p Fu(\))k(+)g Fr(\026)2336 3913 y Fp(m)2399 3901 y Fu(\()7 b(~)-49 b Fr(!)2486 3866 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2662 3901 y Fr(;)14 b(")p Fu(\))p Fo(j)637 4032 y(\024)23 b Fr(C)784 4044 y Fn(0)821 4032 y Fo(j)p Fr(n)p Fo(j)917 3998 y Fv(\000)p Fp(\034)1029 4032 y Fu(+)18 b(3)p Fr(C)6 b(")1258 4044 y Fn(0)1295 4032 y Fo(j)p Fr(n)p Fo(j)p Fr(;)2977 3752 y Fu(\(7)p Fr(:)p Fu(38\))118 4228 y(whic)n(h)28 b(implies)1126 4398 y Fo(j)p Fr(n)p Fo(j)23 b(\025)g(N)1401 4410 y Fn(1)1461 4398 y Fo(\021)1549 4281 y Fq(\022)1663 4342 y Fr(C)1722 4354 y Fn(0)p 1620 4379 183 4 v 1620 4455 a Fu(6)p Fr(C)6 b(")1766 4467 y Fn(0)1813 4281 y Fq(\023)1874 4298 y Fn(1)p Fp(=)p Fn(\()p Fp(\034)1998 4306 y Fi(0)2030 4298 y Fn(+1\))2158 4398 y Fr(:)796 b Fu(\(7)p Fr(:)p Fu(39\))189 4608 y(Besides)27 b(of)g(\(7.11\))g(w)n(e)h(eliminate)f(also)g(the)h(v) -5 b(alues)27 b Fr(")g Fu(v)n(erifying)960 4816 y Fo(j)p Fr(!)1035 4828 y Fn(1)1072 4816 y Fr(n)18 b Fo(\000)g Fr(\026)1273 4828 y Fn(1)1310 4816 y Fu(\()7 b(~)-49 b Fr(!)1397 4781 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1573 4816 y Fr(;)14 b(")p Fu(\))p Fr(n)k Fo(\000)g Fr(m)p Fo(j)23 b(\024)f Fr(C)2097 4828 y Fn(0)2135 4816 y Fo(j)p Fr(n)p Fo(j)2231 4781 y Fv(\000)p Fp(\034)2324 4816 y Fr(;)630 b Fu(\(7)p Fr(:)p Fu(40\))118 5031 y(for)27 b Fr(\034)33 b(>)22 b(\034)437 5043 y Fn(0)493 5031 y Fu(+)c(1)27 b(and)g(for)g(all)g(\()p Fr(n;)14 b(m)p Fu(\))23 b Fo(2)h Fm(Z)1440 4983 y Fn(2)1495 5031 y Fo(n)18 b(f)p Fu(\(0)p Fr(;)c Fu(0\))p Fo(g)p Fu(.)35 b(This)28 b(requires)e(to)h (tak)n(e)g(o\013)g(from)g Fo(E)3029 5001 y Fn(\()p Fp(p)p Fn(\))3147 5031 y Fu(a)118 5138 y(subset)j(whose)f(measure)g(is)h(b)r (ounded)g(b)n(y)f(a)h(constan)n(t)f(times)h Fr(")2164 5098 y Fn(1+)p Fp(\030)2164 5160 y Fn(0)2284 5138 y Fu(,)g(as)g(it)g (can)f(easily)g(c)n(hec)n(k)n(ed)118 5244 y(b)n(y)e(pro)r(ceeding)g(as) g(in)h(the)g(pro)r(of)f(of)g(the)h(\014rst)g(of)f(\(7.11\).)1612 5443 y(20)p eop %%Page: 21 21 21 20 bop 189 559 a Fu(W)-7 b(e)37 b(can)f(b)r(ound)h(the)f(Leb)r (esgue)g(measure)g(of)g(the)h(set)f Fo(I)2063 516 y Fn(\()p Fp(p)p Fn(\))2057 581 y(2)2191 559 y Fu(b)n(y)g(distinguishing,)i(for)e (\014xed)118 665 y(\()p Fr(n;)14 b(`)p Fu(\),)24 b(with)h Fr(`)d Fu(=)h Fr(m)755 635 y Fv(0)788 665 y Fo(\000)10 b Fr(m)23 b(>)g Fu(0,)h(the)g(v)-5 b(alues)23 b Fr(m)g Fo(\024)f Fr(m)1774 677 y Fn(0)1835 665 y Fu(and)i Fr(m)e(>)h(m)2249 677 y Fn(0)2286 665 y Fu(,)i(where)e Fr(m)2643 677 y Fn(0)2703 665 y Fu(is)h(determined)118 772 y(b)n(y)j(the)h(request)f (that)h(one)f(has)h(for)f Fr(m)c(>)f(m)1529 784 y Fn(0)1367 942 y Fu(2)p Fr(K)6 b(\026)1536 954 y Fn(0)1573 942 y Fr(`)p 1367 979 240 4 v 1433 1055 a(m)1506 1031 y Fp(\030)1640 998 y Fo(\024)1779 942 y Fr(C)1838 954 y Fn(0)p 1738 979 179 4 v 1738 1055 a Fu(2)p Fo(j)p Fr(n)p Fo(j)1876 1031 y Fp(\034)1927 998 y Fr(;)1027 b Fu(\(7)p Fr(:)p Fu(41\))118 1230 y(whic)n(h)28 b(giv)n(es)1213 1398 y Fr(m)1286 1410 y Fn(0)1346 1398 y Fu(=)1434 1281 y Fq(\022)1505 1342 y Fu(2)p Fr(\026)1597 1354 y Fn(0)1634 1342 y Fr(K)6 b(`)p Fo(j)p Fr(n)p Fo(j)1842 1312 y Fp(\034)p 1505 1379 378 4 v 1645 1455 a Fr(C)1704 1467 y Fn(0)1892 1281 y Fq(\023)1953 1299 y Fn(1)p Fp(=\030)2071 1398 y Fr(:)883 b Fu(\(7)p Fr(:)p Fu(42\))189 1596 y(Therefore)26 b(for)h Fr(m)c(>)g(m)949 1608 y Fn(0)1014 1596 y Fu(and)k Fo(L)1232 1608 y Fn(0)1297 1596 y Fu(de\014ned)h(b)n(y)506 1767 y Fr(C)565 1779 y Fn(1)603 1767 y Fr(`)p 422 1804 300 4 v 422 1821 a Fo(p)p 491 1821 231 4 v 60 x Fu(1)18 b(+)g Fr(\026)684 1893 y Fn(0)754 1823 y Fo(\024)842 1728 y Fq(\014)842 1777 y(\014)842 1827 y(\014)876 1823 y Fu(~)-48 b Fr(!)925 1780 y Fn(\()p Fp(p)p Fn(\))922 1848 y Fp(m)p Fn(+)p Fp(`)1082 1823 y Fo(\000)24 b Fu(~)-48 b Fr(!)1220 1789 y Fn(\()p Fp(p)p Fn(\))1217 1844 y Fp(m)1310 1728 y Fq(\014)1310 1777 y(\014)1310 1827 y(\014)1360 1823 y Fr(<)30 b Fu(~)-49 b Fr(!)1503 1780 y Fn(\()p Fp(p)p Fn(\))1500 1845 y(1)1593 1823 y Fo(j)p Fr(n)p Fo(j)18 b Fu(+)g(1)23 b Fr(<)g(C)2002 1835 y Fn(2)2039 1823 y Fo(j)p Fr(n)p Fo(j)p Fr(;)180 b Fo(L)2395 1835 y Fn(0)2456 1823 y Fu(=)2553 1767 y Fr(C)2612 1779 y Fn(2)p 2553 1804 97 4 v 2553 1880 a Fr(C)2612 1892 y Fn(1)2660 1823 y Fr(;)294 b Fu(\(7)p Fr(:)p Fu(43\))118 2059 y(where)27 b(\(A8.8\))h(has)f(b)r(een)h(used,)f(one)h(has,)f(from)g(\(7.39\))898 2177 y Fq(\014)898 2226 y(\014)898 2276 y(\014)925 2272 y Fr(!)977 2284 y Fn(1)1014 2272 y Fr(n)18 b Fo(\000)h 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2620 y Fr(C)1529 2632 y Fn(0)p 1429 2657 179 4 v 1429 2733 a Fu(2)p Fo(j)p Fr(n)p Fo(j)1567 2709 y Fp(\034)1641 2676 y Fo(\025)1780 2620 y Fr(C)1839 2632 y Fn(0)p 1739 2657 V 1739 2733 a Fu(2)p Fo(j)p Fr(n)p Fo(j)1877 2709 y Fp(\034)1928 2676 y Fr(;)2977 2486 y Fu(\(7)p Fr(:)p Fu(44\))118 2924 y(so)31 b(that)h(one)f(has)g(to)h(exclude)f(no)h (further)f(v)-5 b(alue)32 b(from)f Fo(E)1996 2894 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2171 2924 y Fu(,)i(pro)n(vided)e(one)g(tak)n (es)f Fr(C)3004 2936 y Fn(0)3074 2924 y Fu(the)118 3031 y(double)22 b(of)g(the)h(original)d(v)-5 b(alue.)35 b(F)-7 b(or)22 b Fr(m)h(<)f(m)1546 3043 y Fn(0)1606 3031 y Fu(one)f(has)h(to)g (exclude)g(a)f(set)i(of)f(measure)f(b)r(ounded)118 3137 y(b)n(y)352 3308 y Fq(X)302 3490 y Fv(j)p Fp(n)p Fv(j\025N)489 3498 y Fi(1)594 3308 y Fq(X)535 3490 y Fp(`)p Fv(\024L)661 3498 y Fi(0)693 3490 y Fv(j)p Fp(n)p Fv(j)848 3308 y Fq(X)788 3490 y Fv(j)p Fp(m)p Fv(j\024)p Fp(m)998 3498 y Fi(0)1085 3331 y Fr(C)1144 3343 y Fn(0)p 1053 3368 160 4 v 1053 3444 a Fo(j)p Fr(n)p Fo(j)1149 3420 y Fp(\034)1187 3403 y Fj(0)1246 3387 y Fo(\024)h Fu(const.)p Fr(C)1605 3399 y Fn(0)1713 3283 y Fv(1)1686 3308 y Fq(X)1657 3486 y Fp(n)p Fn(=)p Fv(N)1803 3494 y Fi(1)1849 3387 y Fr(n)1899 3353 y Fn(1)p Fv(\000)p Fp(\034)2022 3328 y Fj(0)2044 3353 y Fn(+\(1+)p Fp(\034)7 b Fn(\))p Fp(=\030)644 3694 y Fo(\024)22 b Fu(const.)p Fr(C)1003 3706 y Fn(0)1040 3694 y Fo(N)1120 3651 y Fv(\000)p Fp(\034)1210 3626 y Fj(0)1232 3651 y Fn(+1+\()p Fp(\034)7 b Fn(+1\))p Fp(=\030)1108 3716 y Fn(0)1634 3694 y Fu(=)23 b(const.)o Fr(C)1993 3706 y Fn(0)2045 3577 y Fq(\022)2126 3638 y Fr(")2165 3650 y Fn(0)p 2116 3675 97 4 v 2116 3751 a Fr(C)2175 3763 y Fn(0)2222 3577 y Fq(\023)2284 3594 y Fn(\()p Fp(\034)2348 3569 y Fj(0)2369 3594 y Fv(\000)p Fn(2)p Fv(\000)p Fp(\034)7 b Fn(\))p Fp(=)p Fn(\(1+)p Fp(\034)2745 3602 y Fi(0)2776 3594 y Fn(\))644 3894 y Fo(\024)22 b Fu(const.)p Fr(")983 3854 y Fn(1+)p Fp(\030)1099 3829 y Fj(00)983 3916 y Fn(0)1144 3894 y Fr(;)2977 3607 y Fu(\(7)p Fr(:)p Fu(45\))118 4082 y(pro)n(vided)27 b(that)h Fr(\034)685 4052 y Fv(0)736 4082 y Fu(is)g(suc)n(h)f(that)1111 4410 y Fr(\030)1151 4376 y Fv(00)1216 4410 y Fu(=)1314 4305 y Fr(\034)1359 4270 y Fv(0)1401 4305 y Fo(\000)18 b Fu(2)g Fo(\000)1637 4249 y Fu(1)g(+)g Fr(\034)p 1637 4286 189 4 v 1712 4362 a(\030)1855 4305 y Fo(\000)g Fr(\034)1974 4317 y Fn(0)p 1314 4391 698 4 v 1554 4467 a Fu(1)g(+)g Fr(\034)1733 4479 y Fn(0)2044 4410 y Fr(>)23 b Fu(0)p Fr(:)780 b Fu(\(7)p Fr(:)p Fu(46\))189 4672 y(Finally)28 b(w)n(e)g(study)g(the)h(measure)e (of)i(the)f(set)h Fo(I)1713 4629 y Fn(\()p Fp(p)p Fn(\))1707 4694 y(3)1803 4672 y Fu(.)39 b(If)29 b Fr(n)24 b(>)g Fu(0)j(and)2344 4576 y Fq(\014)2344 4626 y(\014)2344 4676 y(\014)2371 4672 y Fo(j)7 b Fu(~)-49 b Fr(!)2449 4629 y Fn(\()p Fp(p)p Fn(\))2446 4682 y Fp(m)2558 4672 y Fu(+)24 b(~)-48 b Fr(!)2693 4684 y Fp(m)2752 4668 y Fj(0)2778 4672 y Fo(j)18 b(\000)25 b Fu(~)-49 b Fr(!)s(n)3007 4576 y Fq(\014)3007 4626 y(\014)3007 4676 y(\014)3059 4672 y Fo(\024)23 b Fu(1)118 4796 y(then)35 b(one)f(has)f(to)h(sum)h(o) n(v)n(er)d Fo(j)p Fr(n)p Fo(j)i(\024)g(N)1399 4808 y Fn(0)1470 4796 y Fu(,)i(with)f Fo(N)1793 4808 y Fn(0)1864 4796 y Fu(giv)n(en)f(b)n(y)g(\(7.17\),)h(while)f(one)g(has)f Fr(m)h(<)125 4912 y Fu(~)-49 b Fr(!)173 4869 y Fn(\()p Fp(p)p Fn(\))170 4922 y Fp(m)288 4912 y Fr(<)32 b Fu(~)-49 b Fr(!)433 4869 y Fn(\()p Fp(p)p Fn(\))430 4936 y Fp(m)489 4920 y Fj(0)542 4912 y Fu(+)26 b(~)-49 b Fr(!)681 4869 y Fn(\()p Fp(p)p Fn(\))678 4922 y Fp(m)797 4912 y Fr(<)31 b Fu(~)-49 b Fr(!)941 4869 y Fn(\()p Fp(p)p Fn(\))938 4934 y(1)1031 4912 y Fr(n)20 b Fu(+)e(1)25 b Fr(<)g Fu(\()7 b(~)-49 b Fr(!)1428 4869 y Fn(\()p Fp(p)p Fn(\))1425 4934 y(1)1538 4912 y Fu(+)19 b(1\))p Fo(j)p Fr(n)p Fo(j)p Fu(,)29 b(so)f(that)h(one)g(has)f(to)h(sum)g(only)g(o)n(v)n(er)e(the) 118 5031 y Fr(m)p Fu('s)k(suc)n(h)f(that)h Fo(j)p Fr(m)p Fo(j)d Fr(<)g Fu(\()7 b(~)-49 b Fr(!)978 4988 y Fn(\()p Fp(p)p Fn(\))975 5054 y(1)1068 5031 y Fr(n)21 b Fu(+)f(1\))p Fo(j)p Fr(n)p Fo(j)p Fu(,)31 b(while)g Fo(j)p Fr(m)1764 5001 y Fv(0)1788 5031 y Fo(j)f Fu(is)h(uniquely)g(determined)g(b)n(y)f (the)h(v)-5 b(alues)118 5138 y(of)p Fr(n)24 b Fu(and)g Fr(m)p Fu(.)36 b(Then)24 b(one)g(can)f(pro)r(ceed)h(as)f(in)i(the)f (previous)f(case)g(and)h(in)h(the)f(end)h(one)e(excludes)118 5244 y(a)k(further)h(subset)f(of)h Fo(E)867 5214 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1070 5244 y Fu(whose)f(Leb)r(esgue)g(measure)g (admits)g(a)g(b)r(ound)h(lik)n(e)g(\(7.26\).)1612 5443 y(21)p eop %%Page: 22 22 22 21 bop 189 559 a Fu(By)27 b(summing)h(together)f(the)h(b)r(ounds)f (for)g Fo(I)1612 516 y Fn(\()p Fp(p)p Fn(\))1606 581 y(1)1703 559 y Fu(,)h(for)f Fo(I)1932 516 y Fn(\()p Fp(p)p Fn(\))1926 581 y(2)2050 559 y Fu(and)g(for)h Fo(I)2390 516 y Fn(\()p Fp(p)p Fn(\))2384 581 y(3)2480 559 y Fu(,)g(then)g(the)g (b)r(ound)1296 763 y(meas\()p Fo(I)1560 729 y Fn(\()p Fp(p)p Fn(\))1651 763 y Fu(\))23 b Fo(\024)g Fr(b")1869 723 y Fp(\030)1901 698 y Fj(0)1923 723 y Fn(+1)1869 785 y(0)2977 763 y Fu(\(7)p Fr(:)p Fu(47\))118 954 y(follo)n(ws)k(for)g (all)g Fr(p)c Fo(\025)g Fu(1.)189 1061 y(W)-7 b(e)30 b(can)g(conclude)g(the)h(pro)r(of)e(of)h(the)h(theorem)e(through)h(the) g(follo)n(wing)f(result)h(\(pro)n(v)n(ed)f(in)118 1167 y(App)r(endix)h(A11\),)g(whic)n(h)f(sho)n(ws)f(that)h(the)h(b)r(ound)g (\(7.47\))e(essen)n(tially)g(extends)h(to)g(the)h(union)118 1273 y(of)e(all)f Fo(I)379 1243 y Fn(\()p Fp(p)p Fn(\))497 1273 y Fu(\(at)h(the)g(cost)f(of)h(taking)f(a)g(larger)f(constan)n(t)g Fr(B)32 b Fu(instead)c(of)f Fr(b)p Fu(\).)118 1450 y Fk(Lemma)j(12.)36 b Fs(De\014ne)29 b Fr(I)921 1420 y Fn(\()p Fp(p)p Fn(\))1041 1450 y Fs(as)h(the)g(set)f(of)i(values)f(in)g Fo(E)1913 1420 y Fn(\()p Fp(p)p Fn(\))2033 1450 y Fs(verifying)i (\(A9.7\).)39 b(Then)31 b(one)f(has)1156 1660 y Fu(meas)1350 1568 y Fq(\020)1400 1660 y Fo([)1455 1626 y Fv(1)1455 1681 y Fp(p)p Fn(=0)1577 1660 y Fo(I)1628 1626 y Fn(\()p Fp(p)p Fn(\))1719 1568 y Fq(\021)1792 1660 y Fo(\024)22 b Fr(B)t(")1985 1620 y Fp(\030)2017 1595 y Fj(0)2040 1620 y Fn(+1)1985 1682 y(0)2128 1660 y Fr(;)826 b Fu(\(7)p Fr(:)p Fu(48\))118 1870 y Fs(for)31 b(two)f(suitable)g(p)l(ositive)h(c) l(onstants)e Fr(B)34 b Fs(and)c Fr(\030)1669 1840 y Fv(0)1692 1870 y Fs(.)189 2047 y Fu(The)e(conclusion)e(is)i(that)g(one)f(has)1189 2239 y(meas)o(\()p Fo(E)7 b Fu(\))24 b(=)f(2)p Fr(")1677 2251 y Fn(0)1727 2146 y Fq(\020)1777 2239 y Fu(1)18 b Fo(\000)g Fr(b")1995 2199 y Fp(\030)1995 2261 y Fn(0)2031 2146 y Fq(\021)2095 2239 y Fr(;)859 b Fu(\(7)p Fr(:)p Fu(49\))118 2444 y(so)27 b(that)h(the)g(pro)r(of)f(of)g(the)h(theorem)g (is)f(complete.)833 2692 y Fw(App)s(endix)38 b(A1.)99 b(Pro)s(of)37 b(of)h(lemma)e(1)118 2869 y Fu(W)-7 b(e)28 b(pro)n(v)n(e)e(inductiv)n(ely)i(the)g(b)r(ound)989 3060 y Fr(N)1065 3026 y Fv(\003)1056 3081 y Fp(h)1103 3060 y Fu(\()p Fr(\022)r Fu(\))c Fo(\024)f Fu(max)o Fo(f)p Fu(0)p Fr(;)14 b Fu(2)p Fr(K)6 b Fu(\()p Fr(\022)r Fu(\)2)1861 3026 y Fn(\(2)p Fv(\000)p Fp(h)p Fn(\))p Fp(=\034)2129 3060 y Fo(\000)18 b Fu(1)p Fo(g)p Fr(;)637 b Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(1\))118 3252 y(where)27 b Fr(N)434 3222 y Fv(\003)425 3275 y Fp(h)472 3252 y Fu(\()p Fr(\022)r Fu(\))i(is)e(the)h(n)n(um)n(b)r(er)f(of)h(non-resonan)n(t)e(lines)h(in) h Fr(L)p Fu(\()p Fr(\022)r Fu(\))g(on)f(scale)g Fr(h)2564 3222 y Fv(0)2610 3252 y Fo(\025)c Fr(h)p Fu(.)189 3358 y(First)33 b(of)g(all)f(note)h(that)h(for)e(a)h(tree)f Fr(\022)k Fu(to)d(ha)n(v)n(e)e(a)i(line)g(on)g(scale)f Fr(h)h Fu(the)g(condition)g Fr(K)6 b Fu(\()p Fr(\022)r Fu(\))32 b Fr(>)118 3464 y Fu(2)160 3434 y Fn(\()p Fp(h)p Fv(\000)p Fn(1\))p Fp(=\034)440 3464 y Fu(is)d(necessary)-7 b(,)27 b(b)n(y)i(the)h(\014rst)e(Diophan)n(tine)h(conditions)g(in)g (\(1.15\).)40 b(This)29 b(means)g(that)118 3571 y(one)f(can)g(ha)n(v)n (e)f Fr(N)692 3541 y Fv(\003)683 3594 y Fp(h)730 3571 y Fu(\()p Fr(\022)r Fu(\))e Fo(\025)f Fu(1)k(only)f(if)i Fr(K)h Fu(=)24 b Fr(K)6 b Fu(\()p Fr(\022)r Fu(\))28 b(is)h(suc)n(h)e(that)i Fr(K)h(>)23 b(k)2363 3583 y Fn(0)2425 3571 y Fo(\021)h Fu(2)2556 3541 y Fn(\()p Fp(h)p Fv(\000)p Fn(1\))p Fp(=\034)2807 3571 y Fu(:)38 b(therefore)118 3677 y(for)27 b(v)-5 b(alues)27 b Fr(K)i Fo(\024)23 b Fr(k)723 3689 y Fn(0)788 3677 y Fu(the)28 b(b)r(ound)g(\(A1.1\))f(is)h (satis\014ed.)189 3783 y(If)d Fr(K)k Fu(=)22 b Fr(K)6 b Fu(\()p Fr(\022)r Fu(\))24 b Fr(>)e(k)792 3795 y Fn(0)830 3783 y Fu(,)j(w)n(e)f(assume)g(that)h(the)g(b)r(ound)g(holds)f(for)g (all)h(trees)f Fr(\022)2539 3753 y Fv(0)2587 3783 y Fu(with)h Fr(K)6 b Fu(\()p Fr(\022)2923 3753 y Fv(0)2946 3783 y Fu(\))24 b Fr(<)e(K)6 b Fu(.)118 3890 y(De\014ne)29 b Fr(E)437 3902 y Fp(h)505 3890 y Fu(=)24 b(2)636 3859 y Fv(\000)p Fn(1)725 3890 y Fu(\(2)799 3859 y Fn(\(2)p Fv(\000)p Fp(h)p Fn(\))p Fp(=\034)1050 3890 y Fu(\))1082 3859 y Fv(\000)p Fn(1)1171 3890 y Fu(:)39 b(so)28 b(w)n(e)g(ha)n(v)n(e) g(to)g(pro)n(v)n(e)f(that)i Fr(N)2236 3859 y Fv(\003)2227 3913 y Fp(h)2274 3890 y Fu(\()p Fr(\022)r Fu(\))c Fo(\024)f Fu(max)p Fo(f)p Fu(0)p Fr(;)14 b(K)6 b Fu(\()p Fr(\022)r Fu(\))p Fr(E)3017 3854 y Fv(\000)p Fn(1)3012 3915 y Fp(h)3124 3890 y Fo(\000)118 3996 y Fu(1)p Fo(g)p Fu(.)189 4102 y(Call)28 b Fr(`)h Fu(the)g(ro)r(ot)f(line)h(of)f Fr(\022)j Fu(and)e Fr(`)1273 4114 y Fn(1)1310 4102 y Fr(;)14 b(:)g(:)g(:)f(;)h(`) 1529 4114 y Fp(m)1621 4102 y Fu(the)29 b Fr(m)c Fo(\025)f Fu(0)k(lines)h(on)f(scale)g Fo(\025)d Fr(h)j Fu(whic)n(h)h(are)f(the) 118 4208 y(closest)21 b(to)h Fr(`)g Fu(\(i.e.)35 b(suc)n(h)22 b(that)g(no)g(other)f(line)h(along)f(the)h(paths)g(connecting)g(the)g (lines)g Fr(`)2870 4220 y Fn(1)2907 4208 y Fr(;)14 b(:)g(:)g(:)f(;)h(`) 3126 4220 y Fp(m)118 4315 y Fu(to)28 b(the)g(ro)r(ot)e(line)i(is)g(on)f (scale)g Fo(\025)22 b Fr(h)p Fu(\).)189 4421 y(If)28 b(the)g(ro)r(ot)f(line)h Fr(`)f Fu(of)g Fr(\022)j Fu(is)e(either)f(on)h (scale)e Fr(<)d(h)28 b Fu(or)e(on)i(scale)e Fo(\025)d Fr(h)28 b Fu(and)f(resonan)n(t,)f(then)1287 4668 y Fr(N)1363 4634 y Fv(\003)1354 4688 y Fp(h)1401 4668 y Fu(\()p Fr(\022)r Fu(\))e(=)1648 4564 y Fp(m)1618 4589 y Fq(X)1624 4766 y Fp(i)p Fn(=1)1752 4668 y Fr(N)1828 4634 y Fv(\003)1819 4688 y Fp(h)1865 4668 y Fu(\()p Fr(\022)1936 4680 y Fp(i)1964 4668 y Fu(\))p Fr(;)937 b Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(2\))118 4925 y(where)26 b Fr(\022)396 4937 y Fp(i)451 4925 y Fu(is)h(the)g(subtree)f(with)i Fr(`)1190 4937 y Fp(i)1244 4925 y Fu(as)e(ro)r(ot)g(line,)h(hence)g(the)g(b)r(ound)g (follo)n(ws)f(b)n(y)h(the)g(inductiv)n(e)118 5031 y(h)n(yp)r(othesis.) 189 5138 y(If)e(the)g(ro)r(ot)f(line)h Fr(`)f Fu(has)h(scale)e Fo(\025)g Fr(h)i Fu(and)f(is)h(non-resonan)n(t,)e(then)i Fr(`)2276 5150 y Fn(1)2313 5138 y Fr(;)14 b(:)g(:)g(:)g(;)g(`)2533 5150 y Fp(m)2620 5138 y Fu(are)24 b(the)h(en)n(tering)118 5244 y(line)j(of)f(a)h(cluster)f Fr(T)12 b Fu(.)1612 5443 y(22)p eop %%Page: 23 23 23 22 bop 189 555 a Fu(By)27 b(denoting)h(again)e(with)i Fr(\022)1110 567 y Fp(i)1165 555 y Fu(the)g(subtree)g(ha)n(ving)e Fr(`)1902 567 y Fp(i)1957 555 y Fu(as)h(ro)r(ot)g(line,)h(one)f(has) 1216 818 y Fr(N)1292 784 y Fv(\003)1283 838 y Fp(h)1330 818 y Fu(\()p Fr(\022)r Fu(\))d(=)e(1)c(+)1720 714 y Fp(m)1689 739 y Fq(X)1695 916 y Fp(i)p Fn(=1)1823 818 y Fr(N)1899 784 y Fv(\003)1890 838 y Fp(h)1937 818 y Fu(\()p Fr(\022)2008 830 y Fp(i)2036 818 y Fu(\))p Fr(;)865 b Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(3\))118 1091 y(so)27 b(that)h(the)g(b)r(ound)g(b)r(ecomes)f(trivial)g(if)h(either)g Fr(m)23 b Fu(=)f(0)27 b(or)g Fr(m)c Fo(\025)g Fu(2.)189 1198 y(If)28 b Fr(m)23 b Fu(=)g(1)k(then)i(one)e(has)g(a)h(cluster)f Fr(T)39 b Fu(with)28 b(t)n(w)n(o)f(external)g(lines)h Fr(`)f Fu(and)g Fr(`)2556 1210 y Fn(1)2593 1198 y Fu(,)h(whic)n(h)g (are)f(b)r(oth)118 1304 y(with)h(scales)f Fo(\025)22 b Fr(h)p Fu(;)28 b(then)451 1511 y Fo(jj)7 b Fu(~)-49 b Fr(!)549 1523 y Fn(1)586 1511 y Fr(n)636 1523 y Fp(`)668 1511 y Fo(j)19 b(\000)24 b Fu(~)-48 b Fr(!)845 1523 y Fp(m)904 1532 y Fh(`)936 1511 y Fo(j)23 b(\024)g Fu(2)1112 1477 y Fv(\000)p Fp(h)p Fn(+1)1290 1511 y Fr(C)1349 1523 y Fn(0)1387 1511 y Fr(;)1590 1441 y Fq(\014)1590 1490 y(\014)1618 1511 y Fo(j)7 b Fu(~)-49 b Fr(!)1693 1523 y Fn(1)1730 1511 y Fr(n)1780 1523 y Fp(`)1808 1531 y Fi(1)1844 1511 y Fo(j)18 b(\000)25 b Fu(~)-49 b Fr(!)2020 1523 y Fp(m)2079 1532 y Fh(`)2104 1544 y Fi(1)2144 1441 y Fq(\014)2144 1490 y(\014)2195 1511 y Fo(\024)23 b Fu(2)2325 1477 y Fv(\000)p Fp(h)p Fn(+1)2503 1511 y Fr(C)2562 1523 y Fn(0)2600 1511 y Fr(;)333 b Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(4\))118 1718 y(and)29 b Fr(n)331 1730 y Fp(`)389 1718 y Fo(6)p Fu(=)c Fr(n)529 1730 y Fp(`)557 1738 y Fi(1)623 1718 y Fu(as)j Fr(T)41 b Fu(is)29 b(not)g(a)g(self-energy)f(graph.)41 b(Then,)30 b(b)n(y)f(\(A1.4\),)g(one)g(has,)g(for)g(suitable)118 1825 y Fr(\021)159 1837 y Fp(`)191 1825 y Fr(;)14 b(\021)269 1837 y Fp(`)297 1845 y Fi(1)357 1825 y Fo(2)23 b(f)p Fu(+)p Fr(;)14 b Fo(\000g)p Fu(,)386 2032 y(2)428 1997 y Fv(\000)p Fp(h)p Fn(+2)606 2032 y Fr(C)665 2044 y Fn(0)726 2032 y Fo(\025)813 1961 y Fq(\014)813 2011 y(\014)848 2032 y Fu(~)-49 b Fr(!)893 2044 y Fn(1)930 2032 y Fu(\()p Fr(n)1012 2044 y Fp(`)1063 2032 y Fo(\000)18 b Fr(n)1196 2044 y Fp(`)1224 2052 y Fi(1)1260 2032 y Fu(\))h(+)f Fr(\021)1435 2044 y Fp(`)1473 2032 y Fu(~)-48 b Fr(!)1519 2044 y Fp(m)1578 2053 y Fh(`)1629 2032 y Fu(+)18 b Fr(\021)1753 2044 y Fp(`)1781 2052 y Fi(1)1824 2032 y Fu(~)-49 b Fr(!)1869 2044 y Fp(m)1928 2053 y Fh(`)1953 2065 y Fi(1)1994 1961 y Fq(\014)1994 2011 y(\014)2044 2032 y Fo(\025)23 b Fr(C)2191 2044 y Fn(0)2228 2032 y Fo(j)p Fr(n)2301 2044 y Fp(`)2352 2032 y Fo(\000)18 b Fr(n)2485 2044 y Fp(`)2513 2052 y Fi(1)2549 2032 y Fo(j)2572 1997 y Fv(\000)p Fp(\034)2665 2032 y Fr(;)268 b Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(5\))118 2239 y(where)34 b(the)h(second)f(Diophan)n(tine)g(conditions)g(in)h (\(1.15\))e(ha)n(v)n(e)g(b)r(een)i(used.)58 b(Hence)34 b Fr(K)6 b Fu(\()p Fr(\022)r Fu(\))23 b Fo(\000)118 2345 y Fr(K)6 b Fu(\()p Fr(\022)266 2357 y Fn(1)303 2345 y Fu(\))31 b Fr(>)f(E)522 2357 y Fp(h)566 2345 y Fu(,)j(whic)n(h,)g (inserted)f(in)n(to)g(\(A1.3\))g(with)h Fr(m)d Fu(=)h(1,)i(giv)n(es,)f (b)n(y)g(using)f(the)i(inductiv)n(e)118 2451 y(h)n(yp)r(othesis,)717 2550 y Fr(N)793 2516 y Fv(\003)784 2571 y Fp(h)831 2550 y Fu(\()p Fr(\022)r Fu(\))24 b(=)e(1)c(+)g Fr(N)1266 2516 y Fv(\003)1257 2571 y Fp(h)1304 2550 y Fu(\()p Fr(\022)1375 2562 y Fn(1)1413 2550 y Fu(\))23 b Fo(\024)g Fu(1)18 b(+)g Fr(K)6 b Fu(\()p Fr(\022)1847 2562 y Fn(1)1884 2550 y Fu(\))p Fr(E)1982 2515 y Fv(\000)p Fn(1)1977 2575 y Fp(h)2090 2550 y Fo(\000)18 b Fu(1)960 2708 y Fo(\024)k Fu(1)c(+)1190 2616 y Fq(\020)1240 2708 y Fr(K)6 b Fu(\()p Fr(\022)r Fu(\))19 b Fo(\000)f Fr(E)1585 2720 y Fp(h)1628 2616 y Fq(\021)1678 2708 y Fr(E)1744 2672 y Fv(\000)p Fn(1)1739 2733 y Fp(h)1851 2708 y Fo(\000)g Fu(1)23 b Fo(\024)g Fr(K)6 b Fu(\()p Fr(\022)r Fu(\))p Fr(E)2335 2672 y Fv(\000)p Fn(1)2330 2733 y Fp(h)2443 2708 y Fo(\000)18 b Fu(1)p Fr(;)2956 2640 y Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(6\))118 2887 y(hence)28 b(the)g(b)r(ound)g(is)f(pro)n(v)n(ed)f(also) h(if)h(the)g(ro)r(ot)f(line)g(is)h(on)f(scale)g Fo(\025)c Fr(h)p Fu(.)189 2994 y(In)32 b(the)g(same)f(w)n(a)n(y)f(one)h(pro)n(v)n (es)f(that,)j(if)f(w)n(e)f(denote)h(with)g Fr(C)2183 3006 y Fp(h)2226 2994 y Fu(\()p Fr(\022)r Fu(\))h(the)f(n)n(um)n(b)r (er)f(of)h(clusters)118 3100 y(on)27 b(scale)g Fr(h)p Fu(,)h(one)f(has)995 3208 y Fr(C)1054 3220 y Fp(h)1097 3208 y Fu(\()p Fr(\022)r Fu(\))d Fo(\024)f Fu(max)o Fo(f)p Fu(0)p Fr(;)14 b Fu(2)p Fr(K)6 b Fu(\()p Fr(\022)r Fu(\)2)1855 3173 y Fn(\(2)p Fv(\000)p Fp(h)p Fn(\))p Fp(=\034)2123 3208 y Fo(\000)18 b Fu(1)p Fo(g)p Fr(:)643 b Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(7\))118 3373 y(F)-7 b(or)32 b(a)g(tree)f(to)i (con)n(tain)e(a)h(cluster)g(on)g(scale)f Fr(h)h Fu(it)h(has)e(to)i(con) n(tain)e Fs(a)j(fortiori)h Fu(a)c(line)i(on)f(that)118 3480 y(scale,)27 b(so)g(that)h(again)e(the)i(b)r(ound)g(\(A1.7\))g(is)f (trivially)g(satis\014ed)g(for)g Fr(K)6 b Fu(\()p Fr(\022)r Fu(\))24 b Fo(\024)e Fr(k)2682 3492 y Fn(0)2743 3480 y Fu(=)g(2)2872 3449 y Fn(\()p Fp(h)p Fv(\000)p Fn(1\))p Fp(=\034)3123 3480 y Fu(.)189 3586 y(F)-7 b(or)24 b Fr(K)6 b Fu(\()p Fr(\022)r Fu(\))23 b(=)g Fr(K)28 b(>)23 b(k)858 3598 y Fn(0)920 3586 y Fu(w)n(e)h(can)h(pro)r(ceed)f(inductiv)n(ely)g (as)g(b)r(efore.)36 b(If)25 b(the)g(no)r(de)f Fr(v)2753 3598 y Fn(0)2791 3586 y Fu(,)h(whic)n(h)g(the)118 3693 y(ro)r(ot)i(line)h Fr(`)f Fu(of)h Fr(\022)i Fu(comes)c(out)i(from,)f (is)h(not)f(in)h(a)f(cluster)h(on)f(scale)g Fr(h)g Fu(the)h(one)f(has) 1299 3955 y Fr(C)1358 3967 y Fp(h)1401 3955 y Fu(\()p Fr(\022)r Fu(\))d(=)1648 3851 y Fp(m)1618 3876 y Fq(X)1624 4053 y Fp(i)p Fn(=1)1752 3955 y Fr(C)1811 3967 y Fp(h)1854 3955 y Fu(\()p Fr(\022)1925 3967 y Fp(i)1953 3955 y Fu(\))p Fr(;)948 b Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(8\))118 4228 y(where)32 b Fr(\022)402 4240 y Fn(1)439 4228 y Fr(;)14 b(:)g(:)g(:)g(;)g(\022)663 4240 y Fp(m)758 4228 y Fu(are)31 b(the)h(subtrees)g(with)g(ro)r(ot)g(in)g Fr(v)1894 4240 y Fn(0)1932 4228 y Fu(;)i(in)e(suc)n(h)g(a)g(case)f(the)i(b)r(ound)f (follo)n(ws)118 4334 y(from)27 b(the)h(inductiv)n(e)g(h)n(yp)r (othesis.)189 4441 y(If)c Fr(v)308 4453 y Fn(0)369 4441 y Fu(is)f(inside)h(a)f(cluster)g Fr(T)34 b Fu(on)23 b(scale)g Fr(h)g Fu(then)h(call)f Fr(\022)1841 4453 y Fn(1)1878 4441 y Fr(;)14 b(:)g(:)g(:)g(;)g(\022)2102 4453 y Fp(m)2188 4441 y Fu(the)24 b(en)n(tering)f(lines)g(of)h Fr(T)12 b Fu(;)24 b(then)118 4547 y(one)j(has)1227 4694 y Fr(C)1286 4706 y Fp(h)1330 4694 y Fu(\()p Fr(\022)r Fu(\))d(=)e(1)c(+)1720 4590 y Fp(m)1689 4615 y Fq(X)1695 4792 y Fp(i)p Fn(=1)1823 4694 y Fr(C)1882 4706 y Fp(h)1925 4694 y Fu(\()p Fr(\022)1996 4706 y Fp(i)2024 4694 y Fu(\))p Fr(;)877 b Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(9\))118 4930 y(and)31 b(the)h(b)r(ound)g(\(A1.7\))g (follo)n(ws)e(again)h(from)g(the)h(inductiv)n(e)f(h)n(yp)r(othesis)g (for)g(either)h Fr(m)d Fu(=)g(0)118 5036 y(or)e Fr(m)c Fo(\025)f Fu(2.)37 b(If)28 b Fr(m)23 b Fu(=)f(1)28 b(then)g(one)f(has) 430 5244 y Fo(j)q(j)7 b Fu(~)-49 b Fr(!)529 5256 y Fn(1)565 5244 y Fr(n)615 5256 y Fp(`)647 5244 y Fo(j)19 b(\000)24 b Fu(~)-48 b Fr(!)824 5256 y Fp(m)883 5265 y Fh(`)915 5244 y Fo(j)24 b(\024)e Fu(2)1091 5209 y Fv(\000)p Fp(h)p Fn(+1)1270 5244 y Fr(C)1329 5256 y Fn(0)1366 5244 y Fr(;)1569 5173 y Fq(\014)1569 5223 y(\014)1597 5244 y Fo(j)7 b Fu(~)-49 b Fr(!)1672 5256 y Fn(1)1709 5244 y Fr(n)1759 5256 y Fp(`)1787 5264 y Fi(1)1823 5244 y Fo(j)19 b(\000)24 b Fu(~)-48 b Fr(!)2000 5256 y Fp(m)2059 5265 y Fh(`)2084 5277 y Fi(1)2124 5173 y Fq(\014)2124 5223 y(\014)2174 5244 y Fo(\024)23 b Fu(2)2304 5209 y Fv(\000)p Fp(h)p Fn(+1)2483 5244 y Fr(C)2542 5256 y Fn(0)2579 5244 y Fr(;)313 b Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(10\))1612 5443 y(23)p eop %%Page: 24 24 24 23 bop 118 555 a Fu(where)27 b Fr(`)c Fo(2)g Fr(L)p Fu(\()p Fr(T)12 b Fu(\))27 b(is)h(on)f(scale)g Fr(h)p Fu(,)h(while)g Fr(`)1452 567 y Fn(1)1516 555 y Fu(is)g(on)f(a)g(scale)g Fr(h)2031 567 y Fp(`)2059 575 y Fi(1)2118 555 y Fr(>)c(h)28 b Fu(\(b)n(y)f(de\014nition)h(of)g(cluster\).)118 662 y(Therefore,)f(if)h Fr(n)644 674 y Fp(`)698 662 y Fu(=)23 b Fr(n)836 632 y Fn(0)836 685 y Fp(`)873 662 y Fu(,)28 b(one)f(has,)g(b)n(y)h(the)g(\014rst)f(Diophan)n(tine)h(condition)f(in) h(\(1.15\),)957 827 y(2)999 792 y Fv(\000)p Fp(h)p Fn(+1)1178 827 y Fr(C)1237 839 y Fn(0)1297 827 y Fo(\025)23 b(jj)7 b Fu(~)-49 b Fr(!)1483 839 y Fn(1)1520 827 y Fr(n)1570 839 y Fp(`)1602 827 y Fo(j)18 b(\000)25 b Fu(~)-49 b Fr(!)1778 839 y Fp(m)1837 848 y Fh(`)1870 827 y Fo(j)23 b(\025)g Fr(C)2063 839 y Fn(0)2100 827 y Fo(j)p Fr(n)2173 792 y Fn(0)2173 847 y Fp(`)2210 827 y Fo(j)2233 792 y Fv(\000)p Fp(\034)2327 827 y Fr(;)565 b Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(11\))118 992 y(while,)28 b(if)g Fr(n)484 1004 y Fp(`)539 992 y Fu(=)23 b Fr(n)677 962 y Fn(0)677 1015 y Fp(`)732 992 y Fu(+)18 b Fr(n)865 1004 y Fp(`)893 1012 y Fi(1)929 992 y Fu(,)28 b(one)f(has)g(for)g(suitable)h Fr(\021)1758 1004 y Fp(`)1790 992 y Fr(;)14 b(\021)1868 1004 y Fp(`)1896 1012 y Fi(1)1955 992 y Fo(2)24 b(f)p Fu(+)p Fr(;)14 b Fo(\000g)p Fu(,)836 1157 y(2)878 1123 y Fv(\000)p Fp(h)p Fn(+2)1056 1157 y Fr(C)1115 1169 y Fn(0)1176 1157 y Fo(\025)1264 1086 y Fq(\014)1264 1136 y(\014)1298 1157 y Fu(~)-49 b Fr(!)1343 1169 y Fn(1)1380 1157 y Fu(\()p Fr(n)1462 1169 y Fp(`)1513 1157 y Fo(\000)18 b Fr(n)1646 1169 y Fp(`)1674 1177 y Fi(1)1710 1157 y Fu(\))h(+)f Fr(\021)1885 1169 y Fp(`)1923 1157 y Fu(~)-48 b Fr(!)1969 1169 y Fp(m)2028 1178 y Fh(`)2079 1157 y Fu(+)18 b Fr(\021)2203 1169 y Fp(`)2231 1177 y Fi(1)2274 1157 y Fu(~)-48 b Fr(!)2320 1169 y Fp(m)2379 1178 y Fh(`)2404 1190 y Fi(1)2444 1086 y Fq(\014)2444 1136 y(\014)1176 1288 y Fo(\025)23 b Fr(C)1323 1300 y Fn(0)1360 1288 y Fo(j)p Fr(n)1433 1300 y Fp(`)1483 1288 y Fo(\000)18 b Fr(n)1616 1300 y Fp(`)1644 1308 y Fi(1)1681 1288 y Fo(j)1704 1254 y Fv(\000)p Fp(\034)1820 1288 y Fo(\025)23 b Fr(C)1967 1300 y Fn(0)2004 1288 y Fo(j)p Fr(n)2077 1254 y Fn(0)2077 1309 y Fp(`)2114 1288 y Fo(j)2137 1254 y Fv(\000)p Fp(\034)2231 1288 y Fr(;)2915 1221 y Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(12\))118 1446 y(b)n(y)k(the)h(second)f(Diophan)n(tine)h(conditions)f (in)h(\(1.15\).)36 b(So)27 b(in)h(b)r(oth)g(cases)f(one)g(has)1198 1543 y Fq(X)1141 1725 y Fp(v)r Fv(2)p Fp(V)15 b Fn(\()p Fp(T)9 b Fn(\))1389 1621 y Fo(j)p Fr(n)1462 1633 y Fp(v)1501 1621 y Fo(j)23 b(\025)g Fu(2)1677 1587 y Fn(\()p Fp(h)p Fv(\000)p Fn(2\))p Fp(=\034)1951 1621 y Fr(>)f(E)2099 1633 y Fp(h)2143 1621 y Fr(;)749 b Fu(\()p Fr(A)p Fu(1)p Fr(:)p Fu(13\))118 1877 y(so)24 b(that)h(\(A1.9\))g(implies)g(the)h(b)r (ound)f(\(A1.7\))g(also)e(for)i Fr(m)e Fu(=)f(1.)36 b(Of)25 b(course)e(one)i(can)f(b)r(ound)i(the)118 1983 y(n)n(um)n(b)r(er)f(of)g (non-resonan)n(t)e(lines)i(on)f(scale)g Fr(h)h Fu(of)g(an)n(y)f(tree)h Fr(\022)i Fu(b)n(y)e Fr(N)2237 1953 y Fv(\003)2228 2006 y Fp(h)2275 1983 y Fu(\()p Fr(\022)r Fu(\))13 b(+)g Fr(C)2530 1995 y Fp(h)2574 1983 y Fu(\()p Fr(\022)r Fu(\))g Fo(\000)g Fr(C)2829 1995 y Fp(h)2873 1983 y Fu(\()p Fr(\022)r Fu(\),)27 b(with)118 2089 y Fr(N)194 2059 y Fv(\003)185 2113 y Fp(h)232 2089 y Fu(\()p Fr(\022)r Fu(\))19 b(+)f Fr(C)498 2101 y Fp(h)542 2089 y Fu(\()p Fr(\022)r Fu(\))24 b Fo(\024)e Fu(4)p Fr(K)6 b Fu(\()p Fr(\022)r Fu(\)2)1024 2059 y Fn(\(2)p Fv(\000)p Fp(h)p Fn(\))p Fp(=\034)1275 2089 y Fu(:)37 b(therefore)26 b(\(4.12\))h(follo)n(ws.)833 2337 y Fw(App)s(endix)38 b(A2.)99 b(Pro)s(of)37 b(of)h(lemma)e(2)118 2514 y Fu(One)29 b(has)g(to)g(sho)n(w)g(inductiv)n(ely)g(that,)h(b)n(y) f(de\014ning)h Fr(N)1898 2484 y Fv(\003)1889 2538 y Fp(h)1936 2514 y Fu(\()p Fr(\022)r Fu(\))g(as)f(the)h(n)n(um)n(b)r(er)f(of)g (non-resonan)n(t)118 2621 y(lines)f(on)f(scale)g Fr(h)670 2591 y Fv(0)716 2621 y Fo(\025)c Fr(h)k Fu(in)h(the)g(renormalized)e (tree)h Fr(\022)r Fu(,)h(one)f(has)989 2786 y Fr(N)1065 2751 y Fv(\003)1056 2806 y Fp(h)1103 2786 y Fu(\()p Fr(\022)r Fu(\))d Fo(\024)f Fu(max)o Fo(f)p Fu(0)p Fr(;)14 b Fu(2)p Fr(K)6 b Fu(\()p Fr(\022)r Fu(\)2)1861 2751 y Fn(\(2)p Fv(\000)p Fp(h)p Fn(\))p Fp(=\034)2129 2786 y Fo(\000)18 b Fu(1)p Fo(g)p Fr(;)637 b Fu(\()p Fr(A)p Fu(2)p Fr(:)p Fu(1\))118 2951 y(and)28 b(that)f(an)h(analogous)d(b)r(ound)j(holds)f (for)g(the)h(n)n(um)n(b)r(er)g Fr(C)2063 2963 y Fp(h)2106 2951 y Fu(\()p Fr(\022)r Fu(\))h(of)e(clusters)g(on)g(scale)g Fr(h)p Fu(.)189 3057 y(Let)41 b(us)g(consider)f(a)g(self-energy)g (graph)f Fr(T)52 b Fu(in)41 b Fr(\022)j Fu(whic)n(h)c(is)h Fs(maximal)p Fu(,)k Fs(i.e.)32 b Fu(whic)n(h)40 b(is)h(not)118 3163 y(con)n(tained)d(in)g(an)n(y)f(other)h(resonance.)66 b(The)39 b(v)-5 b(alue)38 b(of)g(suc)n(h)f(a)h(self-energy)f(graph)g (is)h(giv)n(en)118 3270 y(b)n(y)31 b(the)g(pro)r(duct)g(of)g(the)g(v)-5 b(alues)31 b(of)g(the)g(inner)g(maximal)g(self-energy)e(graphs)h(\(on)h (whic)n(h)g(the)118 3376 y Fo(R)i Fu(op)r(eration)e(is)h(still)g (applied\))g(times)h(the)f(pro)r(duct)g(of)g(propagators)e(corresp)r (onding)g(to)i(the)118 3482 y(remaining)k(lines;)41 b(there)c(is)g(a)f (c)n(hain)h(of)f(propagators)e(and)j(self-energy)e(graphs)h(connecting) 118 3589 y(the)i(external)e(lines)h(suc)n(h)g(that)g(the)h(momen)n(tum) f(\()p Fr(n)1866 3601 y Fp(`)1898 3589 y Fr(;)14 b(m)2008 3601 y Fp(`)2040 3589 y Fu(\))37 b(of)g(an)n(y)g(of)g(suc)n(h)g (propagator)d(or)118 3695 y(self-energy)26 b(graph)h(has)g(the)h(form) 1330 3860 y(\()p Fr(n)1412 3826 y Fn(0)1412 3880 y Fp(`)1468 3860 y Fu(+)18 b Fr(n;)c(m)1711 3826 y Fn(0)1711 3880 y Fp(`)1766 3860 y Fu(+)k Fr(m)p Fu(\))p Fr(;)979 b Fu(\()p Fr(A)p Fu(2)p Fr(:)p Fu(2\))118 4025 y(where)30 b(\()p Fr(n;)14 b(m)p Fu(\))31 b(is)f(the)h(momen)n(tum)f(of)h(the)g(external) e(lines,)i(and)g(\()p Fr(n)2316 3995 y Fn(0)2316 4048 y Fp(`)2353 4025 y Fr(;)14 b(m)2463 3995 y Fn(0)2463 4048 y Fp(`)2500 4025 y Fu(\))31 b(are)e(implicitly)i(de-)118 4131 y(\014ned;)26 b(all)e(the)g(other)g(propagators)d(and)j (self-energy)f(graphs)g(are)g(indep)r(enden)n(t)i(of)f(\()p Fr(n;)14 b(m)p Fu(\).)36 b(Let)118 4237 y(us)i(consider)f(the)h(action) f(of)h Fo(R)g Fu(on)f(the)h(maximal)g(self-energy)e(graphs.)66 b(The)38 b(action)f(of)h Fo(R)118 4344 y Fu(consists)28 b(in)h(writing)f(the)h(self-energy)e(graph)g(as)h(a)g(sum)h(of)f (terms,)h(in)f(whic)n(h)h(one)f(propagator)118 4450 y Fr(g)161 4420 y Fn(\()p Fp(h)226 4429 y Fh(`)254 4420 y Fn(\))285 4450 y Fu(\()7 b(~)-49 b Fr(!)369 4462 y Fn(1)406 4450 y Fr(n)456 4420 y Fn(0)456 4474 y Fp(`)507 4450 y Fu(+)21 b(~)-49 b Fr(!)638 4462 y Fn(1)675 4450 y Fr(n;)14 b(m)835 4420 y Fn(0)835 4474 y Fp(`)886 4450 y Fu(+)g Fr(m)p Fu(\))26 b(is)f(replaced)g(with)h Fr(g)1733 4420 y Fn(\()p Fp(h)1798 4429 y Fh(`)1826 4420 y Fn(\))1856 4450 y Fu(\()7 b(~)-49 b Fr(!)1940 4462 y Fn(1)1978 4450 y Fr(n)2028 4420 y Fn(0)2028 4474 y Fp(`)2079 4450 y Fu(+)21 b(~)-49 b Fr(!)2210 4462 y Fn(1)2247 4450 y Fr(n;)14 b(m)2407 4420 y Fn(0)2407 4474 y Fp(`)2458 4450 y Fu(+)g Fr(m)p Fu(\))g Fo(\000)g Fr(g)2778 4420 y Fn(\()p Fp(h)2843 4429 y Fh(`)2872 4420 y Fn(\))2902 4450 y Fu(\()7 b(~)-49 b Fr(!)2986 4462 y Fn(1)3023 4450 y Fr(n)3073 4420 y Fn(0)3073 4474 y Fp(`)3124 4450 y Fu(+)125 4601 y(~)g Fr(!)170 4613 y Fp(m)233 4601 y Fr(;)14 b(m)343 4570 y Fn(0)343 4624 y Fp(`)401 4601 y Fu(+)20 b Fr(m)p Fu(\))32 b(or)f(the)g(v)-5 b(alue)32 b(of)f(one)g(regularized)e(inner)j (self-energy)e(graph)g Fo(RV)2775 4543 y Fp(h)2814 4513 y Fi(\()p Fh(e)p Fi(\))2814 4572 y Fh(T)2856 4560 y Fj(0)2768 4625 y Fp(T)2816 4608 y Fj(0)2895 4601 y Fu(\()7 b(~)-49 b Fr(!)2979 4613 y Fn(1)3016 4601 y Fr(n)3066 4570 y Fn(0)3066 4624 y Fp(`)3124 4601 y Fu(+)125 4752 y(~)g Fr(!)170 4764 y Fn(1)207 4752 y Fr(n;)14 b(m)367 4722 y Fn(0)367 4775 y Fp(`)404 4752 y Fu(+)q Fr(m)p Fu(\))k(is)h(replaced)f (with)h Fo(RV)1295 4695 y Fp(h)1334 4664 y Fi(\()p Fh(e)p Fi(\))1334 4723 y Fh(T)1375 4711 y Fj(0)1288 4776 y Fp(T)1336 4760 y Fj(0)1415 4752 y Fu(\()7 b(~)-49 b Fr(!)1499 4764 y Fn(1)1536 4752 y Fr(n)1586 4722 y Fn(0)1586 4775 y Fp(`)1624 4752 y Fu(+)7 b(~)-48 b Fr(!)1742 4764 y Fn(1)1778 4752 y Fr(n;)14 b(m)1938 4722 y Fn(0)1938 4775 y Fp(`)1975 4752 y Fu(+)q Fr(m)p Fu(\))q Fo(\000)q(RV)2341 4695 y Fp(h)2380 4664 y Fi(\()p Fh(e)p Fi(\))2380 4723 y Fh(T)2420 4711 y Fj(0)2334 4776 y Fp(T)2382 4760 y Fj(0)2459 4752 y Fu(\()7 b(~)-49 b Fr(!)2543 4764 y Fn(1)2581 4752 y Fr(n)2631 4722 y Fn(0)2631 4775 y Fp(`)2668 4752 y Fu(+)7 b(~)-48 b Fr(!)2786 4764 y Fp(m)2848 4752 y Fr(;)14 b(m)2958 4722 y Fn(0)2958 4775 y Fp(`)2996 4752 y Fu(+)q Fr(m)p Fu(\).)118 4858 y(W)-7 b(e)28 b(can)f(write)352 5003 y Fr(g)395 4969 y Fn(\()p Fp(h)460 4978 y Fh(`)489 4969 y Fn(\))519 5003 y Fu(\()7 b(~)-49 b Fr(!)603 5015 y Fn(1)640 5003 y Fr(n)690 4969 y Fn(0)690 5024 y Fp(`)745 5003 y Fu(+)25 b(~)-49 b Fr(!)880 5015 y Fn(1)917 5003 y Fr(n;)14 b(m)1077 4969 y Fn(0)1077 5024 y Fp(`)1133 5003 y Fu(+)k Fr(m)p Fu(\))g Fo(\000)g Fr(g)1465 4969 y Fn(\()p Fp(h)1530 4978 y Fh(`)1559 4969 y Fn(\))1589 5003 y Fu(\()7 b(~)-49 b Fr(!)s(n)1726 4969 y Fn(0)1726 5024 y Fp(`)1781 5003 y Fu(+)25 b(~)-49 b Fr(!)1916 5015 y Fp(m)1979 5003 y Fr(;)14 b(m)2089 4969 y Fn(0)2089 5024 y Fp(`)2144 5003 y Fu(+)19 b Fr(m)p Fu(\))707 5195 y(=)k(\()7 b(~)-49 b Fr(!)879 5207 y Fn(1)916 5195 y Fr(n)19 b Fo(\000)24 b Fu(~)-48 b Fr(!)1120 5207 y Fp(m)1182 5195 y Fu(\))1228 5082 y Fq(Z)1311 5103 y Fn(1)1275 5271 y(0)1363 5195 y Fu(d)p Fr(t)14 b(@)1511 5139 y(\037)1563 5151 y Fp(h)1602 5160 y Fh(`)1635 5139 y Fu(\()p Fo(j)7 b Fu(~)-49 b Fr(!)1742 5151 y Fn(1)1779 5139 y Fr(n)1829 5109 y Fn(0)1829 5163 y Fp(`)1885 5139 y Fu(+)18 b Fr(t)p Fu(\()7 b(~)-49 b Fr(!)2082 5151 y Fn(1)2119 5139 y Fr(n)18 b Fo(\000)25 b Fu(~)-49 b Fr(!)2322 5151 y Fp(m)2385 5139 y Fu(\))p Fo(j)19 b(\000)24 b Fu(~)-48 b Fr(!)2594 5151 y Fp(m)2657 5139 y Fu(\))p 1511 5176 1178 4 v 1545 5252 a Fo(\000)p Fu(\()7 b(~)-49 b Fr(!)1694 5264 y Fn(1)1731 5252 y Fr(n)1781 5224 y Fn(0)1781 5277 y Fp(`)1837 5252 y Fu(+)18 b Fr(t)p Fu(\()7 b(~)-49 b Fr(!)2034 5264 y Fn(1)2071 5252 y Fr(n)18 b Fo(\000)25 b Fu(~)-49 b Fr(!)2274 5264 y Fp(m)2337 5252 y Fu(\)\))2401 5228 y Fn(2)2457 5252 y Fu(+)25 b(~)-49 b Fr(!)2595 5228 y Fn(2)2592 5273 y Fp(m)2699 5195 y Fr(;)2956 5122 y Fu(\()p Fr(A)p Fu(2)p Fr(:)p Fu(3\))1612 5443 y(24)p eop %%Page: 25 25 25 24 bop 118 555 a Fu(where)27 b Fr(@)33 b Fu(denotes)27 b(the)h(deriv)-5 b(ativ)n(e)27 b(with)h(resp)r(ect)f(to)g(the)h (argumen)n(t)34 b(~)-49 b Fr(!)2400 567 y Fn(1)2437 555 y Fr(n)2487 525 y Fn(0)2487 579 y Fp(`)2542 555 y Fu(+)18 b Fr(t)p Fu(\()7 b(~)-49 b Fr(!)2739 567 y Fn(1)2777 555 y Fr(n)18 b Fo(\000)25 b Fu(~)-49 b Fr(!)2980 567 y Fp(m)3043 555 y Fu(\).)189 708 y(On)29 b(the)h(other)e(hand)i Fo(LV)1017 651 y Fp(h)1056 621 y Fi(\()p Fh(e)p Fi(\))1056 680 y Fh(T)1097 668 y Fj(0)1010 733 y Fp(T)1137 708 y Fu(\()7 b(~)-49 b Fr(!)1221 720 y Fn(1)1258 708 y Fr(n)1308 678 y Fn(0)1308 732 y Fp(`)1365 708 y Fu(+)26 b(~)-49 b Fr(!)1501 720 y Fn(1)1538 708 y Fr(n;)14 b(m)1698 678 y Fn(0)1698 732 y Fp(`)1754 708 y Fu(+)19 b Fr(m)p Fu(\))30 b(is)f(indep)r(enden)n(t)h(of)g Fr(!)2679 720 y Fn(1)2716 708 y Fr(n)f Fu(\(see)g(\(5.1\)\),)118 814 y(so)e(that)385 1048 y Fo(RV)513 991 y Fp(h)552 960 y Fi(\()p Fh(e)p Fi(\))552 1019 y Fh(T)594 1007 y Fj(0)506 1072 y Fp(T)554 1055 y Fj(0)634 1048 y Fu(\()7 b(~)-49 b Fr(!)718 1060 y Fn(1)755 1048 y Fr(n)805 1013 y Fn(0)805 1068 y Fp(`)860 1048 y Fu(+)25 b(~)-48 b Fr(!)996 1060 y Fn(1)1032 1048 y Fr(n;)14 b(m)1192 1013 y Fn(0)1192 1068 y Fp(`)1248 1048 y Fu(+)k Fr(m)p Fu(\))g Fo(\000)g(RV)1665 991 y Fp(h)1704 960 y Fi(\()p Fh(e)p Fi(\))1704 1019 y Fh(T)1746 1007 y Fj(0)1658 1072 y Fp(T)1706 1055 y Fj(0)1786 1048 y Fu(\()7 b(~)-49 b Fr(!)1870 1060 y Fn(1)1907 1048 y Fr(n)1957 1013 y Fn(0)1957 1068 y Fp(`)2012 1048 y Fu(+)25 b(~)-49 b Fr(!)2147 1060 y Fp(m)2210 1048 y Fr(;)14 b(m)2320 1013 y Fn(0)2320 1068 y Fp(`)2376 1048 y Fu(+)k Fr(m)p Fu(\))575 1229 y(=)k Fo(V)720 1172 y Fp(h)759 1141 y Fi(\()p Fh(e)p Fi(\))759 1200 y Fh(T)801 1188 y Fj(0)713 1253 y Fp(T)761 1236 y Fj(0)840 1229 y Fu(\()7 b(~)-49 b Fr(!)924 1241 y Fn(1)961 1229 y Fr(n)1011 1194 y Fn(0)1011 1249 y Fp(`)1067 1229 y Fu(+)24 b(~)-48 b Fr(!)s(n;)14 b(m)1365 1194 y Fn(0)1365 1249 y Fp(`)1420 1229 y Fu(+)k Fr(m)p Fu(\))h Fo(\000)f(V)1768 1172 y Fp(h)1807 1141 y Fi(\()p Fh(e)p Fi(\))1807 1200 y Fh(T)1848 1188 y Fj(0)1761 1253 y Fp(T)1809 1236 y Fj(0)1887 1229 y Fu(\()7 b(~)-49 b Fr(!)1971 1241 y Fn(1)2009 1229 y Fr(n)2059 1194 y Fn(0)2059 1249 y Fp(`)2114 1229 y Fu(+)25 b(~)-49 b Fr(!)2249 1241 y Fp(m)2312 1229 y Fr(;)14 b(m)2422 1194 y Fn(0)2422 1249 y Fp(`)2477 1229 y Fu(+)k Fr(m)p Fu(\))p Fr(;)2956 1114 y Fu(\()p Fr(A)p Fu(2)p Fr(:)p Fu(4\))118 1430 y(and)28 b(w)n(e)f(can)g(reason)f(as)h(ab)r(o)n(v)n(e,)f(writing)h(the)h(inner)g (self-energy)e(graphs)g(as)h(a)g(sum)h(of)g(terms.)189 1538 y(Then)f(the)h(pro)r(of)f(of)g(lemma)g(2)g(pro)r(ceeds)g(exactly)f (as)h(for)g(lemma)g(1,)g(but)h(in)f(\(A1.4\))h(one)f(has)118 1645 y(to)h(replace)33 b(~)-48 b Fr(!)554 1657 y Fn(1)591 1645 y Fr(n)641 1657 y Fp(`)700 1645 y Fu(with)35 b(~)-48 b Fr(!)942 1657 y Fn(1)979 1645 y Fr(n)1029 1615 y Fn(0)1029 1668 y Fp(`)1084 1645 y Fu(+)19 b Fr(t)p Fu(\()7 b(~)-49 b Fr(!)1282 1657 y Fn(1)1319 1645 y Fr(n)18 b Fo(\000)25 b Fu(~)-49 b Fr(!)1522 1657 y Fp(m)1585 1645 y Fu(\),)28 b(when)h Fr(`)e Fu(is)h(along)e(a)i(path)g(of)g(lines)g(connecting)118 1751 y(the)36 b(external)e(lines)h(of)g Fr(T)12 b Fu(.)58 b(But)35 b(this)h(do)r(es)e(not)i(c)n(hange)d(the)j(pro)r(of)e(b)r (ecause)h(when)g(passing)118 1857 y(from)27 b(\(A1.4\))h(to)f(\(A1.5\)) h(the)g(terms)f(dep)r(ending)h(on)g Fr(t)f Fu(cancel)g(out.)833 2105 y Fw(App)s(endix)38 b(A3.)99 b(Pro)s(of)37 b(of)h(lemma)e(3)189 2285 y Fu(By)25 b(using)g(the)g(expression)f(\(A2.3\))h(and)h(\(A2.4\)) f(w)n(e)g(see)f(that)i(the)g(e\013ect)f(of)h Fo(R)f Fu(is)g(to)g(impro) n(v)n(e)118 2406 y(b)n(y)34 b(a)g(factor)f(2)602 2375 y Fv(\000)p Fp(h)693 2345 y Fi(\()p Fh(e)p Fi(\))693 2397 y Fh(T)769 2375 y Fn(+)p Fp(h)859 2383 y Fh(T)944 2406 y Fu(the)h(b)r(ound)h(of)f(the)g(propagator)e(corresp)r(onding)g (to)i(the)g(line)h Fr(`)p Fu(.)56 b(In)118 2512 y(the)32 b(same)f(w)n(a)n(y)-7 b(,)31 b(the)h(di\013erence)f(of)h(propagators)c (in)k(the)g(inner)f(self-energy)f(graph)g(giv)n(es)h(the)118 2618 y(\\gain")1040 2749 y(2)1082 2714 y Fv(\000)p Fp(h)1173 2684 y Fi(\()p Fh(e)p Fi(\))1173 2736 y Fh(T)1250 2714 y Fn(+)p Fp(h)1340 2730 y Fh(T)1381 2718 y Fj(0)1435 2749 y Fo(\024)23 b Fu(2)1565 2714 y Fv(\000)p Fp(h)1656 2684 y Fi(\()p Fh(e)p Fi(\))1656 2736 y Fh(T)1732 2714 y Fn(+)p Fp(h)1822 2722 y Fh(T)1872 2749 y Fu(2)1914 2711 y Fv(\000)p Fp(h)2005 2681 y Fi(\()p Fh(e)p Fi(\))2005 2740 y Fh(T)2046 2728 y Fj(0)2082 2711 y Fn(+)p Fp(h)2172 2726 y Fh(T)2213 2714 y Fj(0)2244 2749 y Fr(;)689 b Fu(\()p Fr(A)p Fu(3)p Fr(:)p Fu(1\))118 2929 y(as)31 b Fr(h)272 2886 y Fn(\()p Fp(e)p Fn(\))272 2954 y Fp(T)320 2937 y Fj(0)388 2929 y Fo(\024)e Fr(h)530 2941 y Fp(T)582 2929 y Fu(.)48 b(A)n(t)31 b(the)h(end)f(\(i\))h(the)g(propagators)c (are)i(deriv)n(ed)g(at)h(most)g(one)g(time;)j(\(ii\))e(the)118 3035 y(n)n(um)n(b)r(er)e(of)f(terms)h(so)f(generated)f(is)i Fo(\024)c Fr(k)d Fu(+)d Fo(j)p Fr(k)p 1588 3048 46 4 v 2 w Fo(j)p Fu(;)31 b(\(iii\))g(to)f(eac)n(h)f(self-energy)f(graph)h Fr(T)41 b Fu(a)29 b(factor)118 3157 y(2)160 3126 y Fv(\000)p Fp(h)251 3096 y Fi(\()p Fh(e)p Fi(\))251 3148 y Fh(T)328 3126 y Fn(+)p Fp(h)418 3134 y Fh(T)495 3157 y Fu(is)f(asso)r(ciated.) 189 3265 y(Assuming)i(that)g Fo(j)p Fr(\027)824 3222 y Fn(\()p Fp(c)p Fn(\))819 3290 y Fp(h;m)941 3265 y Fo(j)d(\024)f Fr(C)6 b(")p Fu(,)31 b(for)e(an)n(y)h Fr(\022)i Fu(one)d(obtains,)h (for)g(a)f(suitable)h(constan)n(ts)p 2954 3198 72 4 v 29 w Fr(D)i Fu(and)p 118 3316 V 118 3383 a Fr(D)189 3395 y Fn(0)226 3383 y Fu(,)289 3610 y Fo(j)p Fu(V)-7 b(al\()p Fr(\022)r Fu(\))p Fo(j)24 b(\024)p 671 3543 V 22 w Fr(D)742 3622 y Fn(0)780 3610 y Fo(j)p Fr(")p Fo(j)865 3576 y Fp(k)q Fn(+)p Fv(j)p Fp(k)p 972 3589 37 4 v 2 w Fv(j)p 1032 3543 72 4 v 1032 3610 a Fr(D)1103 3557 y Fp(k)q Fn(+)p Fv(j)p Fp(k)p 1210 3570 37 4 v 3 w Fv(j)607 3717 y Fq(\020)699 3706 y Fv(1)679 3731 y Fq(Y)670 3909 y Fp(h)p Fn(=0)807 3810 y Fu(exp)948 3717 y Fq(h)987 3810 y Fr(h)14 b Fu(log)g(2)1212 3717 y Fq(\020)1261 3810 y Fu(4)p Fr(k)s Fu(2)1391 3775 y Fv(\000)p Fn(\()p Fp(h)p Fv(\000)p Fn(2\))p Fp(=\034)1711 3810 y Fo(\000)k Fr(C)1853 3822 y Fp(h)1897 3810 y Fu(\()p Fr(T)12 b Fu(\))18 b(+)g Fr(S)2174 3822 y Fp(h)2217 3810 y Fu(\()p Fr(\022)r Fu(\))h(+)f Fr(M)2514 3775 y Fp(\027)2505 3830 y(h)2555 3810 y Fu(\()p Fr(\022)r Fu(\))2660 3717 y Fq(\021i)q(\021)607 3996 y(\020)729 4009 y(Y)670 4191 y Fp(T)9 b Fv(2S)t Fn(\()p Fp(\022)r Fn(\))907 4088 y Fu(2)949 4054 y Fv(\000)p Fp(h)1040 4023 y Fi(\()p Fh(e)p Fi(\))1040 4075 y Fh(T)1117 4054 y Fn(+)p Fp(h)1207 4062 y Fh(T)1257 3996 y Fq(\021)o(\020)1398 3984 y Fv(1)1378 4009 y Fq(Y)1370 4188 y Fp(h)p Fn(=0)1507 4088 y Fu(2)1549 4054 y Fv(\000)p Fp(hM)1709 4029 y Fh(\027)1703 4071 y(h)1745 4054 y Fn(\()p Fp(\022)r Fn(\))1835 3996 y Fq(\021)1884 4088 y Fr(;)2956 3885 y Fu(\()p Fr(A)p Fu(3)p Fr(:)p Fu(2\))118 4386 y(where)27 b(lemma)h(2)f(has)g(b)r(een)h (used)g(in)f(order)g(to)g(b)r(ound)h(the)g(n)n(um)n(b)r(er)f(of)h (lines)g(on)f(scale)g Fr(h)p Fu(,)g(and)1419 4534 y Fq(Y)1360 4716 y Fp(T)9 b Fv(2S)t Fn(\()p Fp(\022)r Fn(\))1597 4613 y Fu(2)1639 4578 y Fv(\000)p Fp(h)1730 4548 y Fi(\()p Fh(e)p Fi(\))1730 4600 y Fh(T)1807 4578 y Fn(+)p Fp(h)1897 4586 y Fh(T)2956 4613 y Fu(\()p Fr(A)p Fu(3)p Fr(:)p Fu(3\))118 4911 y(is)27 b(due)h(to)f(the)h(non)n(trivial)e(action)g(of) i(the)f Fo(R)h Fu(op)r(erator)e(on)h(the)g(self-energy)f(graphs,)g (while)i(the)118 5017 y(factor)1449 5065 y Fv(1)1429 5090 y Fq(Y)1420 5269 y Fp(h)p Fn(=0)1557 5169 y Fu(2)1599 5135 y Fv(\000)p Fp(hM)1759 5110 y Fh(h)1753 5151 y(\027)1797 5135 y Fn(\()p Fp(\022)r Fn(\))2956 5169 y Fu(\()p Fr(A)p Fu(3)p Fr(:)p Fu(4\))1612 5443 y(25)p eop %%Page: 26 26 26 25 bop 118 555 a Fu(tak)n(es)30 b(in)n(to)h(accoun)n(t)f(the)h(2) 1004 525 y Fv(\000)p Fp(h)1129 555 y Fu(factors)f(asso)r(ciated)f(to)i (the)g Fr(\027)5 b Fu(-v)n(ertices)30 b(con)n(tributing)g(a)h(factor) 118 662 y Fr(\027)164 619 y Fn(\()p Fp(c)p Fn(\))159 687 y Fp(h;m)281 662 y Fu(.)37 b(Then)28 b(one)f(has)1035 751 y Fq(\020)1126 740 y Fv(1)1106 765 y Fq(Y)1098 943 y Fp(h)p Fn(=0)1235 844 y Fu(2)1277 809 y Fp(hS)1357 818 y Fh(h)1394 809 y Fn(\()p Fp(\022)r Fn(\))1484 751 y Fq(\021\020)1655 765 y(Y)1597 947 y Fp(T)9 b Fv(2S)t Fn(\()p Fp(\022)r Fn(\))1834 844 y Fu(2)1876 809 y Fv(\000)p Fp(h)1967 779 y Fi(\()p Fh(e)p Fi(\))1967 831 y Fh(T)2048 751 y Fq(\021)2120 844 y Fu(=)23 b(1)p Fr(;)1035 1048 y Fq(\020)1126 1036 y Fv(1)1106 1061 y Fq(Y)1098 1240 y Fp(h)p Fn(=0)1235 1140 y Fu(2)1277 1105 y Fv(\000)p Fp(hC)1416 1114 y Fh(h)1453 1105 y Fn(\()p Fp(\022)r Fn(\))1543 1048 y Fq(\021)o(\020)1714 1061 y(Y)1656 1243 y Fp(T)9 b Fv(2S)t Fn(\()p Fp(\022)r Fn(\))1893 1140 y Fu(2)1935 1105 y Fp(h)1974 1113 y Fh(T)2023 1048 y Fq(\021)2096 1140 y Fo(\024)22 b Fu(1)p Fr(;)2956 1007 y Fu(\()p Fr(A)p Fu(3)p Fr(:)p Fu(5\))118 1399 y(so)27 b(that)h(one)f(\014nds,)h(for)f(suitable)h(constan)n(ts)1598 1378 y(~)1578 1399 y Fr(D)1647 1411 y Fn(0)1712 1399 y Fu(and)1893 1378 y(~)1873 1399 y Fr(D)r Fu(,)275 1504 y Fq(X)179 1686 y Fp(k)p 179 1699 37 4 v 13 w Fn(:)11 b Fv(j)p Fp(k)p 277 1699 V 1 w Fv(j)19 b Fn(\014xed)600 1504 y Fq(X)505 1710 y Fp(\022)r Fv(2)p Fn(\002)635 1674 y Fi(\()p Fh(k)q(;k)p 708 1687 33 4 v 1 w Fi(\))p Fj(R)635 1721 y Fh(n;m)830 1583 y Fo(j)p Fu(V)-7 b(al\()p Fr(\022)r Fu(\))p Fo(j)24 b(\024)1232 1562 y Fu(~)1212 1583 y Fr(D)1281 1595 y Fn(0)1318 1583 y Fo(j)p Fr(")p Fo(j)1403 1549 y Fp(k)q Fn(+)p Fv(j)p Fp(k)p 1510 1562 37 4 v 2 w Fv(j)1590 1562 y Fu(~)1571 1583 y Fr(D)1642 1549 y Fp(k)q Fn(+)p Fv(j)p Fp(k)p 1749 1562 V 2 w Fv(j)1810 1583 y Fu(2)1852 1549 y Fn(\()p Fp(k)q Fn(+)p Fv(j)p Fp(k)p 1985 1562 V 2 w Fv(j)p Fn(\))2071 1583 y Fu(4)2113 1549 y Fp(k)2154 1583 y Fu(2)2196 1549 y Fv(j)p Fp(k)p 2216 1562 V 1 w Fv(j)2299 1583 y Fo(\024)e(j)p Fr(")p Fo(j)2471 1549 y Fp(k)q Fn(+)p Fv(j)p Fp(k)p 2578 1562 V 2 w Fv(j)2639 1583 y Fr(C)2704 1549 y Fp(k)q Fn(+)p Fv(j)p Fp(k)p 2811 1562 V 2 w Fv(j)2872 1583 y Fr(;)61 b Fu(\()p Fr(A)p Fu(3)p Fr(:)p Fu(6\))118 1913 y(where)28 b(2)401 1883 y Fp(k)q Fn(+)p Fv(j)p Fp(k)p 508 1896 V 2 w Fv(j)597 1913 y Fu(is)h(a)f(b)r(ound)h(on)f(the)h(n)n(um)n(b)r(er)f(of)h(trees)f (in)g(some)g(\002)2239 1867 y Fn(\()p Fp(k)q(;k)p 2321 1880 V 2 w Fn(\))2239 1922 y Fp(n;m)2417 1913 y Fu(with)h(\014xed)f Fo(j)p Fr(k)p 2831 1926 46 4 v 3 w Fo(j)p Fu(,)h(4)2994 1883 y Fp(k)3063 1913 y Fu(is)f(a)118 2019 y(b)r(ound)h(on)g(the)g (assignmen)n(ts)f(of)g(the)i(lab)r(els)e(\()p Fr(n)1654 2031 y Fp(v)1694 2019 y Fr(;)14 b(m)1804 2031 y Fp(v)1843 2019 y Fu(\))29 b(for)f(all)h(end-p)r(oin)n(ts)f Fr(v)s Fu(,)i(and)f(\014nally)f(2)3109 1989 y Fv(j)p Fp(k)p 3129 2002 37 4 v 1 w Fv(j)118 2125 y Fu(is)g(a)f(b)r(ound)h(on)f(the)h (lab)r(els)f Fr(c)c Fu(=)g Fr(a;)14 b(b)27 b Fu(for)g(all)h (self-energy)e(graphs.)189 2232 y(Hence,)i(for)f(\014xed)h(\()p Fr(n;)14 b(m)p Fu(\))27 b(one)h(has)822 2347 y Fq(X)835 2526 y Fp(k)q(;k)p 891 2539 V 1026 2347 a Fq(X)956 2553 y Fp(\022)r Fv(2)p Fn(\002)1086 2516 y Fi(\()p Fh(k)q(;k)p 1159 2529 33 4 v Fi(\))1086 2564 y Fh(n;m)1231 2426 y Fo(j)p Fu(V)-7 b(al\()p Fr(\022)r Fu(\))p Fo(j)24 b(\024)f Fr(D)1683 2438 y Fn(0)1719 2426 y Fo(j)p Fr(")p Fo(j)1804 2391 y Fn(\()p Fv(j)p Fp(n)p Fv(j)p Fn(+)p Fv(j)p Fp(m)p Fv(j)p Fn(\))p Fp(=)p Fn(2)2157 2426 y Fr(D)2228 2391 y Fv(j)p Fp(n)p Fv(j)p Fn(+)p Fv(j)p Fp(m)p Fv(j)2462 2426 y Fr(;)471 b Fu(\()p Fr(A)p Fu(3)p Fr(:)p Fu(7\))118 2729 y(as)27 b Fr(k)f Fo(\025)d Fu(min)p Fo(fj)p Fr(n)p Fo(j)p Fr(;)14 b Fo(j)p Fr(m)p Fo(jg)27 b Fu(and)g Fo(j)p Fr(k)p 1062 2742 46 4 v 3 w Fo(j)c(\025)g Fu(0,)k(so)g(that)h(\(5.6\))f (is)g(pro)n(v)n(ed.)833 2977 y Fw(App)s(endix)38 b(A4.)99 b(Pro)s(of)37 b(of)h(lemma)e(4)118 3154 y Fu(First)31 b(of)g(all)g(w)n(e)g(note)g(that)g(the)h(result)f(stated)g(in)g(lemma)g (2)g(still)g(holds,)h(with)g(no)f(c)n(hange,)g(if)118 3260 y(w)n(e)e(allo)n(w)f(an)n(y)h(in)n(teger)g(v)-5 b(alue)29 b(for)g(the)g(mo)r(de)h(lab)r(els)f(\()p Fr(n)1926 3272 y Fp(v)1966 3260 y Fr(;)14 b(m)2076 3272 y Fp(v)2115 3260 y Fu(\))30 b(of)f(the)h(end-p)r(oin)n(ts.)42 b(The)29 b(only)118 3367 y(di\013erence)c(is)f(that)h(the)g(induction)g(has)f (to)g(b)r(e)h(p)r(erformed)g(on)2130 3346 y Fq(e)2110 3367 y Fr(K)5 b Fu(\()p Fr(\022)r Fu(\),)27 b(as)c(it)j(is)e(de\014ned) h(in)g(\(5.10\))118 3473 y(for)33 b Fr(E)312 3485 y Fn(0)349 3473 y Fu(\()p Fr(\022)r Fu(\))g(=)f Fo(;)p Fu(.)54 b(So)33 b(w)n(e)f(can)h(assume)g(that)g(the)h(b)r(ound)f(\(A2.1\))g(holds)g (for)g(an)n(y)f(tree)h Fr(\022)j Fu(with)118 3579 y Fr(E)179 3591 y Fn(0)217 3579 y Fu(\()p Fr(\022)r Fu(\))24 b(=)e Fo(;)p Fu(.)189 3686 y(Then)28 b(w)n(e)f(ha)n(v)n(e)g(to)g(sho)n(w)g (that)h(the)g(b)r(ound)g(\(5.9\))g(holds)f(for)g(trees)g(with)h Fr(E)2598 3698 y Fn(0)2636 3686 y Fu(\()p Fr(\022)r Fu(\))c(=)f Fo(f)p Fr(v)2935 3698 y Fn(0)2972 3686 y Fo(g)g(6)p Fu(=)f Fo(;)p Fu(.)118 3792 y(W)-7 b(e)28 b(mimic)g(the)g(pro)r(of)f(of)h (lemma)f(2)g(\(hence)h(of)g(lemma)f(1\):)37 b(w)n(e)27 b(pro)n(v)n(e)f(the)i(b)r(ound)989 3976 y Fr(N)1065 3942 y Fv(\003)1056 3997 y Fp(h)1103 3976 y Fu(\()p Fr(\022)r Fu(\))c Fo(\024)f Fu(max)o Fo(f)p Fu(0)p Fr(;)14 b Fu(2)1655 3955 y Fq(e)1637 3976 y Fr(K)t Fu(\()p Fr(\022)r Fu(\)2)1859 3942 y Fn(\(2)p Fv(\000)p Fp(h)p Fn(\))p Fp(=\034)2129 3976 y Fo(\000)k Fu(1)p Fo(g)p Fr(;)637 b Fu(\()p Fr(A)p Fu(4)p Fr(:)p Fu(1\))118 4160 y(for)27 b(all)g(trees)h Fr(\022)h Fu(with)g Fr(E)879 4172 y Fn(0)916 4160 y Fu(\()p Fr(\022)r Fu(\))24 b Fo(6)p Fu(=)f Fo(;)p Fu(,)k(again)f(b)n(y)i (induction)g(on)2066 4139 y Fq(e)2046 4160 y Fr(K)5 b Fu(\()p Fr(\022)r Fu(\).)189 4266 y(F)-7 b(or)30 b(an)n(y)g(line)h Fr(`)e Fo(2)f Fr(L)p Fu(\()p Fr(\022)r Fu(\))k(set)f Fr(\021)1176 4278 y Fp(`)1236 4266 y Fu(=)d(1)j(if)g(the)g(line)g(is)g (along)f(the)h(path)g(connecting)f Fr(v)2900 4278 y Fn(0)2969 4266 y Fu(to)h(the)118 4373 y(ro)r(ot)c(and)g Fr(\021)498 4385 y Fp(`)554 4373 y Fu(=)22 b(0)27 b(otherwise,)g(and)h(write)713 4557 y Fr(n)763 4569 y Fp(`)818 4557 y Fu(=)22 b Fr(n)955 4523 y Fn(0)955 4577 y Fp(`)1011 4557 y Fu(+)c Fr(\021)1135 4569 y Fp(`)1181 4557 y Fu(\()p 1213 4511 50 4 v Fr(n)g Fu(+)p 1365 4511 55 4 v 19 w Fr(!)1419 4569 y Fp(m)1482 4557 y Fu(\))c Fr(;)180 b(m)1804 4569 y Fp(`)1859 4557 y Fu(=)23 b Fr(m)2020 4523 y Fn(0)2020 4577 y Fp(`)2075 4557 y Fu(+)18 b Fr(\021)2199 4569 y Fp(`)2245 4557 y Fu(\()p 2278 4511 73 4 v 1 w Fr(m)g Fu(+)g Fr(m)p Fu(\))c Fr(;)362 b Fu(\()p Fr(A)p Fu(4)p Fr(:)p Fu(2\))118 4741 y(whic)n(h)28 b(implicitly)g(de\014nes)g Fr(n)1050 4711 y Fn(0)1050 4765 y Fp(`)1114 4741 y Fu(and)g Fr(m)1349 4711 y Fn(0)1349 4765 y Fp(`)1386 4741 y Fu(.)189 4847 y(De\014ne)h Fr(k)490 4859 y Fn(0)551 4847 y Fu(=)24 b(2)682 4817 y Fn(\()p Fp(h)p Fv(\000)p Fn(1\))p Fp(=\034)933 4847 y Fu(.)39 b(One)28 b(has)f Fr(N)1395 4817 y Fv(\003)1386 4871 y Fp(h)1433 4847 y Fu(\()p Fr(\022)r Fu(\))e(=)f(0)k(for)1869 4826 y Fq(e)1849 4847 y Fr(K)6 b Fu(\()p Fr(\022)r Fu(\))25 b Fr(<)f(k)2188 4859 y Fn(0)2225 4847 y Fu(,)29 b(b)r(ecause)e(if)i(a)f (line)2894 4825 y(\026)2888 4847 y Fr(`)c Fo(2)h Fr(L)p Fu(\()p Fr(\022)r Fu(\))118 4954 y(is)g(indeed)h(on)f(scale)g Fr(h)g Fu(then)h Fo(j)7 b Fu(~)-49 b Fr(!)1105 4966 y Fn(1)1142 4954 y Fr(n)1196 4959 y Fn(\026)1192 4974 y Fp(`)1238 4954 y Fo(\000)20 b Fu(~)-48 b Fr(!)1369 4966 y Fp(m)1432 4970 y Fi(\026)1428 4981 y Fh(`)1460 4954 y Fo(j)23 b Fr(<)g(C)1653 4966 y Fn(0)1691 4954 y Fu(2)1733 4924 y Fn(1)p Fv(\000)p Fp(h)1860 4954 y Fu(,)j(so)f(that)h(\(A4.2\))f (and)g(the)h(Diophan)n(tine)118 5060 y(conditions)h(\(1.15\))g(imply) 1035 5223 y Fq(e)1015 5244 y Fr(K)5 b Fu(\()p Fr(\022)r Fu(\))24 b Fo(\025)1308 5174 y Fq(\014)1308 5223 y(\014)1335 5244 y Fr(n)1385 5210 y Fn(0)1389 5256 y(\026)1385 5272 y Fp(`)1441 5244 y Fu(+)18 b Fr(\021)1569 5249 y Fn(\026)1565 5264 y Fp(`)p 1597 5198 50 4 v 1597 5244 a Fr(n)1647 5174 y Fq(\014)1647 5223 y(\014)1698 5244 y Fr(>)k Fu(2)1827 5210 y Fn(\()p Fp(h)p Fv(\000)p Fn(1\))p Fp(=\034)2101 5244 y Fo(\021)h Fr(k)2232 5256 y Fn(0)2269 5244 y Fr(:)664 b Fu(\()p Fr(A)p Fu(4)p Fr(:)p Fu(3\))1612 5443 y(26)p eop %%Page: 27 27 27 26 bop 118 555 a Fu(Then,)32 b(for)e Fr(K)k Fo(\025)29 b Fr(k)734 567 y Fn(0)771 555 y Fu(,)j(w)n(e)f(assume)f(that)h(the)g(b) r(ound)h(\(A4.1\))e(holds)h(for)f(all)2584 534 y Fq(e)2564 555 y Fr(K)5 b Fu(\()p Fr(\022)r Fu(\))30 b(=)e Fr(K)2945 525 y Fv(0)2996 555 y Fr(<)g(K)6 b Fu(,)118 662 y(and)28 b(w)n(e)f(sho)n(w)f(that)i(it)g(follo)n(ws)f(also)f(for)1456 641 y Fq(e)1437 662 y Fr(K)5 b Fu(\()p Fr(\022)r Fu(\))24 b(=)f Fr(K)6 b Fu(.)189 768 y(If)25 b(the)g(ro)r(ot)f(line)h Fr(`)g Fu(of)g Fr(\022)i Fu(is)e(either)f(on)h(scale)f Fr(<)e(h)j Fu(or)f(on)h(scale)f Fo(\025)e Fr(h)j Fu(and)g(resonan)n(t,) e(the)j(b)r(ound)118 874 y(\(A4.1\))d(follo)n(ws)e(immediately)i(from)f (the)h(b)r(ound)g(\(A2.1\))f(and)h(from)f(the)h(inductiv)n(e)f(h)n(yp)r (othesis.)189 981 y(The)30 b(same)f(o)r(ccurs)f(if)i(the)g(ro)r(ot)f (line)h(is)f(on)h(scale)f Fo(\025)c Fr(h)30 b Fu(and)f(non-resonan)n (t,)f(and,)i(b)n(y)g(calling)118 1087 y Fr(`)153 1099 y Fn(1)190 1087 y Fr(;)14 b(:)g(:)g(:)g(;)g(`)410 1099 y Fp(m)500 1087 y Fu(the)29 b(lines)f(on)g(scale)f Fo(\025)d Fr(h)k Fu(whic)n(h)g(are)g(the)g(closest)g(to)g Fr(`)p Fu(,)g(one)g(has)f Fr(m)d Fo(\025)g Fu(2.)38 b(If)29 b Fr(m)24 b Fu(=)f(0)118 1193 y(then)28 b Fr(N)383 1163 y Fv(\003)374 1217 y Fp(h)421 1193 y Fu(\()p Fr(\022)r Fu(\))c(=)f(1)k(and)888 1172 y Fq(e)868 1193 y Fr(K)6 b Fu(\()p Fr(\022)r Fu(\)2)1092 1163 y Fn(\(2)p Fv(\000)p Fp(h)p Fn(\))p Fp(=\034)1366 1193 y Fo(\025)23 b Fu(1)k(b)r(ecause)g (one)h(m)n(ust)f(ha)n(v)n(e)2400 1172 y Fq(e)2380 1193 y Fr(K)5 b Fu(\()p Fr(\022)r Fu(\))24 b Fo(\025)f Fr(k)2716 1205 y Fn(0)2753 1193 y Fu(.)189 1300 y(So)36 b(the)h(only)f (non-trivial)f(case)h(is)g(when,)j(in)e(the)g(latter)f(case,)i(one)e (has)g Fr(m)h Fu(=)h(1.)63 b(If)37 b(this)118 1406 y(happ)r(ens)31 b Fr(`)482 1418 y Fn(1)551 1406 y Fu(is,)h(b)n(y)f(construction,)g(the) h(ro)r(ot)e(line)i(of)f(a)g(tree)g Fr(\022)2153 1418 y Fn(1)2222 1406 y Fu(suc)n(h)f(that)i Fr(E)2657 1418 y Fn(0)2695 1406 y Fu(\()p Fr(\022)2766 1418 y Fn(1)2803 1406 y Fu(\))e Fo(6)p Fu(=)e Fo(;)p Fu(,)k(and)138 1491 y Fq(e)118 1512 y Fr(K)6 b Fu(\()p Fr(\022)r Fu(\))32 b(=)448 1491 y Fq(e)428 1512 y Fr(K)6 b Fu(\()p Fr(T)12 b Fu(\))21 b(+)757 1491 y Fq(e)737 1512 y Fr(K)6 b Fu(\()p Fr(\022)885 1524 y Fn(1)922 1512 y Fu(\),)35 b(where)d Fr(T)44 b Fu(is)32 b(the)i(cluster)e(whic)n(h)g(has)h Fr(`)f Fu(and)h Fr(`)2525 1524 y Fn(1)2594 1512 y Fu(as)f(external)g (lines)118 1618 y(and)c(w)n(e)f(ha)n(v)n(e)f(de\014ned)1202 1704 y Fq(e)1183 1725 y Fr(K)5 b Fu(\()p Fr(T)12 b Fu(\))23 b Fo(\021)1663 1646 y Fq(X)1494 1828 y Fp(v)r Fv(2)p Fp(E)s Fn(\()p Fp(T)9 b Fn(\))p Fv([)p Fp(E)1820 1836 y Fi(1)1852 1828 y Fn(\()p Fp(T)g Fn(\))1966 1725 y Fo(j)p Fr(n)2039 1737 y Fp(v)2078 1725 y Fo(j)p Fr(;)832 b Fu(\()p Fr(A)p Fu(4)p Fr(:)p Fu(4\))118 1993 y(whic)n(h)28 b(satis\014es)e(the) i(b)r(ound)1080 1972 y Fq(e)1060 1993 y Fr(K)6 b Fu(\()p Fr(T)12 b Fu(\))23 b Fo(\025)f(j)p Fr(n)1445 2005 y Fp(`)1473 2013 y Fi(1)1528 1993 y Fo(\000)c Fr(n)1661 2005 y Fp(`)1693 1993 y Fo(j)p Fu(.)189 2099 y(Moreo)n(v)n(er)25 b(one)i(has)905 2208 y Fq(\014)905 2258 y(\014)932 2279 y Fo(j)7 b Fu(~)-49 b Fr(!)1007 2291 y Fn(1)1044 2279 y Fr(n)1094 2244 y Fn(0)1094 2299 y Fp(`)1150 2279 y Fu(+)24 b(~)-48 b Fr(!)1285 2291 y Fn(1)p 1322 2233 50 4 v 1322 2279 a Fr(n)18 b Fu(+)25 b(~)-49 b Fr(!)1525 2291 y Fp(m)1588 2279 y Fo(j)18 b(\000)25 b Fu(~)-49 b Fr(!)1764 2291 y Fp(m)1823 2300 y Fh(`)1856 2208 y Fq(\014)1856 2258 y(\014)1907 2279 y Fo(\024)22 b Fu(2)2036 2244 y Fv(\000)p Fp(h)2127 2219 y Fj(0)2149 2244 y Fn(+1)2237 2279 y Fr(C)2296 2291 y Fn(0)2334 2279 y Fr(;)905 2353 y Fq(\014)905 2403 y(\014)932 2423 y Fo(j)7 b Fu(~)-49 b Fr(!)1007 2435 y Fn(1)1044 2423 y Fr(n)1094 2389 y Fn(0)1094 2444 y Fp(`)1122 2452 y Fi(1)1177 2423 y Fu(+)24 b(~)-48 b Fr(!)1312 2435 y Fn(1)p 1349 2378 V 1349 2423 a Fr(n)18 b Fu(+)25 b(~)-49 b Fr(!)1552 2435 y Fp(m)1615 2423 y Fo(j)18 b(\000)25 b Fu(~)-49 b Fr(!)1791 2435 y Fp(m)1850 2444 y Fh(`)1875 2456 y Fi(1)1915 2353 y Fq(\014)1915 2403 y(\014)1966 2423 y Fo(\024)23 b Fu(2)2096 2389 y Fv(\000)p Fp(h)2187 2364 y Fj(0)2209 2389 y Fn(+1)2297 2423 y Fr(C)2356 2435 y Fn(0)2393 2423 y Fr(;)2956 2347 y Fu(\()p Fr(A)p Fu(4)p Fr(:)p Fu(5\))118 2616 y(so)k(that,)h(for)f(suitable)h Fr(\021)901 2628 y Fp(`)933 2616 y Fr(;)14 b(\021)1011 2628 y Fp(`)1039 2636 y Fi(1)1098 2616 y Fo(2)24 b(f\000)p Fr(;)14 b Fu(+)p Fo(g)p Fu(,)25 b(w)n(e)j(obtain)822 2816 y(2)864 2782 y Fv(\000)p Fp(h)955 2757 y Fj(0)977 2782 y Fn(+2)1065 2816 y Fr(C)1124 2828 y Fn(0)1185 2816 y Fo(\025)1272 2746 y Fq(\014)1272 2796 y(\014)1306 2816 y Fu(~)-48 b Fr(!)1352 2828 y Fn(1)1389 2816 y Fu(\()p Fr(n)1471 2782 y Fn(0)1471 2837 y Fp(`)1527 2816 y Fo(\000)18 b Fr(n)1660 2782 y Fn(0)1660 2837 y Fp(`)1688 2845 y Fi(1)1724 2816 y Fu(\))g(+)h Fr(\021)1899 2828 y Fp(`)1937 2816 y Fu(~)-48 b Fr(!)1983 2828 y Fp(m)2042 2837 y Fh(`)2093 2816 y Fu(+)18 b Fr(\021)2217 2828 y Fp(`)2245 2836 y Fi(1)2288 2816 y Fu(~)-49 b Fr(!)2333 2828 y Fp(m)2392 2837 y Fh(`)2417 2849 y Fi(1)2457 2746 y Fq(\014)2457 2796 y(\014)1185 2948 y Fo(\025)22 b Fr(C)1331 2960 y Fn(0)1369 2948 y Fo(j)p Fr(n)1442 2913 y Fn(0)1442 2968 y Fp(`)1497 2948 y Fo(\000)c Fr(n)1630 2913 y Fn(0)1630 2968 y Fp(`)1658 2976 y Fi(1)1695 2948 y Fo(j)1718 2913 y Fv(\000)p Fp(\034)1834 2948 y Fo(\021)23 b Fr(C)1981 2960 y Fn(0)2018 2948 y Fo(j)p Fr(n)2091 2960 y Fp(`)2142 2948 y Fo(\000)18 b Fr(n)2275 2960 y Fp(`)2303 2968 y Fi(1)2339 2948 y Fo(j)2362 2913 y Fv(\000)p Fp(\034)2455 2948 y Fr(;)2956 2876 y Fu(\()p Fr(A)p Fu(4)p Fr(:)p Fu(6\))118 3136 y(b)n(y)24 b(the)g(second)g(Diophan)n(tine)g (conditions)f(in)i(\(1.15\),)f(as)f(the)i(quan)n(tities)30 b(~)-49 b Fr(!)2518 3148 y Fn(1)p 2555 3090 V 2555 3136 a Fr(n)11 b Fu(+)18 b(~)-49 b Fr(!)2744 3148 y Fp(m)2831 3136 y Fu(app)r(earing)118 3242 y(in)28 b(\(A4.4\))g(cancel)f(out.)37 b(Therefore)26 b(one)h(obtains)632 3437 y Fr(N)708 3403 y Fv(\003)699 3457 y Fp(h)746 3437 y Fu(\()p Fr(\022)r Fu(\))d Fo(\024)f Fu(1)18 b(+)1126 3416 y Fq(e)1106 3437 y Fr(K)5 b Fu(\()p Fr(\022)1253 3449 y Fn(1)1291 3437 y Fu(\))p Fr(E)1389 3401 y Fv(\000)p Fn(1)1384 3462 y Fp(h)1497 3437 y Fo(\000)1600 3416 y Fq(e)1580 3437 y Fr(K)g Fu(\()p Fr(\022)r Fu(\))p Fr(E)1827 3401 y Fv(\000)p Fn(1)1822 3462 y Fp(h)1936 3437 y Fo(\000)18 b Fu(1)k Fo(\024)2191 3416 y Fq(e)2171 3437 y Fr(K)5 b Fu(\()p Fr(\022)r Fu(\))p Fr(E)2418 3401 y Fv(\000)p Fn(1)2413 3462 y Fp(h)2527 3437 y Fo(\000)18 b Fu(1)p Fr(;)281 b Fu(\()p Fr(A)p Fu(4)p Fr(:)p Fu(7\))118 3632 y(b)n(y)27 b(using)h(the)g(same)f(notations)f(as)h(in)h(\(A1.6\),)g(hence)g(the)g (\014rst)f(b)r(ound)h(in)g(\(A4.1\))f(is)h(pro)n(v)n(ed.)189 3738 y(Analogously)e(one)h(pro)n(v)n(es)f(the)i(b)r(ound)995 3933 y Fr(C)1054 3945 y Fp(h)1097 3933 y Fu(\()p Fr(\022)r Fu(\))c Fo(\024)f Fu(max)o Fo(f)p Fu(0)p Fr(;)14 b Fu(2)1649 3912 y Fq(e)1631 3933 y Fr(K)t Fu(\()p Fr(\022)r Fu(\)2)1853 3899 y Fn(\(2)p Fv(\000)p Fp(h)p Fn(\))p Fp(=\034)2123 3933 y Fo(\000)k Fu(1)p Fo(g)p Fr(;)643 b Fu(\()p Fr(A)p Fu(4)p Fr(:)p Fu(8\))118 4128 y(so)27 b(that)h(\(5.9\))f(follo)n(ws.) 833 4376 y Fw(App)s(endix)38 b(A5.)99 b(Pro)s(of)37 b(of)h(lemma)e(5) 118 4553 y Fu(Consider)29 b(a)g(self-energy)g(graph)f Fr(T)38 b Fo(2)27 b(T)1425 4510 y Fn(\()p Fp(c)p Fn(\))1403 4578 y Fp(h)1510 4553 y Fu(.)44 b(W)-7 b(e)30 b(can)f(consider)g(the)h (tree)g Fr(\022)i Fu(obtained)d(from)h Fr(T)118 4659 y Fu(b)n(y)g(adding)f(to)h(it)h(the)f(outcoming)g(line)g Fr(`)1448 4629 y Fn(1)1448 4682 y Fp(T)1530 4659 y Fu(and)g(replacing)f (the)h(en)n(tering)f(line)i Fr(`)2715 4629 y Fn(2)2715 4682 y Fp(T)2797 4659 y Fu(with)f(a)g(line)118 4766 y(emerging)f(from)g (an)h(end-p)r(oin)n(t)f Fr(v)1207 4778 y Fn(0)1275 4766 y Fu(whic)n(h)g(carries)f(a)i(mo)r(de)g(lab)r(el)f(\()7 b(~)-49 b Fr(!)2364 4778 y Fp(m)2427 4766 y Fr(;)14 b(m)p Fu(\);)31 b(b)n(y)f(construction)118 4872 y(one)d(has)1266 4978 y Fo(V)1324 4944 y Fp(h)1317 4999 y(T)1369 4978 y Fu(\()7 b(~)-49 b Fr(!)1453 4990 y Fp(m)1516 4978 y Fr(;)14 b(m)p Fu(\))23 b(=)g(V)-7 b(al)1889 4941 y Fv(0)1912 4978 y Fu(\()p Fr(\022)r Fu(\))p Fr(;)916 b Fu(\()p Fr(A)p Fu(5)p Fr(:)p Fu(1\))118 5138 y(where)27 b(V)-7 b(al)478 5101 y Fv(0)501 5138 y Fu(\()p Fr(\022)r Fu(\))29 b(di\013ers)e(from)g (\(4.5\))g(as)g(in)h(the)g(\014rst)f(pro)r(duct)g(the)h(line)g Fr(`)2435 5150 y Fn(1)2499 5138 y Fu(coming)f(out)h(from)f Fr(T)118 5244 y Fu(is)h(missing,)f(and)g Fr(E)743 5256 y Fn(0)781 5244 y Fu(\()p Fr(\022)r Fu(\))d(=)e Fo(f)p Fr(v)1079 5256 y Fn(0)1116 5244 y Fo(g)p Fu(.)1612 5443 y(27)p eop %%Page: 28 28 28 27 bop 189 555 a Fu(W)-7 b(e)28 b(w)n(an)n(t)f(to)g(pro)n(v)n(e)f (that)i(one)f(has)950 769 y Fr(N)1026 735 y Fv(\003)1017 790 y Fp(h)1056 773 y Fj(0)1082 769 y Fu(\()p Fr(T)12 b Fu(\))23 b Fo(\024)f Fu(max)p Fo(f)p Fu(0)p Fr(;)14 b Fu(2)p Fr(K)6 b Fu(\()p Fr(T)12 b Fu(\)2)1879 735 y Fn(\(2)p Fv(\000)p Fp(h)2029 710 y Fj(0)2048 735 y Fn(\))p Fp(=\034)2168 769 y Fo(\000)18 b Fu(1)p Fo(g)p Fr(;)598 b Fu(\()p Fr(A)p Fu(5)p Fr(:)p Fu(2\))118 983 y(if)28 b Fr(N)270 953 y Fv(\003)261 1006 y Fp(h)300 990 y Fj(0)326 983 y Fu(\()p Fr(T)12 b Fu(\))28 b(denotes)f(the)h(n)n(um)n(b)r(er)f (of)h(non-resonan)n(t)d(lines)j(on)f(scale)g Fo(\024)c Fr(h)2455 953 y Fv(0)2505 983 y Fu(in)n(ternal)k(to)h Fr(T)12 b Fu(.)189 1092 y(Let)37 b Fr(v)j Fu(b)r(e)e(the)f(no)r(de)g (suc)n(h)f(that)h Fr(`)1332 1062 y Fn(1)1332 1115 y Fp(T)1421 1092 y Fu(comes)g(out)f(from)h Fr(v)s Fu(,)j(and)c(call)h Fr(`)2511 1104 y Fn(1)2548 1092 y Fr(;)14 b(:)g(:)g(:)f(;)h(`)2767 1104 y Fp(s)2798 1112 y Fh(v)2875 1092 y Fu(the)37 b(lines)118 1198 y(en)n(tering)29 b Fr(v)s Fu(;)j(denote)e(with)g Fr(\022)1039 1210 y Fn(1)1076 1198 y Fr(;)14 b(:)g(:)g(:)g(;)g(\022) 1300 1210 y Fp(s)1331 1218 y Fh(v)1401 1198 y Fu(the)30 b(trees)f(whic)n(h)h(ha)n(v)n(e)f Fr(`)2216 1210 y Fn(1)2253 1198 y Fr(;)14 b(:)g(:)g(:)f(;)h(`)2472 1210 y Fp(s)2503 1218 y Fh(v)2573 1198 y Fu(as)29 b(ro)r(ot)g(lines.)44 b(By)118 1304 y(construction)27 b(one)g(has)1256 1458 y Fr(N)1332 1424 y Fv(\003)1323 1479 y Fp(h)1362 1462 y Fj(0)1388 1458 y Fu(\()p Fr(T)12 b Fu(\))23 b(=)1650 1354 y Fp(s)1681 1362 y Fh(v)1624 1380 y Fq(X)1626 1556 y Fp(j)s Fn(=1)1757 1458 y Fr(N)1833 1424 y Fv(\003)1824 1479 y Fp(h)1863 1462 y Fj(0)1889 1458 y Fu(\()p Fr(\022)1960 1470 y Fp(j)1996 1458 y Fu(\))p Fr(;)905 b Fu(\()p Fr(A)p Fu(5)p Fr(:)p Fu(3\))118 1712 y(and)23 b(\(only\))f(one)g(of)h(the)g (subtrees)f Fr(\022)1249 1724 y Fp(j)1284 1712 y Fu(,)i(sa)n(y)d Fr(\022)1508 1724 y Fn(1)1545 1712 y Fu(,)j(con)n(tains)e(the)h(end-p)r (oin)n(t)f Fr(v)2459 1724 y Fn(0)2497 1712 y Fu(.)35 b(Therefore)21 b(in)i(\(5.7\))118 1819 y(w)n(e)k(can)h(b)r(ound)836 1868 y Fp(s)867 1876 y Fh(v)809 1894 y Fq(X)812 2071 y Fp(j)s Fn(=2)943 1973 y Fr(N)1019 1938 y Fv(\003)1010 1993 y Fp(h)1049 1977 y Fj(0)1075 1973 y Fu(\()p Fr(T)12 b Fu(\))23 b Fo(\024)1337 1868 y Fp(s)1368 1876 y Fh(v)1310 1894 y Fq(X)1313 2071 y Fp(j)s Fn(=2)1444 1973 y Fu(max)p Fo(f)p Fu(0)p Fr(;)14 b Fu(2)p Fr(K)6 b Fu(\()p Fr(\022)1910 1985 y Fp(j)1943 1973 y Fu(\)2)2017 1938 y Fn(\(2)p Fv(\000)p Fp(h)2167 1913 y Fj(0)2189 1938 y Fn(\))p Fp(=\034)2309 1973 y Fo(\000)18 b Fu(1)p Fo(g)p Fr(;)457 b Fu(\()p Fr(A)p Fu(5)p Fr(:)p Fu(4\))118 2227 y(b)n(y)27 b(using)h(\(A1.1\),)f (and)h(w)n(e)f(can)g(b)r(ound)1112 2440 y Fr(N)1179 2452 y Fp(h)1218 2436 y Fj(0)1245 2440 y Fu(\()p Fr(\022)1316 2452 y Fn(1)1353 2440 y Fu(\))c Fo(\024)g Fr(K)6 b Fu(\()p Fr(\022)1644 2452 y Fn(1)1681 2440 y Fu(\)2)1755 2406 y Fn(\(2)p Fv(\000)p Fp(h)1905 2381 y Fj(0)1927 2406 y Fn(\))p Fp(=\034)2047 2440 y Fo(\000)18 b Fu(1)p Fr(;)761 b Fu(\()p Fr(A)p Fu(5)p Fr(:)p Fu(5\))118 2654 y(b)n(y)27 b(using)h(\(5.9\))f(\(with)p 870 2608 50 4 v 28 w Fr(n)c Fu(=)g(0\).)189 2763 y(Analogously)j(one)h(can)g(pro)n(v)n(e)f(that)i (one)f(has)947 2977 y Fr(C)1006 2989 y Fp(h)1045 2973 y Fj(0)1071 2977 y Fu(\()p Fr(T)12 b Fu(\))23 b Fo(\024)g Fu(max)o Fo(f)p Fu(0)p Fr(;)14 b Fu(2)p Fr(K)6 b Fu(\()p Fr(\022)1772 2989 y Fp(j)1805 2977 y Fu(\)2)1879 2942 y Fn(\(2)p Fv(\000)p Fp(h)2029 2917 y Fj(0)2051 2942 y Fn(\))p Fp(=\034)2171 2977 y Fo(\000)18 b Fu(1)p Fo(g)p Fr(;)595 b Fu(\()p Fr(A)p Fu(5)p Fr(:)p Fu(6\))118 3190 y(if)28 b Fr(C)253 3202 y Fp(h)292 3186 y Fj(0)319 3190 y Fu(\()p Fr(T)12 b Fu(\))27 b(denotes)h(the)g(n)n(um)n(b)r(er)f(of)g (clusters)g(on)h(scale)f Fo(\024)22 b Fr(h)2062 3160 y Fv(0)2113 3190 y Fu(in)n(ternal)27 b(to)g Fr(T)12 b Fu(.)189 3299 y(Then)28 b(the)g(b)r(ound)g(\(A4.7\))f(follo)n(ws.)833 3547 y Fw(App)s(endix)38 b(A6.)99 b(Pro)s(of)37 b(of)h(lemma)e(6)118 3727 y Fu(By)29 b(using)h(lemma)f(5)g(and)h(the)g(cancellations)e (discussed)h(in)h(App)r(endix)g(A3)g(w)n(e)f(obtain)g(for)g(all)118 3833 y Fr(T)34 b Fo(2)24 b(T)346 3790 y Fn(\()p Fp(c)p Fn(\))325 3858 y Fp(h)362 3976 y Fq(\014)362 4026 y(\014)389 4047 y Fo(V)447 4012 y Fp(h)p Fn(+1)440 4071 y Fp(T)574 4047 y Fu(\()7 b(~)-49 b Fr(!)658 4059 y Fp(m)721 4047 y Fr(;)14 b(m)p Fu(\))863 3976 y Fq(\014)863 4026 y(\014)914 4047 y Fo(\024)22 b Fr(C)1066 4013 y Fp(k)1108 4047 y Fo(j)p Fr(")p Fo(j)1193 4013 y Fp(K)t Fn(\()p Fp(T)9 b Fn(\)+)p Fv(j)p Fp(K)p 1424 4026 60 4 v 3 w Fn(\()p Fp(T)g Fn(\))p Fv(j)902 4143 y(1)882 4168 y Fq(Y)860 4346 y Fp(h)899 4330 y Fj(0)921 4346 y Fn(=)p Fp(h)1025 4246 y Fu(exp)1165 4154 y Fq(h)1205 4246 y Fu(4)p Fr(k)16 b Fu(log)e(2)p Fr(h)1517 4212 y Fv(0)1539 4246 y Fu(2)1581 4212 y Fn(\(2)p Fv(\000)p Fp(h)1731 4187 y Fj(0)1753 4212 y Fn(\))p Fp(=\034)1873 4246 y Fo(\000)k Fr(C)2015 4258 y Fp(h)2054 4242 y Fj(0)2081 4246 y Fu(\()p Fr(T)12 b Fu(\))18 b(+)g Fr(S)2358 4258 y Fp(h)2397 4242 y Fj(0)2423 4246 y Fu(\()p Fr(T)12 b Fu(\))18 b(+)h Fr(M)2740 4212 y Fp(\027)2731 4267 y(h)2770 4250 y Fj(0)2796 4246 y Fu(\()p Fr(T)12 b Fu(\))2921 4154 y Fq(i)846 4433 y(\020)942 4446 y(Y)909 4624 y Fp(T)957 4607 y Fj(0)980 4624 y Fv(\032)p Fp(T)1094 4525 y Fu(2)1136 4491 y Fv(\000)p Fp(h)1227 4460 y Fi(\()p Fh(e)p Fi(\))1227 4512 y Fh(T)1303 4491 y Fn(+)p Fp(h)1393 4499 y Fh(T)1443 4433 y Fq(\021\020)1599 4421 y Fv(1)1579 4446 y Fq(Y)1556 4625 y Fp(h)1595 4608 y Fj(0)1617 4625 y Fn(=)p Fp(h)1721 4525 y Fu(2)1763 4491 y Fv(\000)p Fp(h)1854 4466 y Fj(0)1876 4491 y Fp(M)1945 4466 y Fh(\027)1939 4514 y(h)1973 4502 y Fj(0)2000 4491 y Fn(\()p Fp(T)d Fn(\))2104 4433 y Fq(\021)2154 4525 y Fr(:)779 b Fu(\()p Fr(A)p Fu(6)p Fr(:)p Fu(1\))118 4810 y(The)21 b(main)g(di\013erence)f(with)i(resp)r(ect)e(to)h(the)g (previous)f(case)f(is)i(that,)i(giv)n(en)c(a)i(self-energy)e(graph)118 4916 y Fr(T)34 b Fo(2)24 b(T)346 4873 y Fn(\()p Fp(c)p Fn(\))325 4941 y Fp(h)432 4916 y Fu(,)j(there)h(is)f(at)h(least)f(a)g (line)h Fr(`)23 b Fo(2)g Fr(L)p Fu(\()p Fr(T)12 b Fu(\))27 b(on)g(scale)g Fr(h)2007 4928 y Fp(`)2062 4916 y Fu(=)22 b Fr(h)28 b Fu(and)f(with)h(propagator)1621 5111 y(1)p 1112 5148 1060 4 v 1112 5224 a Fo(\000)7 b Fu(~)-49 b Fr(!)1232 5195 y Fn(2)1229 5246 y(1)1269 5224 y Fu(\()p Fr(n)1351 5195 y Fn(0)1351 5249 y Fp(`)1407 5224 y Fu(+)18 b Fr(\021)1531 5236 y Fp(`)p 1563 5178 55 4 v 1563 5224 a Fr(!)1617 5236 y Fp(m)1681 5224 y Fu(\))1713 5200 y Fn(2)1769 5224 y Fu(+)24 b(~)-48 b Fr(!)1907 5195 y Fn(2)1904 5255 y Fp(m)1963 5235 y Fi(0)1963 5277 y Fh(`)1995 5255 y Fn(+)p Fp(\021)2080 5264 y Fh(`)2109 5255 y Fp(m)2182 5167 y Fr(;)751 b Fu(\()p Fr(A)p Fu(6)p Fr(:)p Fu(2\))1612 5443 y(28)p eop %%Page: 29 29 29 28 bop 118 555 a Fu(where)33 b Fr(\021)405 567 y Fp(`)471 555 y Fu(=)f(1)i(if)g(the)g(line)g Fr(`)f Fu(b)r(elongs)g(to)h(the)g (path)f(of)h(lines)g(connecting)f(the)h(en)n(tering)f(line)118 662 y(\(carrying)39 b(a)h(momen)n(tum)h(\()p Fr(n;)14 b(m)p Fu(\)\))41 b(of)f Fr(T)52 b Fu(with)41 b(the)g(line)g(coming)e (out)i(of)f Fr(T)12 b Fu(,)43 b(and)e Fr(\021)2962 674 y Fp(`)3038 662 y Fu(=)j(0)118 768 y(otherwise.)36 b(Then)28 b(b)n(y)f(\(1.15\))g(one)g(has)740 956 y Fr(C)799 968 y Fn(0)837 956 y Fo(j)p Fr(n)910 921 y Fn(0)910 976 y Fp(`)947 956 y Fo(j)970 921 y Fv(\000)p Fp(\034)1086 956 y Fo(\024)1174 860 y Fq(\014)1174 910 y(\014)1174 960 y(\014)1208 956 y Fu(~)-48 b Fr(!)1254 968 y Fn(1)1290 956 y Fr(n)1340 921 y Fn(0)1340 976 y Fp(`)1396 956 y Fu(+)18 b Fr(\021)1520 968 y Fp(`)1559 956 y Fu(~)-49 b Fr(!)1604 968 y Fp(m)1685 956 y Fo(\006)25 b Fu(~)-49 b Fr(!)1820 974 y Fp(m)1879 954 y Fi(0)1879 996 y Fh(`)1911 974 y Fn(+)p Fp(\021)1996 983 y Fh(`)2026 974 y Fp(m)2089 860 y Fq(\014)2089 910 y(\014)2089 960 y(\014)2139 956 y Fo(\024)23 b Fr(C)2286 968 y Fn(0)2324 956 y Fu(2)2366 921 y Fv(\000)p Fp(h)p Fn(+1)2544 956 y Fr(;)389 b Fu(\()p Fr(A)p Fu(6)p Fr(:)p Fu(3\))118 1154 y(so)35 b(that)h Fo(j)p Fr(n)489 1124 y Fn(0)489 1178 y Fp(`)526 1154 y Fo(j)g(\025)g Fu(2)728 1124 y Fn(\()p Fp(h)p Fv(\000)p Fn(1\))p Fp(=\034)979 1154 y Fu(.)61 b(On)35 b(the)h(other)f(hand)g (one)g(has)g Fo(j)p Fr(n)2189 1124 y Fn(0)2189 1178 y Fp(`)2227 1154 y Fo(j)h(\024)g Fr(K)6 b Fu(\()p Fr(T)12 b Fu(\),)36 b(hence)g Fr(K)6 b Fu(\()p Fr(T)12 b Fu(\))35 b Fo(\025)118 1261 y Fu(2)160 1231 y Fn(\()p Fp(h)p Fv(\000)p Fn(1\))p Fp(=\034)411 1261 y Fu(;)28 b(so)f(w)n(e)g(get)g(the)h(b)r (ound)g(\(A4.8\).)833 1509 y Fw(App)s(endix)38 b(A7.)99 b(Pro)s(of)37 b(of)h(lemma)e(7)118 1686 y Fu(The)29 b(pro)r(of)g(is)g (done)g(b)n(y)g(induction)h(on)1400 1664 y(\026)1399 1686 y Fr(h)p Fu(.)41 b(Let)30 b(us)f(de\014ne)g Fr(J)2065 1643 y Fn(\()p Fp(h)p Fn(\))2057 1696 y Fp(c;m)2196 1686 y Fu(=)c([)p Fo(\000j)p Fr(")p Fo(j)p Fr(;)14 b Fo(j)p Fr(")p Fo(j)p Fu(])29 b(and)g(call)g Fr(J)3004 1656 y Fn(\()p Fp(h)p Fn(\))3124 1686 y Fu(=)118 1802 y Fo(\002)183 1817 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)555 1802 y Fr(J)609 1759 y Fn(\()p Fp(h)p Fn(\))601 1812 y Fp(c;m)742 1802 y Fu(and)e Fr(I)946 1772 y Fn(\()p Fp(h)p Fn(\))1064 1802 y Fu(=)c Fo(\002)1217 1817 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)1589 1802 y Fr(I)1632 1759 y Fn(\()p Fp(h)p Fn(\))1625 1812 y Fp(c;m)1738 1802 y Fu(.)189 1919 y(W)-7 b(e)27 b(supp)r(ose)g(that)g (there)f(exists)h Fr(I)1305 1889 y Fn(\()1332 1873 y(\026)1331 1889 y Fp(h)o Fn(\))1426 1919 y Fu(suc)n(h)g(that,)g(if)g Fr(\027)32 b Fu(spans)26 b Fr(I)2232 1889 y Fn(\()2259 1873 y(\026)2258 1889 y Fp(h)p Fn(\))2354 1919 y Fu(then)h Fr(\027)2584 1924 y Fn(\026)2583 1939 y Fp(h)2653 1919 y Fu(spans)f Fr(J)2933 1889 y Fn(\()2960 1873 y(\026)2959 1889 y Fp(h)p Fn(\))3055 1919 y Fu(and)118 2034 y Fo(j)p Fr(\027)187 1991 y Fn(\()p Fp(c)p Fn(\))182 2059 y Fp(h;m)304 2034 y Fo(j)d(\024)g Fr(C)6 b Fo(j)p Fr(")p Fo(j)22 b Fu(for)732 2012 y(\026)731 2034 y Fr(h)h Fo(\025)g Fr(h)g Fo(\025)f Fu(0;)i(w)n(e)d(w)n(an)n(t)h(to)g(sho)n(w)f(that)h(the)g (same)g(holds)f(for)2595 2012 y(\026)2594 2034 y Fr(h)7 b Fu(+)g(1.)34 b(Let)23 b(us)e(call)138 2144 y(~)118 2165 y Fr(J)172 2135 y Fn(\()199 2120 y(\026)198 2135 y Fp(h)p Fn(+1\))380 2165 y Fu(the)29 b(in)n(terv)-5 b(al)28 b(spanned)h(b)n(y)g Fo(f)p Fr(\027)1355 2122 y Fn(\()p Fp(c)p Fn(\))1351 2183 y(\026)1350 2199 y Fp(h)o Fn(+1)p Fp(;m)1555 2165 y Fo(g)1597 2180 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)1998 2165 y Fu(when)g Fo(f)p Fr(\027)2304 2122 y Fn(\()p Fp(c)p Fn(\))2299 2175 y Fp(m)2389 2165 y Fo(g)2431 2180 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)2833 2165 y Fu(span)f Fr(I)3071 2135 y Fn(\()3098 2120 y(\026)3097 2135 y Fp(h)p Fn(\))3166 2165 y Fu(.)118 2305 y(F)-7 b(or)31 b(an)n(y)g Fo(f)p Fr(\027)520 2262 y Fn(\()p Fp(c)p Fn(\))515 2315 y Fp(m)606 2305 y Fo(g)648 2320 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)1050 2305 y Fo(2)f Fr(I)1178 2275 y Fn(\()1205 2259 y(\026)1204 2275 y Fp(h)p Fn(\))1305 2305 y Fu(one)h(has)h Fo(f)p Fr(\027)1698 2310 y Fn(\026)1697 2325 y Fp(h)o Fn(+1)p Fp(;m)1902 2305 y Fo(g)1944 2320 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)2346 2305 y Fo(2)f Fu([)p Fo(\000)p Fu(2)p Fo(j)p Fr(")p Fo(j)20 b(\000)h Fr(C)6 b Fo(j)p Fr(")p Fo(j)2903 2275 y Fn(2)2940 2305 y Fr(;)14 b Fu(2)p Fo(j)p Fr(")p Fo(j)20 b Fu(+)118 2423 y Fr(C)6 b Fo(j)p Fr(")p Fo(j)268 2393 y Fn(2)305 2423 y Fu(],)40 b(where)c(the)h(b)r(ound)g(\(5.13\))g(has)f(b)r(een)h(used.) 65 b(This)36 b(means)h(that)g Fr(J)2624 2393 y Fn(\()2651 2378 y(\026)2650 2393 y Fp(h)p Fn(+1\))2840 2423 y Fu(is)f(strictly)118 2530 y(con)n(tained)27 b(in)611 2509 y(~)591 2530 y Fr(J)645 2500 y Fn(\()672 2484 y(\026)671 2500 y Fp(h)p Fn(+1\))824 2530 y Fu(.)189 2636 y(On)h(the)g(other)f(hand)h(it)g(is)g(ob)n(vious)f (that)h(there)g(is)f(a)h(one-to-one)e(corresp)r(ondence)g(b)r(et)n(w)n (een)118 2742 y Fo(f)p Fr(\027)206 2699 y Fn(\()p Fp(c)p Fn(\))201 2752 y Fp(m)292 2742 y Fo(g)334 2757 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)733 2742 y Fu(and)h(the)g(sequence)f Fo(f)p Fr(\027)1467 2699 y Fn(\()p Fp(c)p Fn(\))1462 2767 y Fp(h;m)1584 2742 y Fo(g)1626 2757 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)1998 2742 y Fu(,)2049 2720 y(\026)2048 2742 y Fr(h)17 b Fu(+)g(1)22 b Fo(\025)h Fr(h)g Fo(\025)g Fu(0.)36 b(Hence)27 b(there)g(is)f(a)118 2874 y(set)e Fr(I)287 2843 y Fn(\()314 2828 y(\026)313 2843 y Fp(h)p Fn(+1\))489 2874 y Fo(\032)f Fr(I)620 2843 y Fn(\()647 2828 y(\026)646 2843 y Fp(h)p Fn(\))739 2874 y Fu(suc)n(h)h(that,)h(if) g Fo(f)p Fr(\027)1284 2830 y Fn(\()p Fp(c)p Fn(\))1279 2883 y Fp(m)1369 2874 y Fo(g)1411 2889 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)1807 2874 y Fu(spans)f Fr(I)2074 2843 y Fn(\()2101 2828 y(\026)2100 2843 y Fp(h)p Fn(+1\))2253 2874 y Fu(,)h(then)f Fo(f)p Fr(\027)2574 2830 y Fn(\()p Fp(c)p Fn(\))2570 2892 y(\026)2569 2907 y Fp(h)p Fn(+1)p Fp(;m)2775 2874 y Fo(g)2817 2889 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)118 3005 y Fu(spans)j(the)h(in)n(terv)-5 b(al)27 b Fr(J)842 2975 y Fn(\()869 2960 y(\026)868 2975 y Fp(h)p Fn(\))965 3005 y Fu(and,)g(for)g Fr(")h Fu(small)f(enough,)g Fo(j)p Fr(\027)1933 3017 y Fp(h)1976 3005 y Fo(j)1999 3017 y Fv(1)2092 3005 y Fo(\024)c Fr(C)6 b Fo(j)p Fr(")p Fo(j)28 b Fu(for)2486 2984 y(\026)2485 3005 y Fr(h)18 b Fu(+)g(1)23 b Fo(\025)f Fr(h)h Fo(\025)g Fu(0.)189 3112 y(The)28 b(previous)f(computations)g(also)g(sho)n(w)h(that)g(the)g(inductiv)n(e) g(h)n(yp)r(othesis)g(is)g(v)n(eri\014ed)f(also)118 3218 y(for)248 3196 y(\026)247 3218 y Fr(h)e Fu(=)h(0)j(so)f(that)i(w)n(e)f (ha)n(v)n(e)f(pro)n(v)n(ed)f(that)j(there)f(exists)g(a)g(decreasing)e (sets)i(of)h(in)n(terv)-5 b(als)28 b Fr(I)3094 3188 y Fn(\()3121 3173 y(\026)3120 3188 y Fp(h)p Fn(\))118 3324 y Fu(suc)n(h)21 b(that)h(if)g Fo(f)p Fr(\027)631 3281 y Fn(\()p Fp(c)p Fn(\))626 3334 y Fp(m)717 3324 y Fo(g)759 3339 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)1154 3324 y Fo(2)i Fr(I)1276 3294 y Fn(\()1303 3279 y(\026)1302 3294 y Fp(h)p Fn(\))1392 3324 y Fu(then)e(the)h (sequence)e Fo(f)p Fr(\027)2139 3281 y Fn(\()p Fp(c)p Fn(\))2134 3349 y Fp(h;m)2255 3324 y Fo(g)2297 3339 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)2691 3324 y Fu(is)g(w)n(ell)h(de\014ned)118 3461 y(for)i Fr(h)f Fo(\024)402 3439 y Fu(\026)401 3461 y Fr(h)h Fu(and)h(it)g(v)n (eri\014es)e Fo(j)p Fr(\027)1055 3418 y Fn(\()p Fp(c)p Fn(\))1050 3486 y Fp(h;m)1172 3461 y Fo(j)g(\024)g Fr(C)6 b Fo(j)p Fr(")p Fo(j)p Fu(.)36 b(In)24 b(order)g(to)g(pro)n(v)n(e)f (the)i(b)r(ound)g(on)f(the)h(size)f(of)h Fr(I)3083 3418 y Fn(\()3110 3402 y(\026)3109 3418 y Fp(h)p Fn(\))3076 3471 y Fp(c;m)118 3592 y Fu(let)33 b(us)g(denote)g(b)n(y)f Fo(f)p Fr(\027)836 3549 y Fn(\()p Fp(c)p Fn(\))831 3617 y Fp(h;m)952 3592 y Fo(g)994 3607 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)1399 3592 y Fu(and)h Fo(f)p Fr(\027)1654 3549 y Fv(0)p Fn(\()p Fp(c)p Fn(\))1649 3617 y Fp(h;m)1770 3592 y Fo(g)1812 3607 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)2184 3592 y Fu(,)i(0)c Fo(\024)g Fr(h)g Fo(\024)2588 3570 y Fu(\026)2587 3592 y Fr(h)o Fu(,)k(the)e(sequences)118 3724 y(corresp)r(onding)26 b(to)h Fo(f)p Fr(\027)841 3680 y Fn(\()p Fp(c)p Fn(\))836 3733 y Fp(m)927 3724 y Fo(g)969 3739 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)1369 3724 y Fu(and)g Fo(f)p Fr(\027)1618 3680 y Fv(0)p Fn(\()p Fp(c)p Fn(\))1613 3733 y Fp(m)1723 3724 y Fo(g)1765 3739 y Fv(j)p Fp(m)p Fv(j)p Fp(>)p Fn(1)p Fp(;c)p Fn(=)p Fp(a;b)2165 3724 y Fu(in)h Fr(I)2305 3693 y Fn(\()2332 3678 y(\026)2331 3693 y Fp(h)o Fn(\))2400 3724 y Fu(.)36 b(W)-7 b(e)28 b(ha)n(v)n(e)728 3920 y Fr(\027)774 3877 y Fn(\()p Fp(c)p Fn(\))769 3945 y Fp(h)p Fn(+1)p Fp(;m)994 3920 y Fo(\000)18 b Fr(\027)1123 3877 y Fv(0)p Fn(\()p Fp(c)p Fn(\))1118 3945 y Fp(h)p Fn(+1)p Fp(;m)1346 3920 y Fu(=)23 b(2)1490 3828 y Fq(\020)1539 3920 y Fr(\027)1585 3877 y Fn(\()p Fp(c)p Fn(\))1580 3945 y Fp(h;m)1720 3920 y Fo(\000)18 b Fr(\027)1849 3877 y Fv(0)p Fn(\()p Fp(c)p Fn(\))1844 3945 y Fp(h;m)1966 3828 y Fq(\021)2034 3920 y Fu(+)g Fr(\014)2168 3877 y Fn(\()p Fp(c)p Fn(\))2164 3945 y Fp(h;m)2304 3920 y Fo(\000)g Fr(\014)2438 3877 y Fv(0)p Fn(\()p Fp(c)p Fn(\))2434 3945 y Fp(h;m)2556 3920 y Fr(;)377 b Fu(\()p Fr(A)p Fu(7)p Fr(:)p Fu(1\))118 4132 y(where)30 b Fr(\014)412 4089 y Fn(\()p Fp(c)p Fn(\))408 4157 y Fp(h;m)561 4132 y Fu(and)g Fr(\014)776 4089 y Fv(0)p Fn(\()p Fp(c)p Fn(\))772 4157 y Fp(h;m)925 4132 y Fu(are)g(shorthands)f (for)h(the)i(b)r(eta)f(functions.)46 b(Then,)32 b(as)e Fo(j)p Fr(\027)2774 4144 y Fp(k)2836 4132 y Fo(\000)20 b Fr(\027)2967 4102 y Fv(0)2962 4155 y Fp(k)3003 4132 y Fo(j)3026 4144 y Fv(1)3124 4132 y Fo(\024)118 4238 y(j)p Fr(\027)182 4250 y Fp(h)244 4238 y Fo(\000)e Fr(\027)373 4208 y Fv(0)368 4262 y Fp(h)411 4238 y Fo(j)434 4250 y Fv(1)532 4238 y Fu(for)27 b(all)g Fr(k)f Fo(\024)d Fr(h)p Fu(,)k(w)n(e)g(ha)n(v)n(e)748 4442 y Fo(j)p Fr(\027)812 4454 y Fp(h)874 4442 y Fo(\000)18 b Fr(\027)1003 4408 y Fv(0)998 4463 y Fp(h)1041 4442 y Fo(j)1064 4454 y Fv(1)1158 4442 y Fo(\024)1255 4386 y Fu(1)p 1255 4423 42 4 v 1255 4499 a(2)1307 4442 y Fo(j)p Fr(\027)1371 4454 y Fp(h)p Fn(+1)1516 4442 y Fo(\000)g Fr(\027)1645 4408 y Fv(0)1640 4463 y Fp(h)p Fn(+1)1767 4442 y Fo(j)1790 4454 y Fv(1)1879 4442 y Fu(+)g Fr(C)6 b Fo(j)p Fr(")p Fo(j)2112 4408 y Fn(2)2150 4442 y Fo(j)p Fr(\027)2214 4454 y Fp(h)2275 4442 y Fo(\000)18 b Fr(\027)2404 4408 y Fv(0)2399 4463 y Fp(h)2442 4442 y Fo(j)2465 4454 y Fv(1)2536 4442 y Fr(:)397 b Fu(\()p Fr(A)p Fu(7)p Fr(:)p Fu(2\))118 4628 y(Hence)28 b(if)g Fr(")f Fu(is)h(small)f(enough)g(then)h(one)f(has)1024 4798 y Fo(j)p Fr(\027)d Fo(\000)18 b Fr(\027)1241 4763 y Fv(0)1264 4798 y Fo(j)1287 4810 y Fv(1)1381 4798 y Fo(\024)k Fu(\()1500 4725 y Fo(p)p 1570 4725 V 1570 4798 a Fu(2)o(\))1643 4763 y Fv(\000)p Fn(\()1722 4748 y(\026)1721 4763 y Fp(h)p Fn(+1\))1874 4798 y Fo(j)p Fr(\027)1939 4803 y Fn(\026)1938 4818 y Fp(h)2000 4798 y Fo(\000)c Fr(\027)2129 4763 y Fv(0)2125 4810 y Fn(\026)2124 4825 y Fp(h)2167 4798 y Fo(j)2190 4810 y Fv(1)2260 4798 y Fr(:)673 b Fu(\()p Fr(A)p Fu(7)p Fr(:)p Fu(3\))118 4975 y(Since,)31 b(b)n(y)f(de\014nition,)h(if)g Fr(\027)k Fu(spans)30 b Fr(I)1302 4945 y Fn(\()1329 4930 y(\026)1328 4945 y Fp(h)o Fn(\))1397 4975 y Fu(,)g(then)h Fr(\027)1684 4980 y Fn(\026)1683 4995 y Fp(h)1756 4975 y Fu(spans)f(the)g(in)n(terv) -5 b(al)30 b Fr(J)2488 4945 y Fn(\()2515 4930 y(\026)2514 4945 y Fp(h)p Fn(\))2583 4975 y Fu(,)h(of)f(size)f(2)p Fo(j)p Fr(")p Fo(j)p Fu(,)i(the)118 5082 y(size)c(of)h Fr(I)413 5051 y Fn(\()440 5036 y(\026)439 5051 y Fp(h)p Fn(\))536 5082 y Fu(is)f(b)r(ounded)h(b)n(y)f(2)p Fo(j)p Fr(")p Fo(j)p Fu(\()1232 5013 y Fo(p)p 1301 5013 V 69 x Fu(2\))1375 5051 y Fn(\()p Fv(\000)1454 5036 y Fn(\026)1453 5051 y Fp(h)p Fv(\000)p Fn(1\))1607 5082 y Fu(.)1612 5443 y(29)p eop %%Page: 30 30 30 29 bop 833 555 a Fw(App)s(endix)38 b(A8.)99 b(Pro)s(of)37 b(of)h(lemma)e(8)118 733 y Fu(\(i\))28 b(Assume)g Fr(\026)g Fu(to)f(b)r(e)h(Diophan)n(tine)g(with)g(Diophan)n(tine)g(constan)n(ts)e Fr(C)2392 745 y Fn(1)2430 733 y Fr(;)14 b(\034)2503 745 y Fn(1)2540 733 y Fu(,)28 b Fs(i.e.)822 915 y Fo(j)p Fr(\026p)18 b Fu(+)g Fr(q)s Fo(j)23 b Fr(>)g(C)1271 927 y Fn(1)1309 915 y Fo(j)p Fr(p)p Fo(j)1397 880 y Fv(\000)p Fp(\034)1480 888 y Fi(1)1682 915 y Fo(8)p Fu(\()p Fr(p;)14 b(q)s Fu(\))22 b Fo(2)i Fm(Z)2079 867 y Fn(2)2135 915 y Fo(n)18 b(f)p Fu(\(0)p Fr(;)c Fu(0\))p Fo(g)p Fu(;)469 b(\()p Fr(A)p Fu(8)p Fr(:)p Fu(1\))118 1089 y(suc)n(h)24 b Fr(\026)h Fu(ha)n(v)n(e)e(full)j(measure)d(in)i Fm(R)36 b Fu(if)25 b Fr(\034)1328 1101 y Fn(1)1388 1089 y Fr(>)e Fu(1.)35 b(F)-7 b(rom)24 b(\(A8.1\),)i(b)n(y)e(recalling)f(that)i Fr(!)2746 1058 y Fn(2)2743 1109 y Fp(m)2829 1089 y Fu(=)d Fr(m)2989 1058 y Fn(2)3039 1089 y Fu(+)12 b Fr(\026)p Fu(,)118 1203 y(w)n(e)27 b(ha)n(v)n(e)g(that,)h(for)f(all)g(\()p Fr(n;)14 b(m)p Fu(\))23 b Fo(2)h Fm(Z)1269 1155 y Fn(2)1325 1203 y Fo(n)18 b(f)p Fu(\(0)p Fr(;)c Fu(0\))p Fo(g)p Fu(,)420 1413 y Fo(j)p Fu(\()p Fr(!)527 1425 y Fn(1)565 1413 y Fr(n)k Fo(\000)g Fr(m)p Fu(\)\()p Fr(!)905 1425 y Fn(1)942 1413 y Fr(n)h Fu(+)f Fr(m)p Fu(\))p Fo(j)23 b Fu(=)1332 1342 y Fq(\014)1332 1392 y(\014)1360 1413 y Fr(\026n)1460 1378 y Fn(2)1516 1413 y Fu(+)1599 1345 y Fq(\000)1637 1413 y Fr(n)1687 1378 y Fn(2)1742 1413 y Fo(\000)18 b Fr(m)1898 1378 y Fn(2)1935 1345 y Fq(\001)1974 1342 y(\014)1974 1392 y(\014)2024 1413 y Fr(>)2172 1357 y(C)2231 1369 y Fn(1)p 2122 1394 197 4 v 2122 1470 a Fo(j)p Fr(n)p Fo(j)2218 1446 y Fn(2)p Fp(\034)2282 1454 y Fi(1)2328 1413 y Fr(;)420 1637 y Fo(j)p Fu(\()p Fr(!)527 1649 y Fn(1)565 1637 y Fr(n)g Fo(\000)g Fr(!)768 1649 y Fp(m)831 1637 y Fu(\)\()p Fr(!)947 1649 y Fn(1)984 1637 y Fr(n)g Fu(+)g Fr(!)1187 1649 y Fp(m)1250 1637 y Fu(\))p Fo(j)24 b Fu(=)1416 1566 y Fq(\014)1416 1616 y(\014)1444 1637 y Fr(\026)1508 1569 y Fq(\000)1546 1637 y Fr(n)1596 1602 y Fn(2)1651 1637 y Fo(\000)18 b Fu(1)1776 1569 y Fq(\001)1832 1637 y Fu(+)1915 1569 y Fq(\000)1953 1637 y Fr(n)2003 1602 y Fn(2)2059 1637 y Fo(\000)g Fr(m)2215 1602 y Fn(2)2252 1569 y Fq(\001)2290 1566 y(\014)2290 1616 y(\014)2341 1637 y Fr(>)2488 1580 y(C)2547 1592 y Fn(1)p 2438 1617 V 2438 1693 a Fo(j)p Fr(n)p Fo(j)2534 1669 y Fn(2)p Fp(\034)2598 1677 y Fi(1)2645 1637 y Fr(;)2956 1528 y Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(2\))118 1848 y(whic)n(h)28 b(imply)1243 1985 y Fo(j)p Fr(!)1318 1997 y Fn(1)1355 1985 y Fr(n)18 b Fo(\006)g Fr(m)p Fo(j)23 b Fr(>)1815 1929 y(C)1874 1941 y Fn(2)p 1723 1966 281 4 v 1723 2042 a Fo(j)p Fr(n)p Fo(j)1819 2018 y Fn(2)p Fp(\034)1883 2026 y Fi(1)1915 2018 y Fn(+1)2013 1985 y Fr(;)1243 2209 y Fo(j)p Fr(!)1318 2221 y Fn(1)1355 2209 y Fr(n)18 b Fo(\006)g Fr(!)1558 2221 y Fp(m)1621 2209 y Fo(j)23 b Fr(>)1857 2153 y(C)1916 2165 y Fn(2)p 1765 2190 V 1765 2266 a Fo(j)p Fr(n)p Fo(j)1861 2242 y Fn(2)p Fp(\034)1925 2250 y Fi(1)1957 2242 y Fn(+1)2055 2209 y Fr(;)2956 2100 y Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(3\))118 2401 y(from)36 b(whic)n(h)f(the)i(\014rst)e(t)n(w)n(o)g(of)h(\(7.5\))g(follo)n(w)f (pro)n(vided)g(that)h(one)g(has)f Fr(C)2536 2413 y Fn(2)2610 2401 y Fo(\025)i Fr(C)2771 2413 y Fn(0)2844 2401 y Fu(and)f Fr(\034)3050 2413 y Fn(0)3124 2401 y Fo(\025)118 2507 y Fu(2)p Fr(\034)196 2519 y Fn(1)252 2507 y Fu(+)18 b(1)k Fr(>)h Fu(3.)118 2614 y(\(ii\))29 b(Then)f(w)n(e)g(consider)f(the)i (last)f(of)g(\(7.5\).)38 b(Supp)r(ose)28 b(without)h(loss)e(of)h (generalit)n(y)f Fr(n)d(>)f Fu(0)28 b(and)118 2720 y Fr(m)191 2690 y Fv(0)237 2720 y Fo(\025)23 b Fr(m)g(>)g Fu(0.)33 b(W)-7 b(e)19 b(discuss)f(separately)f(the)i(cases)f Fo(j)p Fr(!)1805 2732 y Fn(1)1842 2720 y Fr(n)p Fo(\000)p Fu(\()p Fr(!)2041 2732 y Fp(m)2100 2716 y Fj(0)2127 2720 y Fo(\000)p Fr(!)2244 2732 y Fp(m)2307 2720 y Fu(\))p Fo(j)h Fu(and)f Fo(j)p Fr(!)2608 2732 y Fn(1)2645 2720 y Fr(n)p Fo(\000)p Fu(\()p Fr(!)2844 2732 y Fp(m)2903 2716 y Fj(0)2931 2720 y Fu(+)p Fr(!)3048 2732 y Fp(m)3111 2720 y Fu(\))p Fo(j)p Fu(.)189 2826 y(Let)28 b(us)f(consider)g(\014rst) g(the)h(case)f Fo(j)p Fr(!)1333 2838 y Fn(1)1370 2826 y Fr(n)18 b Fo(\000)g Fu(\()p Fr(!)1605 2838 y Fp(m)1664 2822 y Fj(0)1709 2826 y Fo(\000)g Fr(!)1844 2838 y Fp(m)1907 2826 y Fu(\))p Fo(j)p Fu(;)28 b(if)g Fr(m)2162 2796 y Fv(0)2208 2826 y Fu(=)23 b Fr(m)18 b Fu(+)g Fr(`)28 b Fu(w)n(e)f(can)g(write)756 3037 y Fr(!)808 3049 y Fp(m)p Fn(+)p Fp(`)968 3037 y Fo(\000)18 b Fr(!)1103 3049 y Fp(m)1188 3037 y Fu(=)23 b Fr(`)18 b Fu(+)g Fr(c)1448 3049 y Fp(`;m)1559 3037 y Fr(;)180 b Fu(0)22 b Fr(>)h(c)1950 3049 y Fp(`;m)2083 3037 y Fr(>)g Fo(\000)p Fr(K)6 b(\026)2363 3049 y Fn(0)2447 2980 y Fr(`)p 2409 3018 110 4 v 2409 3094 a(m)2482 3070 y Fp(\030)2528 3037 y Fr(;)405 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(4\))118 3232 y(for)25 b Fr(\030)i Fu(=)c(2)i(and)g(a)g(suitable)g(p)r(ositiv)n(e)g Fr(K)31 b Fu(\(one)25 b(can)g(tak)n(e)g Fr(K)k Fu(=)22 b(2\),)k(and)f(w)n(e)g (consider)g(separately)118 3338 y(the)j(case)f Fo(j)p Fr(c)496 3350 y Fp(`;m)606 3338 y Fo(j)d Fr(<)e(C)799 3350 y Fn(2)837 3338 y Fo(j)p Fr(n)p Fo(j)933 3308 y Fv(\000)p Fn(2)p Fp(\034)1049 3316 y Fi(1)1081 3308 y Fv(\000)p Fn(1)1170 3338 y Fr(=)p Fu(2)27 b(and)g Fo(j)p Fr(c)1501 3350 y Fp(`;m)1612 3338 y Fo(j)c(\025)f Fr(C)1804 3350 y Fn(2)1842 3338 y Fo(j)p Fr(n)p Fo(j)1938 3308 y Fv(\000)p Fn(2)p Fp(\034)2054 3316 y Fi(1)2086 3308 y Fv(\000)p Fn(1)2175 3338 y Fr(=)p Fu(2.)189 3444 y(In)28 b(the)g(\014rst)f(case)g(b)n(y)g(the)h(second)f(of)h(\(A8.3\))f(one)g (has)425 3654 y Fo(j)p Fr(!)500 3666 y Fn(1)537 3654 y Fr(n)18 b Fo(\000)g Fr(!)740 3666 y Fp(m)p Fn(+)p Fp(`)900 3654 y Fu(+)g Fr(!)1035 3666 y Fp(m)1098 3654 y Fo(j)23 b(\025)f(j)q Fr(!)1307 3666 y Fn(1)1343 3654 y Fr(n)d Fo(\000)f Fr(`)p Fo(j)g(\000)g(j)p Fr(c)1713 3666 y Fp(`;m)1823 3654 y Fo(j)24 b Fr(>)1967 3598 y Fu(1)p 1967 3635 42 4 v 1967 3711 a(2)2134 3598 y Fr(C)2193 3610 y Fn(2)p 2042 3635 281 4 v 2042 3711 a Fo(j)p Fr(n)p Fo(j)2138 3687 y Fn(2)p Fp(\034)2202 3695 y Fi(1)2234 3687 y Fn(+1)2356 3654 y Fr(>)2487 3598 y(C)2546 3610 y Fn(0)p 2453 3635 164 4 v 2453 3711 a Fo(j)p Fr(n)p Fo(j)2549 3687 y Fp(\034)2580 3695 y Fi(0)2626 3654 y Fr(;)307 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(5\))118 3870 y(where)27 b(the)h(last)f(inequalit)n(y)h (requires)e Fr(C)1415 3882 y Fn(0)1475 3870 y Fo(\024)d Fr(C)1622 3882 y Fn(2)1660 3870 y Fr(=)p Fu(2.)189 3976 y(In)41 b(the)g(second)f(case)g(w)n(e)g(can)h(assume)f Fo(j)p Fr(!)1608 3988 y Fn(1)1645 3976 y Fr(n)27 b Fo(\000)f Fr(!)1865 3988 y Fp(m)p Fn(+)p Fp(`)2034 3976 y Fu(+)h Fr(!)2178 3988 y Fp(m)2241 3976 y Fo(j)45 b Fr(<)f Fu(1,)g(b)r(ecause)c (otherwise)118 4082 y(\(7.5\))30 b(is)f(trivially)g(satis\014ed:)41 b(hence)30 b(w)n(e)f(ha)n(v)n(e)g(to)g(consider)g(only)g(v)-5 b(alues)30 b(of)f Fr(n;)14 b(m;)g(`)29 b Fu(suc)n(h)h(that)118 4188 y(0)23 b Fr(<)f(!)322 4200 y Fp(m)p Fn(+)p Fp(`)482 4188 y Fo(\000)c Fr(!)617 4200 y Fp(m)703 4188 y Fr(<)23 b(!)843 4200 y Fn(1)880 4188 y Fr(n)18 b Fu(+)g(1)27 b(\(recall)g(that)h(w)n(e)f(are)g(assuming)f Fr(`;)14 b(m;)g(!)2389 4200 y Fn(1)2426 4188 y Fr(n)23 b(>)f Fu(0\).)189 4295 y(The)28 b(condition)f Fo(j)p Fr(c)783 4307 y Fp(`;m)894 4295 y Fo(j)c(\025)f Fr(C)1086 4307 y Fn(2)1124 4295 y Fo(j)p Fr(n)p Fo(j)1220 4265 y Fv(\000)p Fn(2)p Fp(\034)1336 4273 y Fi(1)1368 4265 y Fv(\000)p Fn(1)1457 4295 y Fr(=)p Fu(2)27 b(and)g(the)h(last)f(inequalit)n(y)h(in)f(\(A8.4\))h(imply)1037 4541 y Fo(j)p Fr(m)p Fo(j)23 b(\024)g Fr(M)32 b Fo(\021)1467 4424 y Fq(\022)1538 4485 y Fu(2)p Fr(\026)1630 4497 y Fn(0)1667 4485 y Fr(K)6 b(`)p Fo(j)p Fr(n)p Fo(j)1875 4455 y Fn(2)p Fp(\034)1939 4463 y Fi(1)1970 4455 y Fn(+1)p 1538 4522 521 4 v 1750 4598 a Fr(C)1809 4610 y Fn(2)2068 4424 y Fq(\023)2129 4438 y Fn(1)p Fp(=\030)2247 4541 y Fr(;)686 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(6\))118 4753 y(whereas)26 b(the)i(inequalit)n(y)363 4976 y Fr(!)415 4988 y Fp(m)p Fn(+)p Fp(`)575 4976 y Fo(\000)18 b Fr(!)710 4988 y Fp(m)795 4976 y Fu(=)893 4913 y Fr(!)948 4883 y Fn(2)945 4937 y Fp(m)p Fn(+)p Fp(`)1105 4913 y Fo(\000)g Fr(!)1243 4883 y Fn(2)1240 4934 y Fp(m)p 893 4957 410 4 v 893 5033 a Fr(!)945 5045 y Fp(m)p Fn(+)p Fp(`)1105 5033 y Fu(+)g Fr(!)1240 5045 y Fp(m)1336 4976 y Fu(=)1547 4920 y Fr(`)p 1433 4957 263 4 v 1433 4973 a Fo(p)p 1503 4973 194 4 v 1503 5034 a Fu(1)f(+)i Fr(\026)1715 4920 y Fu(\()q Fr(`)f Fu(+)g Fr(m)p Fu(\))2003 4859 y Fo(p)p 2072 4859 V 61 x Fu(1)g(+)g Fr(\026)g Fu(+)g Fr(m)2439 4859 y Fo(p)p 2509 4859 V 2509 4920 a Fu(1)g(+)g Fr(\026)p 1715 4957 987 4 v 2004 5033 a(!)2056 5045 y Fp(m)p Fn(+)p Fp(`)2216 5033 y Fu(+)g Fr(!)2351 5045 y Fp(m)795 5200 y Fo(\025)1007 5144 y Fr(`)p 893 5181 263 4 v 893 5198 a Fo(p)p 962 5198 194 4 v 60 x Fu(1)g(+)g Fr(\026)1175 5144 y(!)1227 5156 y Fp(m)p Fn(+)p Fp(`)1387 5144 y Fu(+)g Fr(!)1522 5156 y Fp(m)p 1175 5181 410 4 v 1175 5257 a Fr(!)1227 5269 y Fp(m)p Fn(+)p Fp(`)1387 5257 y Fu(+)g Fr(!)1522 5269 y Fp(m)1618 5200 y Fo(\025)1829 5144 y Fr(`)p 1715 5181 263 4 v 1715 5198 a Fo(p)p 1785 5198 194 4 v 1785 5258 a Fu(1)g(+)g Fr(\026)2011 5200 y Fo(\025)2241 5144 y Fr(`)p 2108 5181 300 4 v 2108 5198 a Fo(p)p 2177 5198 231 4 v 60 x Fu(1)g(+)g Fr(\026)2370 5270 y Fn(0)2956 5082 y Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(7\))1612 5443 y(30)p eop %%Page: 31 31 31 30 bop 118 555 a Fu(implies)1168 638 y Fr(`)p 1036 675 300 4 v 1036 691 a Fo(p)p 1105 691 231 4 v 61 x Fu(1)18 b(+)g Fr(\026)1298 764 y Fn(0)1368 694 y Fo(\024)23 b Fr(!)1508 706 y Fp(m)p Fn(+)p Fp(`)1668 694 y Fo(\000)18 b Fr(!)1803 706 y Fp(m)1889 694 y Fo(\024)k Fr(!)2028 706 y Fn(1)2065 694 y Fr(n)d Fu(+)f(1)p Fr(;)674 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(8\))118 895 y(so)27 b(that)h(one)f(m)n(ust)h (ha)n(v)n(e)1003 1101 y Fr(`)23 b(<)f(L)p Fu(\()p Fo(j)p Fr(!)1312 1113 y Fn(1)1349 1101 y Fr(n)p Fo(j)d Fu(+)f(1\))p Fr(;)179 b(L)23 b Fu(=)1968 1028 y Fq(p)p 2051 1028 V 73 x Fu(1)18 b(+)g Fr(\026)2244 1113 y Fn(0)2281 1101 y Fr(:)652 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(9\))189 1308 y(A)n(t)28 b(\014xed)g Fr(n;)14 b(m;)g(m)780 1277 y Fv(0)830 1308 y Fu(w)n(e)27 b(can)g(de\014ne)h(a)f(map)h Fr(t)23 b Fo(!)g Fr(\026)p Fu(\()p Fr(t)p Fu(\))28 b(suc)n(h)g(that)520 1549 y Fr(f)9 b Fu(\()p Fr(\026)p Fu(\()p Fr(t)p Fu(\)\))23 b Fo(\021)g Fr(!)941 1561 y Fn(1)978 1549 y Fu(\()p Fr(\026)p Fu(\()p Fr(t)p Fu(\)\))p Fr(n)c Fo(\000)f Fr(!)1390 1561 y Fp(m)p Fn(+)p Fp(`)1532 1549 y Fu(\()p Fr(\026)p Fu(\()p Fr(t)p Fu(\)\))h(+)f Fr(!)1894 1561 y Fp(m)1957 1549 y Fu(\()p Fr(\026)p Fu(\()p Fr(t)p Fu(\)\))24 b(=)f Fr(t)2350 1493 y(C)2409 1505 y Fn(0)p 2317 1530 164 4 v 2317 1606 a Fo(j)p Fr(n)p Fo(j)2413 1582 y Fp(\034)2444 1590 y Fi(0)2490 1549 y Fr(:)402 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(10\))118 1797 y(Then)36 b(one)g(has,)i(if)e Fo(I)43 b Fu(is)36 b(the)g(set)g(of)g Fr(\026)g Fu(not)g(v)n(erifying)f(the)i (Diophan)n(tine)f(condition)g Fo(j)p Fr(!)3014 1809 y Fn(1)3050 1797 y Fr(n)24 b Fo(\000)118 1904 y Fr(!)170 1916 y Fp(m)p Fn(+)p Fp(`)330 1904 y Fu(+)18 b Fr(!)465 1916 y Fp(m)528 1904 y Fo(j)23 b(\025)g Fr(C)721 1916 y Fn(0)758 1904 y Fo(j)p Fr(n)p Fo(j)854 1874 y Fv(\000)p Fp(\034)937 1882 y Fi(0)973 1904 y Fu(,)880 2082 y Fq(Z)926 2271 y Fv(I)986 2195 y Fu(d)p Fr(\026)g Fu(=)1222 2092 y Fv(1)1195 2116 y Fq(X)1193 2292 y Fp(n)p Fn(=1)1332 2084 y Fp(L)p Fn(\()p Fv(j)p Fp(!)r(n)p Fv(j)p Fn(+1\))1425 2116 y Fq(X)1429 2295 y Fp(`)p Fn(=1)1701 2116 y Fq(X)1651 2298 y Fv(j)p Fp(m)p Fv(j\024)p Fp(M)1885 2082 y Fq(Z)1968 2103 y Fn(1)1931 2271 y Fv(\000)p Fn(1)2034 2195 y Fu(d)p Fr(t)2124 2075 y Fq(\014)2124 2125 y(\014)2124 2175 y(\014)2124 2224 y(\014)2162 2139 y Fu(d)p Fr(\026)p Fu(\()p Fr(t)p Fu(\))p 2162 2176 191 4 v 2219 2252 a(d)p Fr(t)2362 2075 y Fq(\014)2362 2125 y(\014)2362 2175 y(\014)2362 2224 y(\014)2404 2195 y Fr(;)488 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(11\))118 2492 y(where)35 b(d)p Fr(\026)p Fu(\()p Fr(t)p Fu(\))p Fr(=)p Fu(d)p Fr(t)i Fu(can)e(b)r(e)h(obtained)g(b)n(y)f (noting)h(that)g(deriving)f(\(A8.10\))g(with)i(resp)r(ect)e(to)h Fr(t)118 2598 y Fu(giv)n(es)1171 2685 y(d)p Fr(f)9 b Fu(\()p Fr(\026)p Fu(\()p Fr(t)p Fu(\)\))p 1171 2722 305 4 v 1286 2798 a(d)p Fr(t)1509 2741 y Fu(=)1607 2685 y(d)p Fr(f)p 1607 2722 97 4 v 1607 2798 a Fu(d)p Fr(\026)1723 2685 y Fu(d)p Fr(\026)p 1723 2722 V 1733 2798 a Fu(d)p Fr(t)1852 2741 y Fu(=)1983 2685 y Fr(C)2042 2697 y Fn(0)p 1950 2722 164 4 v 1950 2798 a Fo(j)p Fr(n)p Fo(j)2046 2774 y Fp(\034)2077 2782 y Fi(0)2123 2741 y Fr(;)769 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(12\))118 2942 y(Moreo)n(v)n(er)742 3008 y(d)p Fr(f)p 742 3045 97 4 v 742 3121 a Fu(d)p Fr(\026)871 3064 y Fu(=)969 3008 y(1)p 969 3045 42 4 v 969 3121 a(2)1208 3008 y(1)p 1030 3045 398 4 v 1030 3121 a Fr(!)1082 3133 y Fn(1)1119 3121 y Fr(!)1171 3133 y Fp(m)p Fn(+)p Fp(`)1313 3121 y Fr(!)1365 3133 y Fp(m)1451 3064 y Fu(\()p Fr(n!)1585 3076 y Fp(m)1648 3064 y Fr(!)1700 3076 y Fp(m)p Fn(+)p Fp(`)1860 3064 y Fu(+)18 b Fr(!)1995 3076 y Fn(1)2046 3064 y Fu(\()p Fr(!)2130 3076 y Fp(m)p Fn(+)p Fp(`)2290 3064 y Fo(\000)g Fr(!)2425 3076 y Fp(m)2488 3064 y Fu(\)\))871 3283 y Fo(\025)1009 3227 y Fr(n!)1111 3239 y Fp(m)p Fn(+)p Fp(`)1252 3227 y Fr(!)1304 3239 y Fp(m)p 969 3264 439 4 v 969 3340 a Fu(2)p Fr(!)1063 3352 y Fn(1)1099 3340 y Fr(!)1151 3352 y Fp(m)p Fn(+)p Fp(`)1293 3340 y Fr(!)1345 3352 y Fp(m)1440 3283 y Fo(\025)1669 3227 y Fu(1)p 1538 3264 304 4 v 1538 3342 a(2)1580 3281 y Fo(p)p 1649 3281 194 4 v 61 x Fu(1)g(+)g Fr(\026)1875 3283 y Fo(\025)2122 3227 y Fu(1)p 1972 3264 341 4 v 1972 3342 a(2)2014 3281 y Fo(p)p 2083 3281 231 4 v 61 x Fu(1)g(+)g Fr(\026)2276 3354 y Fn(0)2346 3283 y Fu(=)2472 3227 y(1)p 2444 3264 99 4 v 2444 3340 a(2)p Fr(L)2552 3283 y(;)2915 3176 y Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(13\))118 3485 y(so)27 b(that)h(w)n(e)f(\014nd)976 3543 y Fq(Z)1022 3732 y Fv(I)1081 3656 y Fu(d)p Fr(\026)c Fo(\024)g Fu(4)p Fr(L)1446 3552 y Fv(1)1419 3577 y Fq(X)1401 3753 y Fp(n)p Fn(=01)1572 3545 y Fp(L)p Fn(\()p Fv(j)p Fp(!)r(n)p Fv(j)p Fn(+1\))1665 3577 y Fq(X)1669 3756 y Fp(`)p Fn(=1)1942 3577 y Fq(X)1892 3759 y Fv(j)p Fp(m)p Fv(j\024)p Fp(M)2168 3600 y Fr(C)2227 3612 y Fn(0)p 2135 3637 164 4 v 2135 3713 a Fo(j)p Fr(n)p Fo(j)2231 3689 y Fp(\034)2262 3697 y Fi(0)2308 3656 y Fr(:)584 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(14\))118 3911 y(W)-7 b(e)28 b(can)f(b)r(ound)h(\(A8.14\))f(with)h(a)g(constan)n(t)f (times)855 4177 y(\(1)18 b(+)g Fr(\026)1080 4189 y Fn(0)1117 4177 y Fu(\))1149 4143 y Fn(1+1)p Fp(=)p Fn(2)p Fp(\030)1370 4177 y Fr(\026)1420 4134 y Fn(1)p Fp(=\030)1420 4199 y Fn(0)1524 4177 y Fr(C)1583 4189 y Fn(0)1664 4073 y Fv(1)1637 4098 y Fq(X)1634 4274 y Fp(n)p Fn(=1)1773 4177 y Fr(n)1823 4143 y Fv(\000)p Fn(\()p Fp(\034)1932 4151 y Fi(0)1964 4143 y Fv(\000)p Fn(1)p Fv(\000)p Fn(\(2)p Fp(\034)2191 4151 y Fi(1)2223 4143 y Fn(+2\))p Fp(=\030)r Fn(\))2429 4177 y Fr(;)463 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(15\))118 4448 y(whic)n(h)28 b(is)f(prop)r(ortional)f(to)h Fr(C)1077 4460 y Fn(0)1143 4448 y Fu(pro)n(vided)f(that)i(one)f(has) 1263 4682 y Fr(\034)1299 4694 y Fn(0)1360 4682 y Fr(>)c Fu(2)18 b(+)1601 4626 y(2)p Fr(\034)1679 4638 y Fn(1)1734 4626 y Fu(+)g(2)p 1601 4663 259 4 v 1709 4739 a Fr(\030)1892 4682 y(>)k Fu(4)p Fr(:)871 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(16\))118 4925 y(\(iii\))30 b(No)n(w)e(w)n(e)g(pass)g(to)g(the)h (case)f Fo(j)p Fr(!)1274 4937 y Fn(1)1311 4925 y Fr(n)19 b Fo(\000)f Fr(!)1515 4937 y Fp(m)1574 4921 y Fj(0)1620 4925 y Fo(\000)g Fr(!)1755 4937 y Fp(m)1818 4925 y Fo(j)p Fu(.)40 b(W)-7 b(e)29 b(can)f(ha)n(v)n(e)g Fo(j)p Fr(!)2469 4937 y Fn(1)2506 4925 y Fr(n)18 b Fo(\000)g Fr(!)2709 4937 y Fp(m)2768 4921 y Fj(0)2813 4925 y Fo(\000)g Fr(!)2948 4937 y Fp(m)3010 4925 y Fo(j)25 b Fr(<)f Fu(1)118 5031 y(only)34 b(if)h(\()p Fr(!)474 5043 y Fn(1)534 5031 y Fu(+)22 b(1\))p Fr(n)34 b Fo(\025)g Fu(1)22 b(+)h Fr(!)1082 5043 y Fn(1)1118 5031 y Fr(n)34 b(>)g(!)1353 5043 y Fp(m)1412 5027 y Fj(0)1461 5031 y Fu(+)23 b Fr(!)1601 5043 y Fp(m)1698 5031 y Fr(>)33 b(!)1848 5043 y Fp(m)1945 5031 y Fr(>)h Fo(j)p Fr(m)p Fo(j)p Fu(,)i(and)e(for)g(\014xed)g Fr(m)g Fu(there)g(is)g(a)118 5138 y(unique)22 b Fo(j)p Fr(m)478 5108 y Fv(0)502 5138 y Fo(j)g Fu(suc)n(h)f(that)i Fo(j)p Fr(!)978 5150 y Fp(m)1037 5134 y Fj(0)1081 5138 y Fu(+)18 b Fr(!)1216 5150 y Fp(m)1297 5138 y Fo(\000)g Fr(!)s(n)p Fo(j)23 b(\024)g Fu(1)e(\(so)h(that)g(the)g(summation)g(o)n(v)n(er)e Fr(m)2791 5108 y Fv(0)2837 5138 y Fu(giv)n(es)g(only)118 5244 y(a)31 b(factor)g(2\).)48 b(Moreo)n(v)n(er)28 b(for)j Fr(n)e Fu(=)g(1)i(one)g(has)g Fr(!)1683 5256 y Fp(m)1767 5244 y Fu(+)21 b Fr(!)1905 5256 y Fp(m)1964 5240 y Fj(0)2019 5244 y Fr(>)29 b Fu(2)p Fr(!)2207 5256 y Fn(1)2243 5244 y Fu(,)k(so)e(that,)h(if)g Fr(C)2751 5256 y Fn(0)2818 5244 y Fo(\024)d Fu(1)g Fo(\024)g Fr(!)3129 5256 y Fn(1)3166 5244 y Fu(,)1612 5443 y(31)p eop %%Page: 32 32 32 31 bop 118 555 a Fu(the)27 b(Diophan)n(tine)f(condition)f Fo(j)p Fr(!)1167 567 y Fn(1)1204 555 y Fr(n)16 b Fo(\000)e Fr(!)1401 567 y Fp(m)1460 551 y Fj(0)1502 555 y Fo(\000)h Fr(!)1634 567 y Fp(m)1697 555 y Fo(j)23 b(\025)f Fr(C)1889 567 y Fn(0)1927 555 y Fo(j)p Fr(n)p Fo(j)2023 525 y Fv(\000)p Fp(\034)2106 533 y Fi(0)2168 555 y Fu(is)k(automatically)f(satis\014ed) h(for)118 662 y Fr(n)d Fu(=)g(1)189 769 y(A)n(t)28 b(\014xed)g Fr(n;)14 b(m;)g(m)780 739 y Fv(0)802 769 y Fu(,)28 b(with)g Fr(n)23 b Fo(\025)g Fu(2,)k(b)n(y)g(de\014ning)h(a)f(map)g Fr(t)d Fo(!)f Fr(\026)p Fu(\()p Fr(t)p Fu(\))28 b(through)237 1015 y Fr(f)9 b Fu(\()p Fr(\026)p Fu(\()p Fr(t)p Fu(\)\))24 b Fo(\021)e Fr(!)658 1027 y Fn(1)695 1015 y Fu(\()p Fr(\026)p Fu(\()p Fr(t)p Fu(\)\))p Fr(n)e Fo(\000)e Fr(!)1108 1027 y Fp(m)p Fn(+)p Fp(`)1249 1015 y Fu(\()p Fr(\026)p Fu(\()p Fr(t)p Fu(\)\))i Fo(\000)e Fr(!)1612 1027 y Fp(m)1675 1015 y Fu(\()p Fr(\026)p Fu(\()p Fr(t)p Fu(\)\))24 b(=)f Fr(t)2068 959 y(C)2127 971 y Fn(0)p 2034 996 164 4 v 2034 1072 a Fo(j)p Fr(n)p Fo(j)2130 1048 y Fp(\034)2161 1056 y Fi(0)2208 1015 y Fr(;)180 b(t)23 b Fo(2)g Fu([)p Fo(\000)p Fu(1)p Fr(;)14 b Fu(1])p Fr(;)118 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(17\))118 1262 y(and)28 b(using)f(that)h(for)f Fr(n)c Fo(\025)f Fu(2)27 b(one)h(has)831 1495 y Fr(n!)933 1507 y Fp(m)992 1491 y Fj(0)1018 1495 y Fr(!)1070 1507 y Fp(m)1156 1495 y Fo(\025)23 b Fr(n)1304 1439 y(!)1356 1451 y Fp(m)1415 1435 y Fj(0)1459 1439 y Fu(+)18 b Fr(!)1594 1451 y Fp(m)p 1304 1476 354 4 v 1459 1552 a Fu(2)1667 1495 y Fr(!)1719 1507 y Fp(m)1805 1495 y Fo(\025)k Fr(!)1944 1507 y Fp(m)2021 1495 y Fu(\()p Fr(!)2105 1507 y Fp(m)2164 1491 y Fj(0)2209 1495 y Fu(+)c Fr(!)2344 1507 y Fp(m)2407 1495 y Fu(\))c Fr(;)439 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(18\))118 1722 y(w)n(e)27 b(ha)n(v)n(e)781 1792 y(d)p Fr(f)p 781 1829 97 4 v 781 1905 a Fu(d)p Fr(\026)910 1848 y Fu(=)1007 1792 y(1)p 1007 1829 42 4 v 1007 1905 a(2)1218 1792 y(1)p 1069 1829 341 4 v 1069 1905 a Fr(!)1121 1917 y Fn(1)1158 1905 y Fr(!)1210 1917 y Fp(m)1269 1900 y Fj(0)1295 1905 y Fr(!)1347 1917 y Fp(m)1433 1848 y Fu(\()q Fr(n!)1568 1860 y Fp(m)1630 1848 y Fr(!)1682 1860 y Fp(m)p Fn(+)p Fp(`)1842 1848 y Fo(\000)18 b Fr(!)1977 1860 y Fn(1)2028 1848 y Fu(\()p Fr(!)2112 1860 y Fp(m)2171 1844 y Fj(0)2216 1848 y Fu(+)g Fr(!)2351 1860 y Fp(m)2414 1848 y Fu(\)\))910 2078 y Fo(\025)1007 2022 y Fu(1)p 1007 2059 42 4 v 1007 2135 a(4)1069 2022 y Fr(!)1121 2034 y Fp(m)1202 2022 y Fo(\000)g Fr(!)1337 2034 y Fn(1)p 1069 2059 306 4 v 1120 2135 a Fr(!)1172 2147 y Fn(1)1208 2135 y Fr(!)1260 2147 y Fp(m)1394 2022 y Fr(!)1446 2034 y Fp(m)1505 2017 y Fj(0)1549 2022 y Fu(+)h Fr(!)1685 2034 y Fp(m)p 1394 2059 354 4 v 1502 2135 a Fr(!)1554 2147 y Fp(m)1613 2131 y Fj(0)1780 2078 y Fo(\025)1878 2022 y Fu(1)p 1878 2059 42 4 v 1878 2135 a(4)2079 2022 y Fr(!)2134 1991 y Fn(2)2131 2042 y Fp(m)2212 2022 y Fo(\000)f Fr(!)2350 1991 y Fn(2)2347 2042 y(1)p 1939 2059 588 4 v 1939 2135 a Fr(!)1991 2147 y Fn(1)2028 2135 y Fr(!)2080 2147 y Fp(m)2157 2135 y Fu(\()p Fr(!)2241 2147 y Fp(m)2322 2135 y Fu(+)g Fr(!)2457 2147 y Fn(1)2494 2135 y Fu(\))910 2312 y Fo(\025)1028 2256 y Fu(1)p 1007 2293 84 4 v 1007 2369 a(16)1157 2256 y Fr(m)1230 2226 y Fn(2)p 1110 2293 204 4 v 1110 2369 a Fr(!)1165 2345 y Fn(2)1162 2390 y Fp(m)1225 2369 y Fr(!)1277 2381 y Fn(1)1347 2312 y Fo(\025)1465 2256 y Fu(1)p 1445 2293 84 4 v 1445 2369 a(16)1708 2256 y(1)p 1548 2293 362 4 v 1548 2371 a(\(1)g(+)g Fr(\026)p Fu(\))1805 2347 y Fn(3)p Fp(=)p Fn(2)1942 2312 y Fu(=)2108 2256 y(1)p 2040 2293 177 4 v 2040 2369 a(16)p Fr(L)2181 2345 y Fn(3)2227 2312 y Fr(;)2915 2083 y Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(19\))118 2524 y(so)29 b(that)h(w)n(e)f(w)n(e)h(ha)n(v)n(e)e(to)i(exclude)f (further)h(a)f(set)h Fo(I)1785 2494 y Fv(0)1838 2524 y Fu(of)g(v)-5 b(alues)29 b(of)h Fr(\026)g Fu(with)g(measure)f(b)r (ounded)118 2631 y(b)n(y)1006 2683 y Fq(Z)1052 2872 y Fv(I)1094 2855 y Fj(0)1134 2796 y Fu(d)p Fr(\026)23 b Fo(\024)g Fu(32)p Fr(L)1482 2762 y Fn(3)1561 2693 y Fv(1)1534 2717 y Fq(X)1531 2893 y Fp(n)p Fn(=1)1816 2717 y Fq(X)1670 2899 y Fv(j)p Fp(m)p Fv(j)p Fp(<)p Fn(\()p Fp(!)r Fn(+1\))p Fv(j)p Fp(n)p Fv(j)2138 2740 y Fr(C)2197 2752 y Fn(0)p 2105 2777 164 4 v 2105 2853 a Fo(j)p Fr(n)p Fo(j)2201 2829 y Fp(\034)2232 2837 y Fi(0)2278 2796 y Fr(;)614 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(20\))118 3050 y(whic)n(h)28 b(is)f(b)r(ounded)h(b)n(y)f(a)h(constan)n(t)e(times)1218 3315 y(\(1)18 b(+)g Fr(\026)1443 3327 y Fn(0)1480 3315 y Fu(\))1513 3274 y Fn(2)1564 3315 y Fr(C)1623 3327 y Fn(0)1704 3212 y Fv(1)1677 3237 y Fq(X)1674 3412 y Fp(n)p Fn(=1)1813 3315 y Fr(n)1863 3281 y Fv(\000)p Fp(\034)1946 3289 y Fi(0)1978 3281 y Fn(+1)2066 3315 y Fr(;)826 b Fu(\()p Fr(A)p Fu(8)p Fr(:)p Fu(21\))118 3595 y(whic)n(h)28 b(is)f(prop)r(ortional)f(to)h Fr(C)1077 3607 y Fn(0)1143 3595 y Fu(if)h Fr(\034)1255 3607 y Fn(0)1320 3595 y Fu(satisfes)f (\(A8.16\).)833 3843 y Fw(App)s(endix)38 b(A9.)99 b(Pro)s(of)37 b(of)h(lemma)e(9)118 4022 y Fs(Pr)l(o)l(of.)j Fu(\(i\))28 b(W)-7 b(e)28 b(shall)f(pro)n(v)n(e)f(inductiv)n(ely)i(on)f Fr(p)g Fu(the)h(b)r(ounds)g(\(7.7\).)37 b(F)-7 b(rom)27 b(\(7.4\))g(w)n(e)g(ha)n(v)n(e)239 4231 y Fo(j)7 b Fu(~)-49 b Fr(!)317 4197 y Fn(\()p Fp(p)p Fn(\))314 4252 y Fp(m)407 4231 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))k Fo(\000)25 b Fu(~)-49 b Fr(!)758 4197 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))755 4252 y Fp(m)933 4231 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fo(j)23 b(\024)g Fr(C)6 b Fo(j)p Fr(\027)1391 4243 y Fp(m)1454 4231 y Fu(\()h(~)-49 b Fr(!)1541 4197 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1716 4231 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fr(;)g(")p Fu(\))k Fo(\000)g Fr(\027)2161 4243 y Fp(m)2224 4231 y Fu(\()7 b(~)-49 b Fr(!)2311 4197 y Fn(\()p Fp(p)p Fv(\000)p Fn(2\))2487 4231 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fr(;)g(")p Fu(\))p Fo(j)p Fr(;)120 b Fu(\()p Fr(A)p Fu(9)p Fr(:)p Fu(1\))118 4451 y(as)27 b(w)n(e)g(can)g(b)r(ound)h Fo(j)7 b Fu(~)-49 b Fr(!)828 4408 y Fn(\()p Fp(p)p Fn(\))825 4461 y Fp(m)918 4451 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))19 b(+)24 b(~)-48 b Fr(!)1270 4408 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1267 4461 y Fp(m)1445 4451 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fo(j)23 b(\025)f Fu(1)27 b(for)h Fr(")22 b Fo(2)i(E)2161 4421 y Fn(\()p Fp(p)p Fn(\))2251 4451 y Fu(.)189 4558 y(W)-7 b(e)28 b(set,)g(for)f Fo(j)p Fr(m)p Fo(j)c Fr(>)f Fu(1)315 4778 y(\001)p Fr(\027)425 4790 y Fp(h;m)570 4778 y Fo(\021)h Fr(\027)699 4790 y Fp(h;m)821 4778 y Fu(\()7 b(~)-49 b Fr(!)908 4744 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1083 4778 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fr(;)g(")p Fu(\))k Fo(\000)g Fr(\027)1528 4790 y Fp(h;m)1650 4778 y Fu(\()7 b(~)-49 b Fr(!)1737 4744 y Fn(\()p Fp(p)p Fv(\000)p Fn(2\))1912 4778 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fr(;)g(")p Fu(\))23 b(=)47 b(lim)2325 4828 y Fp(q)r Fv(!1)2504 4778 y Fu(\001)p Fr(\027)2619 4735 y Fn(\()p Fp(q)r Fn(\))2614 4803 y Fp(h;m)2736 4778 y Fr(;)197 b Fu(\()p Fr(A)p Fu(9)p Fr(:)p Fu(2\))118 5022 y(where)27 b(w)n(e)g(ha)n(v)n(e)g(used)g(the)h(notations)f(\(6.7\))h (to)f(de\014ne)692 5241 y(\001)p Fr(\027)807 5198 y Fn(\()p Fp(q)r Fn(\))802 5266 y Fp(h;m)947 5241 y Fu(=)c Fr(\027)1081 5198 y Fn(\()p Fp(q)r Fn(\))1076 5266 y Fp(h;m)1198 5241 y Fu(\()7 b(~)-49 b Fr(!)1285 5207 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1460 5241 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fr(;)g(")p Fu(\))k Fo(\000)g Fr(\027)1910 5198 y Fn(\()p Fp(q)r Fn(\))1905 5266 y Fp(h;m)2027 5241 y Fu(\()7 b(~)-49 b Fr(!)2114 5207 y Fn(\()p Fp(p)p Fv(\000)p Fn(2\))2289 5241 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fr(;)g(")p Fu(\))p Fr(:)341 b Fu(\()p Fr(A)p Fu(9)p Fr(:)p Fu(3\))1612 5443 y(32)p eop %%Page: 33 33 33 32 bop 189 555 a Fu(W)-7 b(e)28 b(w)n(an)n(t)f(to)g(pro)n(v)n(e)f (inductiv)n(ely)i(on)f Fr(q)k Fu(the)d(b)r(ound)820 675 y Fq(\014)820 725 y(\014)820 775 y(\014)848 771 y Fu(\001)p Fr(\027)963 728 y Fn(\()p Fp(q)r Fn(\))958 796 y Fp(h;m)1080 675 y Fq(\014)1080 725 y(\014)1080 775 y(\014)1131 771 y Fo(\024)22 b Fr(C)6 b(")1322 736 y Fn(2)1359 771 y Fo(k)h Fu(~)-49 b Fr(!)1456 736 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1631 771 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))k Fo(\000)25 b Fu(~)-49 b Fr(!)1982 736 y Fn(\()p Fp(p)p Fv(\000)p Fn(2\))2157 771 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fo(k)2394 783 y Fv(1)2464 771 y Fr(;)469 b Fu(\()p Fr(A)p Fu(9)p Fr(:)p Fu(4\))118 986 y(for)27 b(some)g(constan)n(t)g Fr(C)6 b Fu(,)28 b(uniformly)g(in)f Fr(q)s Fu(,)h Fr(h)g Fu(and)f Fr(m)p Fu(.)118 1092 y(\(ii\))k(F)-7 b(or)30 b Fr(q)h Fu(=)d(0)i(the)h(b)r(ound)g(\(A9.4\))g(is)f(trivially)g (satis\014ed.)45 b(Then)31 b(assume)f(that)h(\(A9.4\))f(hold)118 1199 y(for)d(all)g Fr(q)400 1168 y Fv(0)447 1199 y Fr(<)c(q)s Fu(.)189 1305 y(F)-7 b(or)23 b(simplicit)n(y)g(w)n(e)g(set)30 b(~)-49 b Fr(!)26 b Fu(=)j(~)-48 b Fr(!)1172 1275 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1347 1305 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))p Fr(;)g(")p Fu(\))23 b(and)29 b(~)-48 b Fr(!)1885 1275 y Fv(0)1931 1305 y Fu(=)29 b(~)-49 b Fr(!)2073 1275 y Fn(\()p Fp(p)p Fv(\000)p Fn(2\))2248 1305 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\).)36 b(W)-7 b(e)23 b(can)g(write,)h(from)118 1411 y(\(6.7\))j(and)h(\(6.6\),)f(for)g Fo(j)p Fr(m)p Fo(j)c Fr(>)g Fu(1)232 1696 y(\001)p Fr(\027)347 1653 y Fn(\()p Fp(q)r Fn(\))342 1721 y Fp(h;m)487 1696 y Fu(=)g Fo(\000)656 1577 y Fn(\026)655 1593 y Fp(h)o Fv(\000)p Fn(1)657 1618 y Fq(X)654 1796 y Fp(k)q Fn(=)p Fp(h)794 1696 y Fu(2)836 1662 y Fv(\000)p Fp(k)q Fv(\000)p Fn(2)1013 1604 y Fq(\020)1062 1696 y Fr(\014)1113 1653 y Fn(\()p Fp(q)r Fn(\))1109 1721 y Fp(k)q(;m)1229 1696 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(";)g Fo(f)p Fr(\027)1517 1653 y Fn(\()p Fp(q)r Fv(\000)p Fn(1\))1512 1721 y Fp(k)1548 1705 y Fj(0)1690 1696 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))p Fo(g)p Fu(\))k Fo(\000)g Fr(\014)2111 1653 y Fn(\()p Fp(q)r Fn(\))2107 1721 y Fp(k)q(;m)2226 1696 y Fu(\()7 b(~)-49 b Fr(!)2313 1662 y Fv(0)2336 1696 y Fr(;)14 b(";)g Fo(f)p Fr(\027)2537 1653 y Fn(\()p Fp(q)r Fv(\000)p Fn(1\))2532 1721 y Fp(k)2568 1705 y Fj(0)2710 1696 y Fu(\()7 b(~)-49 b Fr(!)2797 1662 y Fv(0)2820 1696 y Fr(;)14 b(")p Fu(\))p Fo(g)p Fu(\))3002 1604 y Fq(\021)3052 1696 y Fr(;)2956 1875 y Fu(\()p Fr(A)p Fu(9)p Fr(:)p Fu(5\))118 1991 y(where)28 b(w)n(e)g(recall)f(that)i Fr(\014)936 1948 y Fn(\()p Fp(q)r Fn(\))932 2016 y Fp(k)q(;m)1052 1991 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(";)g Fo(f)p Fr(\027)1340 1948 y Fn(\()p Fp(q)r Fv(\000)p Fn(1\))1335 2016 y Fp(k)1371 1999 y Fj(0)1512 1991 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))p Fo(g)p Fu(\))29 b(dep)r(end)g(only)f(on)g Fr(\027)2444 1948 y Fn(\()p Fp(q)r Fv(\000)p Fn(1\))2439 2016 y Fp(k)2475 1999 y Fj(0)2618 1991 y Fu(\()7 b(~)-49 b Fr(!)s(;)14 b(")p Fu(\))28 b(with)h Fr(k)3077 1961 y Fv(0)3124 1991 y Fo(\024)118 2097 y Fr(k)21 b Fo(\000)d Fu(1,)28 b(and)f(w)n(e)g(can)h(set)355 2304 y Fr(\014)406 2261 y Fn(\()p Fp(q)r 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Fp(k)2021 4125 y Fj(0)2163 4116 y Fu(\()7 b(~)-49 b Fr(!)2250 4082 y Fv(0)2274 4116 y Fr(;)14 b(")p Fu(\))p Fo(g)p Fu(\))2456 4021 y Fq(\014)2456 4070 y(\014)2456 4120 y(\014)957 4290 y Fo(\024)23 b Fr(D)1114 4302 y Fn(1)1151 4290 y Fr(")1190 4256 y Fn(2)1251 4290 y Fu(sup)1240 4360 y Fp(h)1279 4344 y Fj(0)1302 4360 y Fv(\025)p Fn(0)1441 4290 y Fu(sup)1401 4364 y Fv(j)p Fp(m)1480 4347 y Fj(0)1502 4364 y Fv(j\025)p Fn(1)1620 4290 y Fu(\001)p Fr(\027)1735 4247 y Fn(\()p Fp(q)r Fv(\000)p Fn(1\))1730 4315 y Fp(h)1769 4298 y Fj(0)1792 4315 y Fp(;m)1871 4298 y Fj(0)1932 4290 y Fo(\024)g Fr(D)2089 4302 y Fn(1)2126 4290 y Fr(C)6 b(")2230 4256 y Fn(4)2281 4290 y Fo(k)g Fu(~)-49 b Fr(!)2377 4256 y Fv(0)2419 4290 y Fo(\000)24 b Fu(~)-48 b Fr(!)s Fo(k)2598 4315 y Fv(1)2682 4290 y Fr(;)2956 4224 y Fu(\()p Fr(A)p Fu(9)p Fr(:)p Fu(8\))118 4581 y(where)26 b(w)n(e)h(ha)n(v)n(e)e(used)i(that)g(for)f (all)h Fr(q)1318 4551 y Fv(0)1368 4581 y Fu(the)g(trees)f(con)n (tributing)g(to)h Fr(\014)2332 4538 y Fn(\()p Fp(q)2390 4513 y Fj(0)2413 4538 y Fn(\))2328 4606 y Fp(k)q(;m)2474 4581 y Fu(con)n(tain)f(at)h(least)f(t)n(w)n(o)118 4688 y(end-p)r(oin)n(ts,)g(so)g(that)g(the)h(\014rst)f(non)n(v)-5 b(anishing)25 b(con)n(tribution)h(to)g(the)h(left)g(hand)f(side)g(of)g (\(A9.8\))118 4794 y(is)i(to)f(second)g(order.)118 4900 y(\(iv\))35 b(W)-7 b(e)35 b(are)f(left)h(with)g(the)g(\014rst)f(term)h (in)f(\(A9.7\).)58 b(W)-7 b(e)35 b(write)f Fr(\014)2299 4857 y Fn(\()p Fp(c)p Fn(\)\()p Fp(q)r Fn(\))2295 4925 y Fp(k)q(;m)2504 4900 y Fu(as)g(in)h(\(5.8\),)h(with)f Fr(k)118 5031 y Fu(replacing)28 b Fr(h)p Fu(;)j(for)e(eac)n(h)g Fr(T)37 b Fo(2)27 b(T)1129 4988 y Fn(\()p Fp(c)p Fn(\))1108 5057 y Fp(k)1245 5031 y Fu(w)n(e)i(can)g(order)g(the)h(lines)f(in)h Fr(L)p Fu(\()p Fr(T)12 b Fu(\))29 b(and)g(construct)g(a)g(set)h(of)118 5138 y Fr(k)161 5150 y Fp(T)246 5138 y Fu(subsets)j Fr(L)596 5150 y Fn(1)633 5138 y Fu(\()p Fr(T)12 b Fu(\))p Fr(;)i(:)g(:)g(:)f(;)h (L)999 5150 y Fp(k)1034 5158 y Fh(T)1084 5138 y Fu(\()p Fr(T)e Fu(\))32 b(of)h Fr(L)p Fu(\()p Fr(T)12 b Fu(\),)33 b(with)g Fo(j)p Fr(L)1853 5150 y Fp(j)1888 5138 y Fu(\()p Fr(T)12 b Fu(\))p Fo(j)31 b Fu(=)g Fr(j)5 b Fu(,)34 b(in)f(the)g(follo) n(wing)f(w)n(a)n(y)-7 b(.)51 b(Set)118 5244 y Fr(L)175 5256 y Fn(1)212 5244 y Fu(\()p Fr(T)12 b Fu(\))35 b(=)f Fo(;)p Fu(,)j Fr(L)630 5256 y Fn(2)667 5244 y Fu(\()p Fr(T)12 b Fu(\))34 b(=)h Fr(`)961 5256 y Fn(1)998 5244 y Fu(,)i(if)e Fr(`)1176 5256 y Fn(1)1248 5244 y Fu(is)g(a)f(line)h (connected)g(to)g(the)g(outcoming)f(line)h(of)g Fr(T)12 b Fu(,)36 b(and,)1612 5443 y(33)p eop %%Page: 34 34 34 33 bop 118 555 a Fu(inductiv)n(ely)33 b(for)g Fr(k)726 567 y Fp(T)811 555 y Fo(\025)f Fu(3)h(and)g(2)f Fo(\024)g Fr(j)37 b Fo(\024)32 b Fr(k)1532 567 y Fp(T)1607 555 y Fo(\000)22 b Fu(1,)34 b Fr(L)1850 567 y Fp(j)s Fn(+1)1968 555 y Fu(\()p Fr(T)12 b Fu(\))32 b(=)g Fr(L)2279 567 y Fp(j)2314 555 y Fu(\()p Fr(T)12 b Fu(\))22 b Fo([)g Fr(`)2573 567 y Fp(j)2608 555 y Fu(,)35 b(where)d(the)i(line)118 662 y 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y Fq(\020)901 4265 y Fr(g)944 4206 y Fn(\()p Fp(h)1009 4215 y Fh(`)1034 4228 y(j)1069 4206 y Fn(\))941 4290 y Fp(`)969 4298 y Fh(j)1099 4265 y Fu(\()7 b(~)-49 b Fr(!)1186 4231 y Fv(0)1209 4265 y Fu(\))19 b Fo(\000)f Fr(g)1386 4206 y Fn(\()p Fp(h)1451 4215 y Fh(`)1476 4228 y(j)1510 4206 y Fn(\))1383 4290 y Fp(`)1411 4298 y Fh(j)1540 4265 y Fu(\()7 b(~)-49 b Fr(!)s Fu(\))1659 4173 y Fq(\021)1723 4265 y Fu(V)-7 b(al)1843 4228 y Fv(0)1866 4265 y Fu(\()p Fr(\022)1937 4277 y Fn(1)1975 4265 y Fu(\))2007 4173 y Fq(\020)2109 4161 y Fp(s)2071 4186 y Fq(Y)2070 4363 y Fp(i)p Fn(=2)2192 4265 y Fu(V)g(al\()p Fr(\022)2383 4277 y Fp(i)2411 4265 y Fu(\))2443 4173 y Fq(\021)2492 4120 y(\014)2492 4169 y(\014)2492 4219 y(\014)2492 4269 y(\014)2492 4319 y(\014)868 4482 y Fo(\024)22 b Fr(C)1020 4448 y Fp(K)t Fn(\()p Fp(T)9 b Fn(\))1185 4482 y Fr(D)1256 4448 y Fn(2)p Fp(K)t Fn(\()p Fp(T)g Fn(\))1453 4482 y Fo(j)p Fr(")p Fo(j)1538 4448 y Fp(K)t Fn(\()p Fp(T)g Fn(\))1716 4482 y Fo(k)d Fu(~)-49 b Fr(!)1812 4448 y Fv(0)1853 4482 y Fo(\000)25 b Fu(~)-49 b Fr(!)s Fo(k)2033 4507 y Fv(1)536 4635 y Fo(\024)22 b Fr(C)688 4601 y Fp(K)t Fn(\()p Fp(T)9 b Fn(\))853 4635 y Fr(D)924 4601 y Fn(2)p Fp(K)t Fn(\()p Fp(T)g Fn(\))1121 4635 y Fo(j)p Fr(")p Fo(j)1206 4601 y Fp(K)t Fn(\()p Fp(T)g Fn(\))p Fp(=)p Fn(2)1437 4635 y Fo(j)p Fr(")p Fo(j)1522 4601 y Fn(2)1555 4576 y Fi(\()p Fh(k)q Fj(\000)p Fi(1\))p Fh(=\034)1773 4601 y Fp(=)p Fn(2)1858 4635 y Fo(k)d Fu(~)-49 b Fr(!)1954 4601 y Fv(0)1996 4635 y Fo(\000)24 b Fu(~)-48 b Fr(!)r Fo(k)2175 4660 y Fv(1)2259 4635 y Fr(;)2915 4413 y Fu(\()p Fr(A)p Fu(9)p Fr(:)p Fu(14\))118 4819 y(where)21 b Fr(K)6 b Fu(\()p Fr(T)12 b Fu(\))21 b(is)g(de\014ned)h(in)f(the)h(statemen)n (t)g(of)f(lemma)g(5,)i(and)e(the)h(last)f(inequalit)n(y)g(is)g (obtained)118 4925 y(as)27 b(explained)g(at)h(the)g(end)g(of)f(the)h (pro)r(of)f(of)h(lemma)f(6.)118 5031 y(\(vi\))36 b(Supp)r(ose)f(no)n(w) g(that)g(there)g(are)f(self-energy)g(graphs)g(with)i(outcoming)e(line)i (in)f Fr(L)2983 5043 y Fn(0)3055 5031 y Fu(and)118 5138 y(en)n(tering)26 b(line)h(outside)g Fr(L)937 5150 y Fn(0)974 5138 y Fu(;)g(in)h(suc)n(h)e(a)h(case)f(w)n(e)g(split)i(the)f (renormalized)e(part)i(of)g(the)g(v)-5 b(alue)27 b(of)118 5244 y(the)k(maximal)e Fr(T)666 5214 y Fv(0)719 5244 y Fu(among)g(suc)n(h)h(self-energy)e(graphs)h(\(note)h(that)h(there)f (cannot)f(b)r(e)i(t)n(w)n(o)e(suc)n(h)1612 5443 y(36)p eop %%Page: 37 37 37 36 bop 118 555 a Fu(maximal)29 b(self-energy)f(graphs\))g(as)h Fo(RV)1399 567 y Fp(T)1447 551 y Fj(0)1500 555 y Fu(=)d Fo(V)1642 567 y Fp(T)1690 551 y Fj(0)1737 555 y Fo(\000)19 b(L)p Fr(V)1926 567 y Fp(T)1974 551 y Fj(0)2001 555 y Fu(,)30 b(so)f(obtaining)g(t)n(w)n(o)g(con)n(tributions.)118 662 y(Moreo)n(v)n(er)18 b(if)j(in)g(the)g(con)n(tribution)f(in)h(whic)n (h)f Fo(V)1608 674 y Fp(T)1656 658 y Fj(0)1704 662 y Fu(w)n(as)f(selected)i(there)f(is)h(still)f(other)g(self-energy)118 768 y(graphs)27 b(con)n(tained)g(in)i Fr(T)922 738 y Fv(0)972 768 y Fu(with)g(outcoming)f(line)g(in)g Fr(L)1880 780 y Fn(0)1945 768 y Fu(and)g(en)n(tering)g(line)g(outside)g Fr(L)2930 780 y Fn(0)2995 768 y Fu(again)118 874 y(w)n(e)34 b(split)g(the)h(maximal)e Fr(T)994 844 y Fv(00)1070 874 y Fu(among)g(them)h(as)g Fo(RV)1792 886 y Fp(T)1840 870 y Fj(00)1919 874 y Fu(=)f Fo(V)2068 886 y Fp(T)2116 870 y Fj(00)2184 874 y Fo(\000)22 b(L)p Fr(V)2376 886 y Fp(T)2424 870 y Fj(00)2504 874 y Fu(and)34 b(so)f(on.)56 b(A)n(t)35 b(the)118 981 y(end)23 b(w)n(e)g(obtain)g(a)f(sum)i(of)f(con)n (tributions)f(suc)n(h)g(that)i(no)e(renormalized)g(self-energy)g(graph) f(can)118 1087 y(ha)n(v)n(e)k(the)i(outcoming)f(line)g(in)h Fr(L)1163 1099 y Fn(0)1226 1087 y Fu(and)f(the)h(incoming)f(line)g (outside)g Fr(L)2379 1099 y Fn(0)2416 1087 y Fu(.)37 b(If)26 b(w)n(e)g(are)g(left)h(with)f(a)118 1193 y(lo)r(calized)j (self-energy)f(graph)g Fo(LV)1230 1203 y Fn(~)1217 1218 y Fp(T)1270 1193 y Fu(,)i(with)g(external)e(lines)h(one)g(in)h Fr(L)2337 1205 y Fn(0)2403 1193 y Fu(and)g(the)f(other)g(outside)118 1300 y Fr(L)175 1312 y Fn(0)212 1300 y Fu(,)d(then)g(it)f(can)g(b)r(e)h (factorized)f(a)n(w)n(a)n(y)e(and)i(the)h(remaining)e(part)h(can)g(b)r (e)h(b)r(ounded)f(as)g(in)h(item)118 1406 y(\(v\))f(ab)r(o)n(v)n(e.)35 b(In)24 b(order)g(to)g(b)r(ound)h Fo(L)p Fr(V)1289 1418 y Fp(T)1337 1402 y Fj(0)1389 1406 y Fu(w)n(e)f(note)h(that)g(it)g(has)f (exactly)g(the)h(form)f(\(A9.9\),)h Fs(i.e.)32 b Fu(it)118 1522 y(con)n(tains)h(a)h(pro)r(duct)g(of)g(propagators)d Fr(g)1452 1478 y Fn(\()p Fp(h)1517 1487 y Fh(`)1545 1478 y Fn(\))1449 1547 y Fp(`)1575 1522 y Fu(\()7 b(~)-49 b Fr(!)1662 1491 y Fv(0)1686 1522 y Fu(\),)36 b(a)d(pro)r(duct)h(of)g (propagators)d Fr(g)2778 1478 y Fn(\()p Fp(h)2843 1487 y Fh(`)2872 1478 y Fn(\))2775 1547 y Fp(`)2902 1522 y Fu(\()7 b(~)-49 b Fr(!)s Fu(\))34 b(and)118 1657 y(a)e(di\013erence)f (of)h(propagators)d Fr(g)1171 1599 y Fn(\()p Fp(h)1236 1608 y Fh(`)1261 1621 y(j)1295 1599 y Fn(\))1168 1682 y Fp(`)1196 1690 y Fh(j)1325 1657 y Fu(\()7 b(~)-49 b Fr(!)1412 1627 y Fv(0)1435 1657 y Fu(\))22 b Fo(\000)f Fr(g)1618 1599 y Fn(\()p Fp(h)1683 1608 y Fh(`)1708 1621 y(j)1742 1599 y Fn(\))1615 1682 y Fp(`)1643 1690 y Fh(j)1772 1657 y Fu(\()7 b(~)-49 b Fr(!)s Fu(\),)33 b(so)f(that)g(it)g(can)f(b)r (ounded)i(exactly)e(as)118 1764 y(ab)r(o)n(v)n(e.)189 1871 y(A)n(t)25 b(the)g(end,)h(for)e(a)g(\014xed)h(c)n(hoice)f(of)h (the)g(scales,)f(the)h(total)g(n)n(um)n(b)r(er)f(of)h(terms)g(so)f (obtained)g(is)118 1977 y(b)r(ounded)g(b)n(y)582 1956 y(~)564 1977 y Fr(C)629 1947 y Fp(k)670 1977 y Fu(,)g(for)e(some)h (constan)n(t)1392 1956 y(~)1373 1977 y Fr(C)7 b Fu(,)24 b(so)e(that)i(a)f(b)r(ound)2094 1956 y(~)2075 1977 y Fr(C)2140 1947 y Fp(k)2205 1977 y Fu(times)g(the)g(righ)n(t)g(hand)g (side)g(of)118 2083 y(\(A9.14\))j(follo)n(ws)g(again.)36 b(In)27 b(the)g(course)e(of)i(the)g(ab)r(o)n(v)n(e)f(pro)r(cedure)g(w)n (e)g(are)g(lo)r(osing)g(the)h(factor)118 2204 y(2)160 2174 y Fv(\000)p Fp(h)251 2143 y Fi(\()p Fh(e)p Fi(\))251 2195 y Fh(T)328 2174 y Fn(+)p Fp(h)418 2182 y Fh(T)467 2204 y Fu(,)g(see)f(\(A3.1\),)g(for)g(all)g(the)g(self)h(energy)e (graphs)f Fr(T)2049 2174 y Fv(0)2072 2204 y Fr(;)14 b(T)2170 2174 y Fv(00)2211 2204 y Fr(;)g(:::)27 b Fu(with)f(an)g(external)f (line)i(in)118 2311 y Fr(L)175 2323 y Fn(0)240 2311 y Fu(and)h(another)f(outside)h Fr(L)1051 2323 y Fn(0)1088 2311 y Fu(;)h(but)f(b)n(y)g(de\014nition)h(suc)n(h)f(clusters)f(are)g (a)h(sequence)g(of)g(clusters)118 2417 y(con)n(tained)e(one)g(in)g(the) h(other,)f(hence)g(w)n(e)g(ha)n(v)n(e)f(at)h(most)g(an)g(extra)g (factor)f(2)2568 2387 y Fp(h)2637 2417 y Fu(for)h(eac)n(h)f(cluster)118 2523 y(with)j(scale)f Fr(h)p Fu(,)h(hence)f(the)h(\014nal)g(b)r(ound)g (is)f(still)h(giv)n(en)f(b)n(y)g(\(A3.2\))h(with)g(log)14 b(4)27 b(replacing)f(log)15 b(2.)118 2630 y(\(vii\))27 b(A)g(similar,)f(but)h(simpler,)g(analysis)e(holds)h(in)h(the)g(case)f Fr(m)d Fu(=)g(1;)j(in)h(that)g(case)e Fr(\027)2853 2642 y Fn(1)2917 2630 y Fu(is)i(giv)n(en)118 2736 y(b)n(y)g(\(2.7\).)118 2843 y(\(viii\))k(No)n(w)g(w)n(e)f(can)g(dra)n(w)g(the)h(conclusion)e (from)i(the)g(construction)f(ab)r(o)n(v)n(e.)44 b(W)-7 b(e)31 b(can)g(b)r(ound)118 2879 y Fq(\015)118 2929 y(\015)171 2950 y Fu(~)-49 b Fr(!)219 2919 y Fn(\()p Fp(p)p Fn(\))309 2950 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))k Fo(\000)25 b Fu(~)-49 b Fr(!)660 2919 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))835 2950 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))1030 2879 y Fq(\015)1030 2929 y(\015)1076 2983 y Fv(1)1166 2950 y Fu(with)20 b(a)e(constan)n(t)h (times)g Fo(j)p Fr(")p Fo(j)2032 2919 y Fn(2)2088 2950 y Fu(times)h(the)f(same)g(expression)f(with)118 3065 y Fr(p)32 b Fu(replaced)e(with)j Fr(p)21 b Fo(\000)f Fu(1,)33 b Fs(i.e.)1107 2994 y Fq(\015)1107 3044 y(\015)1160 3065 y Fu(~)-49 b Fr(!)1208 3035 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1383 3065 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))k Fo(\000)25 b Fu(~)-49 b Fr(!)1734 3035 y Fn(\()p Fp(p)p Fv(\000)p Fn(2\))1909 3065 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))2104 2994 y Fq(\015)2104 3044 y(\015)2150 3098 y Fv(1)2220 3065 y Fu(,)33 b(so)e(that,)i(b)n(y)f(the)g(inductiv)n(e) 118 3171 y(h)n(yp)r(othesis,)27 b(the)h(b)r(ound)g(\(7.7\))g(follo)n (ws.)777 3419 y Fw(App)s(endix)38 b(A10.)99 b(Pro)s(of)37 b(of)h(lemma)e(10)118 3597 y Fu(As)28 b Fr(\027)282 3609 y Fp(m)368 3597 y Fu(=)23 b Fr(\027)497 3609 y Fv(\000)p Fp(m)639 3597 y Fu(w)n(e)k(can)h(consider)e(only)h Fr(m)c Fo(\025)g Fu(1.)36 b(F)-7 b(or)27 b Fr(m)c Fo(\025)g Fu(2)k(one)g(has)351 3804 y Fr(\027)392 3816 y Fp(m)479 3804 y Fu(=)22 b Fr(\027)612 3770 y Fn(\()p Fp(a)p Fn(\))607 3825 y Fp(m)723 3804 y Fo(\000)c Fr(\027)852 3770 y Fn(\()p Fp(b)p Fn(\))847 3825 y Fp(m)938 3804 y Fr(;)179 b(\027)1186 3770 y Fn(\()p Fp(a)p Fn(\))1181 3825 y Fp(m)1302 3804 y Fu(=)23 b Fr(O)r Fu(\()p Fr(")1526 3770 y Fn(2)1563 3804 y Fu(\))p Fr(;)181 b(\027)1845 3770 y Fn(\()p Fp(b)p Fn(\))1840 3825 y Fp(m)1953 3804 y Fu(=)23 b Fr(O)r Fu(\()p Fr(")2177 3770 y Fn(min)o Fv(f)p Fn(2)p Fp(m;M)6 b Fv(\000)p Fn(1)p Fv(g)2626 3804 y Fu(\))p Fr(;)234 b Fu(\()p Fr(A)p Fu(10)p Fr(:)p Fu(1\))118 4012 y(and,)28 b(if)g(2)p Fr(m)22 b(>)h(M)9 b Fu(,)27 b(a)g(trivial)g(computation)h(giv)n(es)351 4277 y Fr(\027)397 4243 y Fn(\()p Fp(a)p Fn(\))392 4298 y Fp(m)512 4277 y Fu(=)23 b(\()p Fo(\000)p Fu(1\))771 4243 y Fn(\()p Fp(M)6 b Fv(\000)p Fn(1\))p Fp(=)p Fn(2)1048 4277 y Fr(M)1151 4160 y Fq(\022)1300 4228 y Fr(M)27 b Fo(\000)18 b Fu(1)1226 4327 y(\()p Fr(M)28 b Fo(\000)18 b Fu(1\))p Fr(=)p Fu(2)1620 4160 y Fq(\023)1682 4177 y Fn(2)1733 4160 y Fq(\022)1818 4221 y Fu(1)p 1804 4258 71 4 v 1804 4334 a(4)p Fr(i)1884 4160 y Fq(\023)1945 4177 y Fp(M)6 b Fv(\000)p Fn(1)2117 4277 y Fr(")2156 4243 y Fp(M)g Fv(\000)p Fn(1)2333 4277 y Fu(+)18 b Fr(O)r Fu(\()p Fr(")2552 4243 y Fp(M)2627 4277 y Fu(\))p Fr(;)233 b Fu(\()p Fr(A)p Fu(10)p Fr(:)p Fu(2\))118 4523 y(where)28 b(the)h(factor)f Fr(M)37 b Fu(is)29 b(due)f(to)h(the)g(p)r(ossible)f(c) n(hoices)f(of)i(the)g(line)g(en)n(tering)f(the)g(self-energy)118 4630 y(graph)20 b(con)n(tributing)g(to)h Fr(\027)949 4642 y Fp(m)1012 4630 y Fu(,)h(while)f(the)g(square)f(of)g(the)h (binomial)g(factor)f(is)h(due)g(to)f(the)h(p)r(ossible)118 4736 y(c)n(hoices)h(of)h(the)g(lab)r(els)g(\()p Fr(n)932 4748 y Fn(1)969 4736 y Fr(;)14 b(m)1079 4748 y Fn(1)1116 4736 y Fu(\))p Fr(;)g(:)g(:)g(:)g(;)g Fu(\()p Fr(n)1415 4748 y Fp(M)6 b Fv(\000)p Fn(1)1574 4736 y Fr(;)14 b(m)1684 4748 y Fp(M)6 b Fv(\000)p Fn(1)1842 4736 y Fu(\))23 b(of)g(the)g(t)n(w) n(o)f(no)r(des)h(in)n(ternal)f(to)h(the)g(self-)118 4842 y(energy)k(graph)f(suc)n(h)h(that)h Fr(n)1036 4854 y Fn(1)1092 4842 y Fu(+)18 b Fr(:)c(:)g(:)k Fu(+)g Fr(n)1423 4854 y Fp(M)6 b Fv(\000)p Fn(1)1605 4842 y Fu(=)22 b(0)27 b(and)h Fr(m)1996 4854 y Fn(1)2051 4842 y Fu(+)18 b Fr(:)c(:)g(:)19 b Fu(+)f Fr(m)2406 4854 y Fp(M)6 b Fv(\000)p Fn(1)2587 4842 y Fu(=)23 b(0.)189 4949 y(F)-7 b(or)27 b Fr(m)c Fu(=)g(1)k(one)g(has)402 5198 y Fr(\027)443 5210 y Fn(1)504 5198 y Fu(=)22 b(\()p Fo(\000)p Fu(1\))762 5164 y Fn(\()p Fp(M)6 b Fv(\000)p Fn(1\))p Fp(=)p Fn(2)1054 5081 y Fq(\022)1274 5149 y Fr(M)1129 5248 y Fu(\()p Fr(M)27 b Fo(\000)18 b Fu(1\))p Fr(=)p Fu(2)1523 5081 y Fq(\023)1584 5099 y Fn(2)1639 5198 y Fo(\001)1681 5081 y Fq(\022)1766 5142 y Fu(1)p 1752 5179 V 1752 5255 a(4)p Fr(i)1832 5081 y Fq(\023)1893 5099 y Fp(M)6 b Fv(\000)p Fn(1)2066 5198 y Fr(")2105 5164 y Fp(M)g Fv(\000)p Fn(1)2281 5198 y Fu(+)19 b Fr(O)r Fu(\()p Fr(")2501 5164 y Fp(M)2575 5198 y Fu(\))p Fr(;)285 b Fu(\()p Fr(A)p Fu(10)p Fr(:)p Fu(3\))1612 5443 y(37)p eop %%Page: 38 38 38 37 bop 118 555 a Fu(where)21 b(the)h(square)e(of)h(the)h(com)n (binatorial)e(factor)g(is)i(due)f(to)h(the)g(p)r(ossible)f(c)n(hoices)f (of)h(the)h(lab)r(els)118 662 y(\()p Fr(n)200 674 y Fn(1)237 662 y Fr(;)14 b(m)347 674 y Fn(1)385 662 y Fu(\))p Fr(;)g(:)g(:)g(:)f (;)h Fu(\()p Fr(n)683 674 y Fp(M)757 662 y Fr(;)g(m)867 674 y Fp(M)941 662 y Fu(\))20 b(of)g(the)g Fr(M)28 b Fu(no)r(des)20 b(in)n(ternal)f(to)h(the)g(tree)g(suc)n(h)f(that)h Fr(n)2638 674 y Fn(1)2678 662 y Fu(+)s Fr(:)14 b(:)g(:)s Fu(+)s Fr(n)2964 674 y Fp(M)3060 662 y Fu(=)22 b(1)118 768 y(and)28 b Fr(m)353 780 y Fn(1)408 768 y Fu(+)18 b Fr(:)c(:)g(:)k Fu(+)g Fr(m)762 780 y Fp(M)859 768 y Fu(=)23 b(1.)189 874 y(Therefore,)29 b(for)f(2)p Fr(m)e(>)f(M)9 b Fu(,)29 b(the)h(lo)n(w)n(er)e(b)r(ound)h(on)g Fo(j)p Fr(n\013)1956 886 y Fp(M)2050 874 y Fo(\000)19 b Fr(\014)2181 886 y Fp(M)2254 874 y Fo(j)p Fu(,)30 b Fr(n)c Fo(2)g Fm(Z)n Fu(,)j(is)h(an)f(easy)f(c)n(hec)n(k)118 981 y(from)f(the)h (explicit)g(expressions)e(in)i(\(A10.2\))f(and)g(\(A10.3\),)h(whic)n(h) f(yield)365 1345 y Fo(j)p Fr(n\013)491 1357 y Fp(M)583 1345 y Fo(\000)18 b Fr(\014)713 1357 y Fp(M)786 1345 y Fo(j)24 b(\025)985 1289 y Fr(M)p 930 1326 201 4 v 930 1402 a Fu(4)972 1378 y Fp(M)6 b Fv(\000)p Fn(1)1154 1228 y Fq(\022)1303 1295 y Fr(M)27 b Fo(\000)18 b Fu(1)1229 1395 y(\()p Fr(M)27 b Fo(\000)18 b Fu(1\))p Fr(=)p Fu(2)1623 1228 y Fq(\023)1684 1245 y Fn(2)1735 1100 y Fq(\014)1735 1150 y(\014)1735 1200 y(\014)1735 1250 y(\014)1735 1300 y(\014)1735 1349 y(\014)1735 1399 y(\014)1735 1449 y(\014)1735 1499 y(\014)1763 1345 y Fr(n)g Fo(\000)2085 1289 y Fr(M)p 1924 1326 412 4 v 1924 1356 a Fq(\022)1995 1417 y Fr(M)27 b Fu(+)18 b(1)p 1995 1454 233 4 v 2091 1530 a(2)2238 1356 y Fq(\023)2299 1374 y Fn(2)2346 1100 y Fq(\014)2346 1150 y(\014)2346 1200 y(\014)2346 1250 y(\014)2346 1300 y(\014)2346 1349 y(\014)2346 1399 y(\014)2346 1449 y(\014)2346 1499 y(\014)2388 1345 y Fo(j)p Fr(")p Fo(j)2472 1303 y Fp(M)6 b Fv(\000)p Fn(1)2645 1345 y Fr(;)247 b Fu(\()p Fr(A)p Fu(10)p Fr(:)p Fu(4\))118 1713 y(where)30 b(the)h(term)g(in)g (the)g(last)f(paran)n(theses)f(is)i(greater)d(than)j(1)p Fr(=)p Fu(4)f(\(and)g(it)h(equalt)g(to)f(1)p Fr(=)p Fu(4)g(for)118 1820 y Fr(M)i Fu(=)22 b(3)28 b(and)f Fr(n)c Fu(=)g(1\).)189 1926 y(If)k(2)p Fr(m)22 b Fo(\024)h Fr(M)35 b Fu(b)r(ecomes)26 b(a)f(little)j(more)d(in)n(v)n(olv)n(ed,)g(as)h(one)g(has)f(to)i(tak)n (e)e(in)n(to)h(accoun)n(t)f(also)h(the)118 2032 y(terms)33 b Fr(\027)401 1989 y Fn(\()p Fp(b)p Fn(\))396 2042 y Fp(m)487 2032 y Fu(,)h(whic)n(h)f(has)g(a)g(dominan)n(t)g(con)n (tribution)f(giv)n(en)g(b)n(y)h(the)h(same)e(expression)g(as)h(for)118 2139 y Fr(\027)164 2095 y Fn(\()p Fp(a)p Fn(\))159 2148 y Fp(m)284 2139 y Fu(times)28 b(the)g(factor)1362 2313 y(\()p Fo(\000)p Fu(1\))1533 2283 y Fv(\000)p Fp(m)p 307 2350 2397 4 v 307 2364 a Fq(\022)378 2425 y Fr(M)f Fo(\000)18 b Fu(2)p 378 2462 233 4 v 473 2538 a(2)639 2481 y(+)g(1)764 2364 y Fq(\023)838 2481 y Fr(:)c(:)g(:)949 2364 y Fq(\022)1020 2425 y Fr(M)27 b Fo(\000)18 b Fu(2)p 1020 2462 V 1115 2538 a(2)1281 2481 y(+)g Fr(m)1437 2364 y Fq(\023)c(\022)1583 2425 y Fr(M)27 b Fo(\000)18 b Fu(2)p 1583 2462 V 1678 2538 a(2)1844 2481 y Fo(\000)g Fu(1)1969 2364 y Fq(\023)2043 2481 y Fr(:)c(:)g(:)2154 2364 y Fq(\022)2225 2425 y Fr(M)27 b Fo(\000)18 b Fu(2)p 2225 2462 V 2321 2538 a(2)2486 2481 y Fo(\000)g Fr(m)2642 2364 y Fq(\023)2713 2369 y Fr(;)179 b Fu(\()p Fr(A)p Fu(10)p Fr(:)p Fu(5\))118 2721 y(but)33 b(it)g(is)f(easy)g(to)g(realize)f(that)i(the)g(same)e (conlusions)h(still)h(hold)f(\(with)h(1)p Fr(=)p Fu(4)e(replaced)h (with)118 2827 y(1)p Fr(=)p Fu(9,)27 b(whic)n(h)i(is)f(the)h(v)-5 b(alue)28 b(of)h(the)g(di\013erence)f Fo(j)p Fr(n\013)1714 2839 y Fp(M)1807 2827 y Fo(\000)18 b Fr(\014)1937 2839 y Fp(M)2011 2827 y Fo(j)29 b Fu(computed)f(at)h Fr(M)k Fu(=)24 b(5)k(and)g Fr(n)c Fu(=)g(1)118 2933 y(for)j Fr(m)c Fu(=)g(1\).)777 3181 y Fw(App)s(endix)38 b(A11.)99 b(Pro)s(of)37 b(of)h(lemma)e(11)118 3359 y Fu(In)28 b(order)e(to)i(pro) n(v)n(e)e(the)i(b)r(ounds)f(\(7.15\))g(note)h(that)220 3485 y Fq(\014)220 3535 y(\014)220 3585 y(\014)248 3581 y Fr(g)291 3522 y Fn(\()p Fp(h)356 3531 y Fh(`)381 3544 y(j)415 3522 y Fn(\))288 3606 y Fp(`)316 3614 y Fh(j)445 3581 y Fu(\()7 b(~)-49 b Fr(!)532 3546 y Fv(0)555 3581 y Fu(\))19 b Fo(\000)f Fr(g)732 3522 y Fn(\()p Fp(h)797 3531 y Fh(`)822 3544 y(j)856 3522 y Fn(\))729 3606 y Fp(`)757 3614 y Fh(j)887 3581 y Fu(\()7 b(~)-49 b Fr(!)s Fu(\))18 b Fo(\000)g Fr(@)1156 3593 y Fn(~)-38 b Fp(!)1199 3581 y Fr(g)1242 3522 y Fn(\()p Fp(h)1307 3531 y Fh(`)1332 3544 y(j)1367 3522 y Fn(\))1239 3606 y Fp(`)1267 3614 y Fh(j)1397 3581 y Fu(\()7 b(~)-49 b Fr(!)s Fu(\)\()7 b(~)-49 b Fr(!)21 b Fo(\000)k Fu(~)-49 b Fr(!)1759 3546 y Fv(0)1782 3581 y Fu(\))1814 3485 y Fq(\014)1814 3535 y(\014)1814 3585 y(\014)1865 3581 y Fo(\024)23 b Fr(C)d(n)2082 3546 y Fn(2)2082 3601 y Fp(`)2110 3609 y Fh(j)2145 3581 y Fu(2)2187 3541 y Fv(\000)p Fn(3)p Fp(h)2311 3550 y Fh(`)2336 3563 y(j)2388 3581 y Fo(k)6 b Fu(~)-48 b Fr(!)2485 3546 y Fv(0)2526 3581 y Fo(\000)25 b Fu(~)-49 b Fr(!)s Fo(k)2705 3535 y Fn(2)2705 3606 y Fv(1)2790 3581 y Fr(;)102 b Fu(\()p Fr(A)p Fu(11)p Fr(:)p Fu(1\))118 3791 y(from)27 b(the)h(compact)f(supp)r(ort)h(prop)r(erties)f(of)g(the)h(propagator.) 189 3897 y(Let)e(us)g(consider)f(the)h(quan)n(tit)n(y)f Fr(\027)5 b Fu(\()i(~)-49 b Fr(!)1367 3867 y Fv(0)1391 3897 y Fr(;)14 b(")p Fu(\))h Fo(\000)f Fr(\027)5 b Fu(\()i(~)-49 b Fr(!)t(;)14 b(")p Fu(\))h Fo(\000)f Fr(@)1978 3909 y Fn(~)-38 b Fp(!)2021 3897 y Fr(\027)5 b Fu(\()i(~)-49 b Fr(!)s(;)14 b(")p Fu(\))g(\()p Fr(!)k Fo(\000)d Fr(!)2513 3867 y Fv(0)2536 3897 y Fu(\),)26 b(where)g Fr(@)2905 3909 y Fn(~)-38 b Fp(!)2948 3897 y Fr(\027)5 b Fu(\()i(~)-49 b Fr(!)s(;)14 b(")p Fu(\))118 4004 y(denotes)29 b(the)g(deriv)-5 b(ativ)n(e)28 b(in)i(the)f(sense)f(of)h(Whitney)-7 b(,)31 b(and)d(note)h(that)g(it)h(can)e(b)r(e)i(expressed)e(as)118 4131 y(a)h(sum)h(o)n(v)n(er)e(trees)h(eac)n(h)f(one)i(con)n(taining)e (a)h(line)h(with)g(propagator)d Fr(g)2387 4072 y Fn(\()p Fp(h)2452 4081 y Fh(`)2477 4094 y(j)2511 4072 y Fn(\))2384 4156 y Fp(`)2412 4164 y Fh(j)2541 4131 y Fu(\()7 b(~)-49 b Fr(!)2628 4100 y Fv(0)2652 4131 y Fu(\))20 b Fo(\000)f Fr(g)2831 4072 y Fn(\()p Fp(h)2896 4081 y Fh(`)2921 4094 y(j)2955 4072 y Fn(\))2828 4156 y Fp(`)2856 4164 y Fh(j)2985 4131 y Fu(\()7 b(~)-49 b Fr(!)s Fu(\))20 b Fo(\000)118 4283 y Fr(@)167 4295 y Fn(~)-38 b Fp(!)210 4283 y Fr(g)253 4224 y Fn(\()p Fp(h)318 4233 y Fh(`)343 4246 y(j)378 4224 y Fn(\))250 4308 y Fp(`)278 4316 y Fh(j)408 4283 y Fu(\()7 b(~)-49 b Fr(!)s Fu(\))14 b(\()7 b(~)-49 b Fr(!)21 b Fo(\000)k Fu(~)-49 b Fr(!)784 4253 y Fv(0)807 4283 y Fu(\),)28 b(b)n(y)f(pro)r(ceeding)g(as)g(in)h(the)g(pro)r(of)f (of)g(lemma)h(9.)36 b(Then)28 b(w)n(e)f(\014nd)299 4507 y Fo(k)o Fr(\027)5 b Fu(\()i(~)-49 b Fr(!)473 4472 y Fv(0)497 4507 y Fr(;)14 b(")p Fu(\))k Fo(\000)g Fr(\027)5 b Fu(\()i(~)-49 b Fr(!)s(;)14 b(")p Fu(\))19 b Fo(\000)f Fr(@)1098 4519 y Fn(~)-38 b Fp(!)1141 4507 y Fr(\027)5 b Fu(\()i(~)-49 b Fr(!)s(;)14 b(")p Fu(\))g(\()7 b(~)-49 b Fr(!)21 b Fo(\000)k Fu(~)-49 b Fr(!)1639 4472 y Fv(0)1662 4507 y Fu(\))p Fo(k)1736 4531 y Fv(1)1829 4507 y Fo(\024)23 b Fr(C)1976 4519 y Fn(1)2014 4507 y Fr(")2053 4472 y Fn(2\()p Fp(M)6 b Fv(\000)p Fn(1\))2310 4507 y Fo(k)g Fu(~)-49 b Fr(!)2406 4472 y Fv(0)2448 4507 y Fo(\000)24 b Fu(~)-48 b Fr(!)r Fo(k)2627 4460 y Fn(2)2627 4531 y Fv(1)2711 4507 y Fr(;)181 b Fu(\()p Fr(A)p Fu(11)p Fr(:)p Fu(2\))118 4703 y(and)28 b(the)g(\014rst)f(b)r(ound)h(in)g(\(7.15\).) 189 4810 y(In)34 b(order)e(to)i(pro)n(v)n(e)e(the)i(\014rst)f(b)r(ound) h(in)g(\(7.15\))f(w)n(e)g(pro)r(ceed)g(b)n(y)h(induction)g(b)n(y)f (assuming)118 4916 y(that)28 b(it)g(holds)f(for)34 b(~)-49 b Fr(!)780 4886 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))955 4916 y Fu(;)28 b(then)g(from)f(\(7.4\))h(w)n(e)f(ha)n(v)n(e)407 5113 y(2)7 b(~)-49 b Fr(!)504 5079 y Fn(\()p Fp(p)p Fn(\))593 5113 y Fr(@)637 5125 y Fp(")679 5113 y Fu(~)h Fr(!)728 5079 y Fn(\()p Fp(p)p Fn(\))725 5133 y Fp(m)818 5113 y Fu(\()p Fr(";)14 b(!)s Fu(\))23 b(=)f Fo(\000)p Fr(@)1232 5125 y Fp(")1267 5113 y Fr(\027)1308 5125 y Fp(m)1371 5113 y Fu(\()7 b(~)-49 b Fr(!)1458 5079 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1455 5133 y Fp(m)1634 5113 y Fu(\()p Fr(";)14 b(!)s Fu(\))p Fr(;)g(")p Fu(\))978 b(\()p Fr(A)p Fu(11)p Fr(:)p Fu(3\))596 5244 y(=)22 b Fo(\000)p Fr(@)797 5256 y Fn(~)-38 b Fp(!)840 5244 y Fr(\027)881 5256 y Fp(m)944 5244 y Fu(\()7 b(~)-49 b Fr(!)1031 5210 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1028 5265 y Fp(m)1206 5244 y Fu(\()p Fr(";)14 b(!)s Fu(\))p Fr(;)g(")p Fu(\))g Fr(@)1567 5256 y Fp(")1609 5244 y Fu(~)-48 b Fr(!)1658 5210 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1655 5265 y Fp(m)1832 5244 y Fu(\()p Fr(";)14 b(!)s Fu(\))19 b Fo(\000)f Fr(@)2173 5256 y Fp(")2208 5244 y Fr(\027)2249 5256 y Fp(m)2312 5244 y Fu(\()7 b(~)-49 b Fr(!)2399 5210 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2396 5265 y Fp(m)2575 5244 y Fu(\()p Fr(";)14 b(!)s Fu(\))p Fr(;)g(")p Fu(\))p Fr(;)1612 5443 y Fu(38)p eop %%Page: 39 39 39 38 bop 118 555 a Fu(hence)28 b(from)f(the)h(inductiv)n(e)g(h)n(yp)r (othesis)f(and)g(the)h(pro)n(v)n(ed)e(b)r(ound)i(in)g(\(7.15\),)f(w)n (e)g(obtain)254 691 y Fq(\015)254 741 y(\015)254 791 y(\015)307 786 y Fu(~)-49 b Fr(!)355 752 y Fn(\()p Fp(p)p Fn(\))445 786 y Fu(\()p Fr(!)s(;)14 b(")608 752 y Fv(0)631 786 y Fu(\))19 b Fo(\000)25 b Fu(~)-49 b Fr(!)820 752 y Fn(\()p Fp(p)p Fn(\))910 786 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))k Fo(\000)g Fr(@)1250 798 y Fp(")1292 786 y Fu(~)-48 b Fr(!)1341 752 y Fn(\()p Fp(p)p Fn(\))1431 786 y Fu(\()p Fr(!)s(;)14 b(")p Fu(\))g(\()p Fr(")k Fo(\000)g Fr(")1851 752 y Fv(0)1874 786 y Fu(\))1906 691 y Fq(\015)1906 741 y(\015)1906 791 y(\015)1952 845 y Fv(1)2046 786 y Fo(\024)k Fr(C)6 b Fo(j)p Fr(")p Fo(j)2283 752 y Fp(M)g Fv(\000)p Fn(3)2456 786 y Fo(j)p Fr(")2518 752 y Fv(0)2559 786 y Fo(\000)18 b Fr(")p Fo(j)2704 740 y Fn(2)2755 786 y Fr(;)137 b Fu(\()p Fr(A)p Fu(11)p Fr(:)p Fu(4\))118 1021 y(so)27 b(that)h(also)e(the)i(second)f(b)r(ound)h(in)g(\(7.15\))f (follo)n(ws.)777 1270 y Fw(App)s(endix)38 b(A12.)99 b(Pro)s(of)37 b(of)h(lemma)e(12)118 1449 y Fu(First)29 b(of)g(all)g(w)n(e)g(c)n(hec)n (k)g(that,)g(if)h(w)n(e)f(call)g Fr(")1484 1406 y Fn(\()p Fp(p)p Fn(\))1484 1472 y Fp(j)1574 1449 y Fu(\()p Fr(n)p Fu(\))h(the)f(cen)n(ters)g(of)g(the)g(in)n(terv)-5 b(als)29 b Fo(I)2770 1406 y Fn(\()p Fp(p)p Fn(\))2764 1472 y Fp(j)2860 1449 y Fu(\()p Fr(n)p Fu(\),)i(with)118 1555 y Fr(j)d Fu(=)23 b(1)p Fr(;)14 b Fu(2)p Fr(;)g Fu(3,)26 b(then)i(one)f(has)1149 1586 y Fq(\014)1149 1636 y(\014)1149 1686 y(\014)1176 1682 y Fr(")1215 1639 y Fn(\()p Fp(p)p Fn(+1\))1215 1705 y Fp(j)1389 1682 y Fu(\()p Fr(n)p Fu(\))19 b Fo(\000)f Fr(")1644 1639 y Fn(\()p Fp(p)p Fn(\))1644 1705 y Fp(j)1734 1682 y Fu(\()p Fr(n)p Fu(\))1848 1586 y Fq(\014)1848 1636 y(\014)1848 1686 y(\014)1899 1682 y Fo(\024)23 b Fr(D)r(")2097 1642 y Fp(p)2097 1704 y Fn(0)2135 1682 y Fr(;)757 b Fu(\()p Fr(A)p Fu(12)p Fr(:)p Fu(1\))118 1867 y(for)27 b(a)g(suitable)h(constan)n(t)f Fr(D)r Fu(.)189 1975 y(The)h(cen)n(ter)f Fr(")646 1932 y Fn(\()p Fp(p)p Fn(\))646 1997 y(1)736 1975 y Fu(\()p Fr(n)p Fu(\))h(is)f(de\014ned)h (b)n(y)f(the)h(condition)160 2199 y Fr(!)212 2211 y Fn(1)249 2199 y Fr(n)13 b Fo(\000)g Fr(\026)440 2211 y Fn(1)475 2199 y Fu(\()7 b(~)-49 b Fr(!)562 2165 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))738 2199 y Fu(\()p Fr(")809 2156 y Fn(\()p Fp(p)p Fn(\))809 2221 y(1)899 2199 y Fu(\()p Fr(n)p Fu(\)\))p Fr(;)14 b(")1121 2156 y Fn(\()p Fp(p)p Fn(\))1121 2221 y(1)1211 2199 y Fu(\()p Fr(n)p Fu(\)\))p Fr(n)f Fo(\000)g Fr(!)1550 2211 y Fp(m)1625 2199 y Fu(+)g Fr(\026)1753 2211 y Fp(m)1815 2199 y Fu(\()7 b(~)-49 b Fr(!)1902 2165 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2077 2199 y Fu(\()p Fr(")2148 2156 y Fn(\()p Fp(p)p Fn(\))2148 2221 y(1)2239 2199 y Fu(\()p Fr(n)p Fu(\)\))p Fr(;)14 b(")2461 2156 y Fn(\()p Fp(p)p Fn(\))2461 2221 y(1)2551 2199 y Fu(\()p Fr(n)p Fu(\)\))24 b(=)e(0)p Fr(;)42 b Fu(\()p Fr(A)p Fu(12)p Fr(:)p Fu(2\))118 2412 y(where)21 b(Whitney)h(extensions)e(are) h(considered)f(outside)h Fo(E)1944 2381 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2119 2412 y Fu(;)j(then,)f(b)n(y)e(subtracting)g (\(A12.2\))118 2518 y(from)27 b(the)h(equiv)-5 b(alen)n(t)28 b(expression)e(for)h Fr(p)18 b Fu(+)g(1,)27 b(w)n(e)h(ha)n(v)n(e)225 2652 y Fq(\020)275 2744 y Fr(\026)325 2756 y Fn(1)362 2744 y Fu(\()7 b(~)-49 b Fr(!)449 2710 y Fn(\()p Fp(p)p Fn(\))539 2744 y Fu(\()p Fr(")610 2701 y Fn(\()p Fp(p)p Fn(+1\))610 2766 y(1)785 2744 y Fu(\()p Fr(n)p Fu(\)\))p Fr(;)14 b(")1007 2701 y Fn(\()p Fp(p)p Fn(+1\))1007 2766 y(1)1181 2744 y Fu(\()p Fr(n)p Fu(\)\))19 b Fo(\000)f Fr(\026)1479 2756 y Fn(1)1517 2744 y Fu(\()7 b(~)-49 b Fr(!)1604 2710 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))1779 2744 y Fu(\()p Fr(")1850 2701 y Fn(\()p Fp(p)p Fn(\))1850 2766 y(1)1940 2744 y Fu(\()p Fr(n)p Fu(\)\))p Fr(;)14 b(")2162 2701 y Fn(\()p Fp(p)p Fn(\))2162 2766 y(1)2253 2744 y Fu(\()p Fr(n)p Fu(\)\))2399 2652 y Fq(\021)2463 2744 y Fr(n)402 b Fu(\()p Fr(A)p Fu(12)p Fr(:)p Fu(3\))562 2927 y Fo(\000)645 2835 y Fq(\020)695 2927 y Fr(\026)745 2939 y Fp(m)808 2927 y Fu(\()7 b(~)-49 b Fr(!)895 2893 y Fn(\()p Fp(p)p Fn(\))985 2927 y Fu(\()p Fr(")1056 2884 y Fn(\()p Fp(p)p Fn(+1\))1056 2949 y(1)1230 2927 y Fu(\()p Fr(n)p Fu(\)\))p Fr(;)14 b(")1452 2884 y Fn(\()p Fp(p)p Fn(+1\))1452 2949 y(1)1627 2927 y Fu(\()p Fr(n)p Fu(\)\))19 b Fo(\000)f Fr(\026)1925 2939 y Fp(m)1988 2927 y Fu(\()7 b(~)-49 b Fr(!)2075 2893 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))2250 2927 y Fu(\()p Fr(")2321 2884 y Fn(\()p Fp(p)p Fn(\))2321 2949 y(1)2412 2927 y Fu(\()p Fr(n)p Fu(\)\))p Fr(;)14 b(")2634 2884 y Fn(\()p Fp(p)p Fn(\))2634 2949 y(1)2724 2927 y Fu(\()p Fr(n)p Fu(\)\))2870 2835 y Fq(\021)2943 2927 y Fu(=)23 b(0)p Fr(:)189 3185 y Fu(In)e(\(7.28\))e(w)n(e)h(can)g (write,)i(b)n(y)e(setting)g(for)g(simplicit)n(y)h Fr(")h Fu(=)h Fr(")2061 3141 y Fn(\()p Fp(p)p Fn(\))2061 3207 y(1)2151 3185 y Fu(\()p Fr(n)p Fu(\),)f Fr(")2349 3154 y Fv(0)2395 3185 y Fu(=)h Fr(")2522 3141 y Fn(\()p Fp(p)p Fn(+1\))2522 3207 y(1)2696 3185 y Fu(\()p Fr(n)p Fu(\),)29 b(~)-49 b Fr(!)2910 3154 y Fv(0)2956 3185 y Fu(=)30 b(~)-49 b Fr(!)3099 3154 y Fn(\()p Fp(p)p Fn(\))118 3291 y Fu(and)34 b(~)-48 b Fr(!)25 b Fu(=)k(~)-48 b Fr(!)500 3261 y Fn(\()p Fp(p)p Fv(\000)p Fn(1\))675 3291 y Fu(,)247 3492 y(\()q Fr(\026)330 3504 y Fn(1)367 3492 y Fu(\()7 b(~)-49 b Fr(!)454 3458 y Fv(0)477 3492 y Fu(\()p Fr(")548 3458 y Fv(0)571 3492 y Fu(\))p Fr(;)14 b(")679 3458 y Fv(0)703 3492 y Fu(\))k 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Fu(\))p Fr(;)14 b(")987 5214 y Fv(0)1011 5248 y Fu(\))k Fo(\000)g Fr(\026)1194 5260 y Fp(m)1257 5248 y Fu(\()7 b(~)-49 b Fr(!)s Fu(\()p Fr(")p Fu(\))p Fr(;)14 b(")1523 5214 y Fv(0)1547 5248 y Fu(\))k(+)g(\()p Fr(\026)1762 5260 y Fp(m)1826 5248 y Fu(\()7 b(~)-49 b Fr(!)s Fu(\()p Fr(")p Fu(\))p Fr(;)14 b(")2092 5214 y Fv(0)2115 5248 y Fu(\))19 b Fo(\000)f Fr(\026)2299 5260 y Fp(m)2362 5248 y Fu(\()7 b(~)-49 b Fr(!)s Fu(\()p Fr(")p Fu(\))p Fr(;)14 b(")p Fu(\)\))p Fr(;)2915 5183 y Fu(\()p Fr(A)p Fu(12)p Fr(:)p Fu(7\))1612 5443 y(39)p eop %%Page: 40 40 40 39 bop 118 555 a Fu(and,)28 b(from)f(lemma)g(11,)509 734 y Fo(j)p Fr(\026)582 746 y Fp(m)645 734 y Fu(\()7 b(~)-49 b Fr(!)732 699 y Fv(0)755 734 y Fu(\()p Fr(")826 699 y Fv(0)849 734 y Fu(\))p Fr(;)14 b(")957 699 y Fv(0)981 734 y Fu(\))k Fo(\000)g Fr(\026)1164 746 y Fp(m)1227 734 y Fu(\()7 b(~)-49 b Fr(!)s Fu(\()p Fr(")1385 699 y Fv(0)1409 734 y Fu(\))p Fr(;)14 b(")1517 699 y Fv(0)1540 734 y Fu(\))p Fo(j)23 b(\024)g Fr(C)6 b Fo(j)p Fr(")p 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b(")p Fu(\))24 b(=)1738 1160 y Fq(\000)1776 1227 y Fu(2)p Fr(\014)1865 1239 y Fp(M)1939 1227 y Fr(")18 b Fu(+)g Fr(O)r Fu(\()p Fr(")2215 1193 y Fn(2)2253 1227 y Fu(\))2285 1160 y Fq(\001)2337 1227 y Fu(\()p Fr(")g Fo(\000)g Fr(")2548 1193 y Fv(0)2571 1227 y Fu(\))p Fr(:)289 b Fu(\()p Fr(A)p Fu(12)p Fr(:)p Fu(9\))118 1409 y(Then)28 b(w)n(e)f(get,)h(b)n(y)f(lemma)g(9,)1091 1509 y Fq(\014)1091 1559 y(\014)1091 1609 y(\014)1119 1605 y Fr(")1158 1562 y Fn(\()p Fp(p)p Fn(+1\))1158 1627 y(1)1332 1605 y Fu(\()p Fr(n)p Fu(\))19 b Fo(\000)f Fr(")1587 1562 y Fn(\()p Fp(p)p Fn(\))1587 1627 y(1)1677 1605 y Fu(\()p Fr(n)p Fu(\))1791 1509 y Fq(\014)1791 1559 y(\014)1791 1609 y(\014)1842 1605 y Fo(\024)23 b Fr(C)6 b(")2034 1565 y Fp(p)p Fn(+)p Fp(M)2034 1627 y Fn(0)2193 1605 y Fr(;)657 b Fu(\()p Fr(A)p Fu(12)p Fr(:)p Fu(10\))118 1805 y(for)27 b(a)g(suitable)h(p)r(ositiv)n(e)f(constan)n(t)g Fr(C)6 b Fu(.)37 b(This)28 b(pro)n(v)n(es)d(the)j(b)r(ound)g (\(A12.1\).)189 1911 y(De\014ne)g Fr(p)488 1923 y 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Fu(\))f(as)g(the)g(set)g(of)h(v)-5 b(alues)25 b Fr(")h Fu(suc)n(h)g(that)h(\(7.11\),)e(\(7.12\))h(and)g(\(7.13\))118 2765 y(are)j(satis\014ed)h(with)g Fr(C)831 2777 y Fn(0)899 2765 y Fu(replaced)f(with)h(2)p Fr(C)1521 2777 y Fn(0)1558 2765 y Fu(.)44 b(By)30 b(the)g(de\014nition)h(\(A12.11\))e(all)h(the)g (in)n(terv)-5 b(als)118 2881 y Fo(I)169 2838 y Fn(\()p Fp(p)p Fn(\))163 2904 y Fp(j)260 2881 y Fu(\()p Fr(n)p Fu(\))29 b(fall)g(inside)g(the)h(union)e(of)h(the)h(in)n(terv)-5 b(als)28 b Fo(J)1804 2838 y Fn(\(0\))1789 2904 y Fp(j)1894 2881 y Fu(\()p Fr(n)p Fu(\))p Fr(;)14 b(:)g(:)g(:)g(;)g Fo(J)2264 2838 y Fn(\()p Fp(p)2324 2846 y Fi(0)2357 2838 y Fn(\))2249 2904 y Fp(j)2387 2881 y Fu(\()p Fr(n)p Fu(\))29 b(as)f(so)r(on)h(as)f Fr(p)d(>)g(p)3129 2893 y Fn(0)3166 2881 y Fu(.)118 2987 y(This)j(means)f(that,)h(b)n(y)f(calling)1159 3225 y Fo(I)j Fu(=)1347 3121 y Fv(1)1334 3146 y Fq([)1321 3322 y Fp(p)p Fn(=0)1491 3146 y Fq([)1453 3341 y Fp(n)p Fv(2)p Fl(Z)1586 3313 y Fh(d)1699 3146 y Fq([)1634 3323 y Fp(j)s Fn(=1)p Fp(;)p Fn(2)p Fp(;)p Fn(3)1869 3225 y Fo(I)1920 3182 y Fn(\()p Fp(p)p Fn(\))1914 3248 y Fp(j)2010 3225 y Fu(\()p Fr(n)p Fu(\))p Fr(;)726 b Fu(\()p Fr(A)p Fu(12)p Fr(:)p Fu(13\))118 3511 y(so)27 b(that)h Fo(E)i Fu(=)23 b Fo(E)613 3481 y Fn(\(0\))721 3511 y Fo(n)18 b(I)6 b Fu(,)28 b(w)n(e)f(can)g(b)r(ound)h(meas)o(\()p Fo(I)6 b Fu(\))29 b(with)257 3779 y(meas)o(\()p Fo(I)6 b Fu(\))24 b Fo(\024)713 3700 y Fq(X)664 3882 y Fv(j)p Fp(n)p Fv(j\025N)851 3890 y Fi(0)920 3700 y Fq(X)897 3877 y Fp(j)s Fn(=1)p Fp(;)p Fn(2)1078 3779 y Fu(meas)o(\()p Fo(I)1341 3745 y Fn(\()p Fp(p)p Fn(\))1432 3779 y Fu(\()p Fr(n)p Fu(\)\))g Fo(\024)f Fu(const.)1966 3700 y Fq(X)1916 3882 y Fv(j)p Fp(n)p Fv(j\025N)2103 3890 y Fi(0)2173 3700 y Fq(X)2149 3877 y Fp(j)s Fn(=1)p Fp(;)p Fn(2)2331 3668 y Fp(p)2365 3676 y Fi(0)2397 3668 y Fn(\()p Fp(n;j)s Fn(\))2376 3700 y Fq(X)2377 3876 y Fp(p)p Fn(=0)2576 3723 y Fu(2)p Fr(C)2677 3735 y Fn(0)p 2565 3760 160 4 v 2565 3836 a Fo(j)p Fr(n)p Fo(j)2661 3812 y Fp(\034)2699 3795 y Fj(0)576 4025 y Fo(\024)693 3946 y Fq(X)664 4125 y Fp(n)p Fn(=)p Fv(N)810 4133 y Fi(0)856 4025 y Fr(n)906 3991 y Fv(\000)p Fp(\034)996 3966 y Fj(0)1036 4025 y Fu(log)14 b Fr(n)23 b Fo(\024)f Fr(b")1392 3985 y Fn(1+)p Fp(\030)1392 4047 y Fn(0)1512 4025 y Fr(;)2873 3899 y Fu(\()p Fr(A)p Fu(12)p Fr(:)p Fu(14\))118 4288 y(with)29 b Fr(\034)353 4258 y Fv(0)405 4288 y Fu(v)n(erifying)e(\(7.33\))g(and)h (\(7.46\),)f(and)h(a)g(constan)n(t)f Fr(\030)32 b Fu(smaller)27 b(than)h Fr(\030)2568 4258 y Fv(0)2620 4288 y Fu(in)g(order)f(to)h(tak) n(e)118 4394 y(in)n(to)j(accoun)n(t)f(the)h(logarithmic)f(corrections)f (due)j(to)f(\(A12.12\).)46 b(Then)31 b(the)g(b)r(ound)h(\(7.48\))e(is) 118 4500 y(pro)n(v)n(ed.)1391 4890 y Fw(References)118 5067 y Fu([1])78 b(D.)25 b(Bam)n(busi,)f Fs(Lyapunov)j(c)l(enter)f(the) l(or)l(em)h(for)g(some)g(nonline)l(ar)g(PDE's:)38 b(a)27 b(simple)284 5174 y(pr)l(o)l(of)p Fu(,)i(Ann.)38 b(Scuola)27 b(Norm.)36 b(Sup.)h(Pisa)27 b(Cl.)37 b(Sci.)g(\(4\))28 b Fk(29)f Fu(\(2000\),)f(823{837.)1612 5443 y(40)p eop %%Page: 41 41 41 40 bop 118 555 a Fu([2])78 b(J.)24 b(Bourgain,)e Fs(Construction)k (of)h(p)l(erio)l(dic)h(solutions)e(of)h(nonline)l(ar)g(wave)g(e)l (quations)284 662 y(in)j(higher)h(dimension)p Fu(,)e(Geom.)37 b(F)-7 b(unct.)38 b(Anal.)f Fk(5)27 b Fu(\(1995\),)f(629{639.)118 836 y([3])78 b(W.)26 b(Craig)d(and)i(C.E.)g(W)-7 b(a)n(yne,)25 b Fs(Newton)-8 b('s)27 b(metho)l(d)h(and)g(p)l(erio)l(dic)i(solutions)d (of)h(non-)284 942 y(line)l(ar)j(wave)f(e)l(quations)p Fu(,)e(Comm.)37 b(Pure)27 b(Appl.)37 b(Math.)g Fk(46)27 b Fu(\(1993\),)g(1409{1501.)118 1117 y([4])78 b(L.H.)40 b(Eliasson,)h Fs(A)n(bsolutely)g(c)l(onver)l(gent)g(series)g(exp)l (ansions)h(for)f(quasi-p)l(erio)l(dic)284 1223 y(motions)p Fu(,)28 b(Math.)37 b(Ph)n(ys.)f(Electron.)g(J.)27 b Fk(2)h Fu(\(1996\),)e(P)n(ap)r(er)g(4.)118 1397 y([5])78 b(G.)40 b(Galla)n(v)n(otti,)i Fs(Twistless)g(KAM)f(tori)p Fu(,)j(Comm.)73 b(Math.)h(Ph)n(ys.)f Fk(164)39 b Fu(\(1994\),)284 1504 y(145{154.)118 1657 y([6])78 b(G.)21 b(Galla)n(v)n(otti)f(and)g(G.)i (Gen)n(tile,)g Fs(Hyp)l(erb)l(olic)j(low-dimensional)h(invariant)e (tori)g(and)284 1764 y(summation)35 b(of)h(diver)l(gent)g(series)p Fu(,)g(Comm.)55 b(Math.)h(Ph)n(ys.)e Fk(227)33 b Fu(\(2002\),)h(no.)54 b(3,)284 1870 y(421{460.)118 2023 y([7])78 b(G.)28 b(Gen)n(tile,)f Fs(Quasi-p)l(erio)l(dic)k(solutions)f(for)g(two-level)h(systems)p Fu(,)c(Preprin)n(t,)f(2002.)118 2193 y([8])78 b(G.)30 b(Gen)n(tile)g(and)g(V.)g(Mastropietro,)f Fs(Metho)l(ds)k(for)g(the)f (analysis)h(of)g(the)f(Lindste)l(dt)284 2300 y(series)c(for)g(KAM)g (tori)g(and)f(r)l(enormalizability)j(in)e(classic)l(al)h(me)l(chanics.) 39 b(A)27 b(r)l(eview)284 2406 y(with)j(some)h(applic)l(ations)p Fu(,)f(Rev.)37 b(Math.)g(Ph)n(ys.)e Fk(8)28 b Fu(\(1996\),)e(393{444.) 118 2580 y([9])78 b(G.)32 b(Gen)n(tile)h(and)f(V.)g(Mastropietro,)g Fs(Construction)h(of)i(p)l(erio)l(dic)h(solutions)e(of)h(the)284 2686 y(nonline)l(ar)e(wave)g(e)l(quation)f(under)g(str)l(ong)f(irr)l (ationality)j(c)l(onditions)f(by)g(the)f(Lins-)284 2793 y(dte)l(dt)e(series)g(metho)l(d)p Fu(,)f(Preprin)n(t,)d(2002.)118 2962 y([10])36 b(G.)43 b(Gen)n(tile)g(and)f(V.)h(Mastropietro,)h Fs(A)n(nderson)f(lo)l(c)l(alization)j(for)e(the)g(Holstein)284 3069 y(mo)l(del)p Fu(,)29 b(Comm.)37 b(Math.)g(Ph)n(ys.)e Fk(215)27 b Fu(\(2000\),)g(no.)36 b(1,)28 b(69{103.)118 3243 y([11])36 b(G.)20 b(Gen)n(tile)g(and)f(V.)h(Mastropietro,)g Fs(R)l(enormalization)j(Gr)l(oup)g(for)g(one-dimension-)284 3349 y(al)31 b(fermions)g(A)f(r)l(eview)h(on)f(mathematic)l(al)i(r)l (esults)p Fu(,)27 b(Ph)n(ys.)38 b(Rep.)g Fk(352)27 b Fu(\(2001\),)g(no.)284 3456 y(4{6,)f(273{437.)118 3625 y([12])36 b(F.)26 b(Harary)e(and)h(E.M.)h(P)n(almer,)e Fs(Gr)l(aphic)l(al)31 b(enumer)l(ation)p Fu(,)26 b(Academic)f(Press,)g (New)284 3732 y(Y)-7 b(ork-London,)26 b(1973.)35 b(xiv+271)26 b(pp.)118 3901 y([13])36 b(S.B.)h(Kuksin,)i Fs(Ne)l(arly)g(inte)l(gr)l (able)f(in\014nite-dimensional)i(Hamiltonian)f(systems)p Fu(,)284 4008 y(Lecture)27 b(Notes)h(in)g(Mathematics)f(1556,)f (Springer,)g(Berlin,)h(1994.)118 4178 y([14])36 b(A.)28 b(M.)f(Ly)n(apuno)n(v,)f Fs(Pr)l(obl)n(\022)-40 b(eme)31 b(g)n(\023)-40 b(en)n(\023)g(er)l(al)30 b(de)g(la)g(stabilit)n(\023)-40 b(e)30 b(du)f(mouvement)p Fu(,)e(Ann.)38 b(Sc.)284 4284 y(F)-7 b(ac.)37 b(T)-7 b(oulouse)26 b Fk(2)i Fu(\(1907\),)e(203{474.) 118 4458 y([15])36 b(J.)e(P\177)-42 b(osc)n(hel,)35 b Fs(Quasi-p)l(erio)l(dic)j(solutions)e(for)h(a)g(nonline)l(ar)f(wave)h (e)l(quation)p Fu(,)g(Com-)284 4564 y(men)n(t.)g(Math.)g(Helv.)g Fk(71)27 b Fu(\(1996\),)g(no.)36 b(2,)28 b(269{296.)118 4739 y([16])36 b(C.E.)27 b(W)-7 b(a)n(yne,)27 b Fs(Perio)l(dic)k(and)f (quasi-p)l(erio)l(dic)i(solutions)d(of)h(nonline)l(ar)g(wave)g(e)l (qua-)284 4845 y(tions)g(via)h(KAM)e(the)l(ory)p Fu(,)g(Comm.)37 b(Math.)g(Ph)n(ys.)f Fk(127)26 b Fu(\(1990\),)h(no.)36 b(3,)28 b(479{528.)118 5019 y([17])36 b(A.)h(W)-7 b(einstein,)40 b Fs(Normal)f(mo)l(des)g(for)g(nonline)l(ar)f(Hamiltonian)i(systems)p Fu(,)f(In)n(v)n(en)n(t.)284 5126 y(Math.)e Fk(20)27 b Fu(\(1973\),)g(47{57.)1612 5443 y(41)p eop %%Page: 42 42 42 41 bop 118 555 a Fu([18])36 b(H.)c(Whitney)-7 b(,)32 b Fs(A)n(nalytic)i(extensions)e(of)i(di\013er)l(ential)g(functions)f (de\014ne)l(d)h(in)f(close)l(d)284 662 y(sets)p Fu(,)28 b(T)-7 b(rans.)35 b(Amer.)i(Math.)g(So)r(c.)g Fk(36)27 b Fu(\(1934\),)g(no.)36 b(1,)27 b(63{89.)1612 5443 y(42)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0307170315994--