Content-Type: multipart/mixed; boundary="-------------0009061145911" This is a multi-part message in MIME format. ---------------0009061145911 Content-Type: text/plain; name="00-341.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-341.keywords" generalized fractal dimensions, Renyi dimensions, entropy, Hausdorff dimension, Packing dimension ---------------0009061145911 Content-Type: application/postscript; name="Dim.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Dim.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: Dim.dvi %%Pages: 35 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips Dim.dvi -o %DVIPSParameters: dpi=1200, compressed %DVIPSSource: TeX output 2000.09.06:1834 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false 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Fl(\026)2216 11133 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\),)56 b Fm(\034)2820 11067 y Fz(\006)2799 11178 y Fl(\026)2937 11133 y Fo(\()p Fm(q)7 b Fo(\))58 b(and)g Fm(D)3732 11067 y Fz(\006)3727 11178 y Fl(\026)3849 11133 y Fo(\()p Fm(q)7 b Fo(\)?)79 b(After)58 b(summ)-5 b(urazing)55 b(some)h(w)-5 b(ell-kno)g(wn)3439 11631 y(3)p eop %%Page: 4 4 4 3 bop -201 1158 a Fo(results)68 b(\(mainly)g(Theorem)e(3.1\),)j(w)-5 b(e)69 b(study)e(for)h(whic)-5 b(h)69 b(set)f(of)f Fm(q)7 b Fo('s)68 b(the)f(functions)h Fm(\034)5860 1092 y Fz(\006)5839 1203 y Fl(\026)5977 1158 y Fo(\()p Fm(q)7 b Fo(\))p Fm(;)97 b(D)6511 1092 y Fz(\006)6506 1203 y Fl(\026)6629 1158 y Fo(\()p Fm(q)7 b Fo(\))67 b(are)-201 1384 y(\014nite)55 b(and)g Fm(D)762 1318 y Fz(\006)757 1429 y Fl(\026)880 1384 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(2)g Fo([0)p Fm(;)30 b Fo(1].)77 b(W)-15 b(e)55 b(shall)h(tak)-5 b(e)54 b(the)h(adv)-10 b(an)-5 b(tage)54 b(of)g(the)h(equiv)-10 b(alence)55 b(b)5 b(et)-5 b(w)g(een)54 b(the)h(in)-5 b(tegral)-201 1610 y 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b(haracterization)60 b(of)g(the)h(imp)5 b(ortan)-5 b(t)59 b(quan)-5 b(tit)g(y)60 b Fm(q)5412 3615 y Fz(\003)5405 3726 y Fl(\026)5559 3681 y Fo(de\014ned)g(as)1883 4101 y Fm(q)1971 4026 y Fz(\003)1964 4146 y Fl(\026)2108 4101 y Fo(:=)110 b(inf)14 b Fk(f)p Fm(q)67 b Fk(j)60 b Fm(\034)3083 4026 y Ff(+)3062 4146 y Fl(\026)3200 4101 y Fo(\()p Fm(q)7 b Fo(\))50 b Fm(<)g Fo(+)p Fk(1g)100 b(2)h Fo([)p Fk(\0001)p Fm(;)30 b Fo(1])g Fm(:)81 4504 y Fo(Next,)69 b(w)-5 b(e)67 b(study)f(the)h(regularit)-5 b(y)68 b(of)e(b)5 b(oth)67 b(the)g Fm(D)3647 4438 y Fz(\006)3642 4549 y Fl(\026)3764 4504 y Fo(\()p Fm(q)7 b Fo(\))66 b(and)h(the)g(related)g(functions)f Fm(\034)6244 4438 y Fz(\006)6223 4549 y Fl(\026)6362 4504 y Fo(\()p Fm(q)7 b Fo(\))66 b(on)h(the)-201 4730 y(set)58 b(\()p Fm(q)240 4664 y Fz(\003)233 4775 y Fl(\026)326 4730 y Fm(;)30 b Fo(+)p Fk(1)p Fo(\),)57 b(where)h(all)i(these)d(functions)h(are)g (\014nite.)80 b(W)-15 b(e)57 b(sho)-5 b(w)59 b(that)e(on)i(\()p Fm(q)5325 4664 y Fz(\003)5318 4775 y Fl(\026)5411 4730 y Fm(;)30 b Fo(+)p Fk(1)p Fo(\))57 b(b)5 b(oth)58 b Fm(\034)6472 4664 y Ff(+)6451 4775 y Fl(\026)6589 4730 y Fo(\()p Fm(q)7 b Fo(\))57 b(and)-201 4956 y Fm(\034)-100 4890 y Fz(\000)-121 5001 y Fl(\026)17 4956 y Fo(\()p Fm(q)7 b Fo(\))55 b(are)g(con)-5 b(tin)g(uous.)79 b(More)55 b(precisely)h(w)-5 b(e)55 b(sho)-5 b(w)56 b(\(P)-5 b(oin)g(t)55 b(\(i\))g(of)g(Theorem)f(3.2\))h (that)g(for)g(an)-5 b(y)55 b Fm(A)c(>)f(q)7083 4890 y Fz(\003)7076 5001 y Fl(\026)-201 5182 y Fo(there)60 b(exists)g Fm(K)13 b Fo(\()p Fm(A)p Fo(\))60 b(suc)-5 b(h)60 b(that)1491 5585 y Fk(8)30 b Fm(q)7 b(;)30 b(r)55 b Fk(2)c Fo([)p Fm(A;)30 b Fo(+)p Fk(1)p Fo(\))p Fm(;)210 b Fk(j)p Fm(\034)3176 5509 y Fz(\006)3293 5585 y Fo(\()p Fm(q)7 b Fo(\))39 b Fk(\000)h Fm(\034)3845 5509 y Fz(\006)3962 5585 y Fo(\()p Fm(r)5 b Fo(\))p Fk(j)50 b(\024)g Fm(K)13 b Fo(\()p Fm(A)p Fo(\))p Fk(j)p Fm(q)46 b Fk(\000)39 b Fm(r)5 b Fk(j)p Fm(:)-201 5988 y Fo(This)42 b(in)f(turn)g(implies)g(the)f(con)-5 b(tin)g(uit)g(y)41 b(of)f(b)5 b(oth)41 b(the)g(functions)f Fm(D)4313 5922 y Ff(+)4308 6033 y Fl(\026)4430 5988 y Fo(\()p Fm(q)7 b Fo(\))40 b(and)h Fm(D)5190 5922 y Fz(\000)5185 6033 y Fl(\026)5308 5988 y Fo(\()p Fm(q)7 b Fo(\))40 b(on)h(\()p Fm(q)5970 5922 y Fz(\003)5963 6033 y Fl(\026)6056 5988 y Fm(;)60 b Fo(1\))q Fk([)q Fo(\(1)p Fm(;)g Fo(+)p Fk(1)p Fo(\).)-201 6213 y(Some)51 b(consequences)g(on)h(the)g(deriv)-10 b(ativ)-5 b(es)52 b(of)g Fm(\034)3112 6147 y Ff(+)3091 6258 y Fl(\026)3230 6213 y Fo(\()p Fm(q)7 b Fo(\))51 b(and)h Fm(\034)3957 6147 y Fz(\000)3936 6258 y Fl(\026)4075 6213 y Fo(\()p Fm(q)7 b Fo(\))51 b(are)i(then)e(deriv)-5 b(ed)53 b(\(P)-5 b(oin)g(t)52 b(\(ii\))h(and)f(\(iii\))-201 6439 y(of)69 b(Theorem)g(3.2,)i(Corollary)g(3.3\).)107 b(Some)68 b(regularit)-5 b(y)70 b(results)f(\(di\013eren)-5 b(tiabilit)g(y\))71 b(w)-5 b(ere)70 b(obtained)g(for)-201 6665 y(particular)g(classes)e(of)h(measures)e(that)h(arise)i(in)f (dynamical)f(systems,)g(lik)-5 b(e)70 b(quasi-Bernoulli)g(measure)-201 6891 y([20],)63 b(but)h(not)g(in)g(full)h(generalit)-5 b(y)-15 b(.)91 b(Note)63 b(that)h(the)f(Lipsc)-5 b(hitz)64 b(con)-5 b(tin)g(uit)g(y)64 b(of)f Fm(\034)5412 6825 y Ff(+)5391 6936 y Fl(\026)5529 6891 y Fo(\()p Fm(q)7 b Fo(\))64 b(is)g(already)g(kno)-5 b(wn)-201 7117 y(as)58 b(a)g(direct)g(consequence)e(of)i(its)g(con)-5 b(v)g(exit)g(y)-15 b(.)78 b(That)58 b(the)g(result)g(remains)e(true)i(for)f Fm(\034)5723 7051 y Fz(\000)5702 7162 y Fl(\026)5841 7117 y Fo(\()p Fm(q)7 b Fo(\))57 b(w)-5 b(as)57 b(not)h(clear)-201 7342 y(at)i(all.)81 7568 y(W)-15 b(e)51 b(then)g(establish)g(some)e (inequalities)k(that)d(mak)-5 b(e)49 b(more)h(precise)h(the)g(b)5 b(eha)-5 b(viour)52 b(of)e(the)h(functions)-201 7794 y Fm(\034)-100 7728 y Fz(\006)-121 7839 y Fl(\026)17 7794 y Fo(\()p Fm(q)7 b Fo(\))62 b(at)f Fk(\0061)h Fo(\(Prop)5 b(osition)62 b(3.3\).)83 b(Finally)-15 b(,)63 b(w)-5 b(e)63 b(discuss)e(the)h(relation)g(b)5 b(et)-5 b(w)g(een)62 b(the)g(lo)5 b(cal)62 b(exp)5 b(onen)-5 b(ts)-201 8020 y(of)60 b(the)g(measure)f Fm(\026)h Fo(and)h(the)f(b)5 b(eha)-5 b(viour)61 b(of)f Fm(D)3072 7954 y Fz(\006)3067 8065 y Fl(\026)3189 8020 y Fo(\()p Fm(q)7 b Fo(\))60 b(at)g Fk(\0061)g Fo(\(Prop)5 b(ositions)60 b(3.4)g(and)h(3.5\).)81 8344 y(Section)80 b(4)f(is)g(dev)-5 b(oted)79 b(to)g(the)g(study)g(of)f (the)h(critical)i(p)5 b(oin)-5 b(t)79 b Fm(q)89 b Fo(=)81 b(1.)136 b(As)80 b(pro)-5 b(v)g(en)79 b(in)h(Section)f(3,)-201 8569 y Fm(D)-45 8503 y Ff(+)-50 8614 y Fl(\026)73 8569 y Fo(\()p Fm(q)7 b Fo(\))64 b(and)h Fm(D)881 8503 y Fz(\000)876 8614 y Fl(\026)999 8569 y Fo(\()p Fm(q)7 b Fo(\))64 b(are)h(con)-5 b(tin)g(uous)65 b(on)g(\()p Fm(q)2932 8503 y Fz(\003)2925 8614 y Fl(\026)3018 8569 y Fm(;)30 b Fo(+)p Fk(1)p Fo(\))64 b(except,)h(ma)-5 b(yb)5 b(e,)64 b(at)h(the)g(p)5 b(oin)-5 b(t)65 b Fm(q)g Fo(=)57 b(1.)94 b(W)-15 b(e)65 b(made)-201 8795 y(the)f(c)-5 b(hoice)65 b(to)f(de\014ne)g(the)g(information)f (dimension)h(\(or)g(en)-5 b(trop)g(y)64 b(dimension\))f Fm(D)5639 8729 y Fz(\006)5634 8840 y Fl(\026)5757 8795 y Fo(\(1\))56 b(:=)g Fm(h)6400 8729 y Fz(\006)6400 8840 y Fl(\026)6518 8795 y Fo(,)65 b(as)f(it)h(is)-201 9021 y(usually)e(done)e(in)h(the)g(literature)g([12)o(],)f([30].)83 b(But)61 b(there)h(is)g(no)f(need)h(for)f Fm(D)5126 8955 y Fz(\006)5121 9066 y Fl(\026)5244 9021 y Fo(\(1\))g(to)g(b)5 b(e)62 b(equal)g(to)f(either)-201 9247 y Fm(D)-45 9181 y Fz(\006)-50 9292 y Fl(\026)73 9247 y Fo(\(1)35 b Fk(\000)f Fo(0\))57 b(or)i Fm(D)1042 9181 y Fz(\006)1037 9292 y Fl(\026)1159 9247 y Fo(\(1)35 b(+)f(0\).)79 b(This)59 b(is)f(actually)g(not)g(true)f(in)i(full)f(generalit)-5 b(y)-15 b(.)80 b(A)58 b(relation)h(that)e(certainly)-201 9473 y(holds)k(is)g(the)f(follo)-5 b(wing)62 b(\(Prop)5 b(osition)60 b(4.2\):)2144 9876 y Fm(D)2300 9801 y Fz(\006)2295 9921 y Fl(\026)2418 9876 y Fo(\(1)40 b(+)g(0\))50 b Fk(\024)g Fm(D)3362 9801 y Fz(\006)3357 9921 y Fl(\026)3480 9876 y Fo(\(1\))f Fk(\024)h Fm(D)4110 9801 y Fz(\006)4105 9921 y Fl(\026)4228 9876 y Fo(\(1)40 b Fk(\000)g Fo(0\))p Fm(:)-201 10279 y Fo(In)58 b(addition)g(to)g(that,)f(as)h(it)g(is)g (kno)-5 b(wn,)58 b(man)-5 b(y)56 b(other)i(dimensions)e(arise)i(in)h (the)e(t)-5 b(w)g(o)58 b(in)-5 b(terv)-10 b(als)58 b([)p Fm(D)6714 10213 y Fz(\006)6709 10324 y Fl(\026)6831 10279 y Fo(\(1)35 b(+)-201 10504 y(0\))p Fm(;)30 b(D)198 10438 y Fz(\006)193 10549 y Fl(\026)315 10504 y Fo(\(1)43 b Fk(\000)f Fo(0\)],)64 b(and)h(in)g(particular)g(the)f(Hausdor\013)h (and)f(pac)-5 b(king)65 b(dimensions.)90 b(One)64 b(sho)-5 b(ws)64 b(\(Prop)5 b(o-)-201 10730 y(sition)61 b(4.1\))f(that)894 11133 y Fm(D)1050 11058 y Fz(\006)1045 11178 y Fl(\026)1168 11133 y Fo(\(1)40 b(+)g(0\))50 b Fk(\024)g Fm(\026)70 b Fk(\000)40 b Fo(ess)p Fm(:)p Fo(inf)j Fm(\015)2948 11058 y Fz(\006)2938 11178 y Fl(\026)3066 11133 y Fo(\()p Fm(x)p Fo(\))50 b Fk(\024)g Fm(\026)70 b Fk(\000)40 b Fo(ess)p Fm(:)p Fo(sup)29 b Fm(\015)4599 11058 y Fz(\006)4589 11178 y Fl(\026)4717 11133 y Fo(\()p Fm(x)p Fo(\))50 b Fk(\024)g Fm(D)5361 11058 y Fz(\006)5356 11178 y Fl(\026)5479 11133 y Fo(\(1)40 b Fk(\000)f Fo(0\))p Fm(;)3439 11631 y Fo(4)p eop %%Page: 5 5 5 4 bop -201 1158 a Fo(where)67 b Fm(\015)435 1092 y Fz(\006)425 1203 y Fl(\026)553 1158 y Fo(\()p Fm(x)p Fo(\))f(are)g(the)g(lo)-5 b(w)g(er)67 b(and)f(upp)5 b(er)66 b(lo)5 b(cal)67 b(exp)5 b(onen)-5 b(ts.)96 b(It)66 b(is)h(w)-5 b(ell)67 b(kno)-5 b(wn)66 b(that)g Fm(\026)30 b Fo(ess)p Fm(:)p Fo(sup)f Fm(\015)6806 1092 y Fz(\006)6796 1203 y Fl(\026)6924 1158 y Fo(\()p Fm(x)p Fo(\))-201 1384 y(are)79 b(equal)g(to)f(resp)5 b(ectiv)-5 b(ely)79 b(the)f(Hausdor\013) h(and)f(pac)-5 b(king)79 b(dimension)f(of)g Fm(\026)g Fo([11)o(],)83 b([20)o(].)134 b(Finally)80 b(the)-201 1610 y(picture)61 b(is)g(sho)-5 b(wn)60 b(to)g(b)5 b(e)61 b(complete.)78 b(Indeed)60 b(one)h(alw)-5 b(a)g(ys)60 b(has)1030 2011 y Fm(\026;)30 b Fo(ess)p Fm(:)p Fo(inf)43 b Fm(\015)1851 1936 y Fz(\000)1841 2056 y Fl(\026)1969 2011 y Fo(\()p Fm(x)p Fo(\))50 b Fk(\024)g Fm(D)2613 1936 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b(and)i(their)g(discretized)f(equiv)-10 b(alen)-5 b(t)48 b Fm(S)6631 4920 y Fl(\026)6724 4893 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\).)-201 5119 y(These)89 b(results)g(ma)-5 b(y)87 b(b)5 b(e)89 b(seen)f(as)h(an)g(illustration)h(of)f(the)f(in)-5 b(terest)89 b(of)f(the)h(equiv)-10 b(alence)89 b(pro)-5 b(v)g(ed)89 b(in)-201 5345 y(Section)61 b(2)f(that)g(enables)g(one)h (to)f(w)-5 b(ork)61 b(with)g(simpli\014ed)f(to)5 b(ols.)81 5668 y(In)48 b(Section)g(6,)i(as)d(an)h(illustration)h(of)e(their)i(b)5 b(eha)-5 b(viour)48 b(w)-5 b(e)48 b(pro)-5 b(vide)48 b(calculations)g(of)g(the)f(generalized)-201 5894 y(fractal)52 b(dimensions)f Fm(D)1441 5828 y Fz(\006)1436 5939 y Fl(\026)1559 5894 y Fo(\()p Fm(q)7 b Fo(\))51 b(for)h(some)e(sp)5 b(eci\014c)52 b(examples.)76 b(The)52 b(\014rst)g(example)f(is)i(a)f (complete)e(analysis)-201 6120 y(of)80 b(a)g(discrete)g(measure)e (\(with)i(one)g(accum)-5 b(ulating)79 b(p)5 b(oin)-5 b(t)80 b(in)h(its)f(supp)5 b(ort\))79 b(whic)-5 b(h)81 b(p)5 b(ossesses)78 b(a)i(non)-201 6346 y(trivial)63 b(sp)5 b(ectrum)60 b(of)h(dimensions.)82 b(The)62 b(existence)f(of)g (suc)-5 b(h)61 b(measures)f(is)i(of)f(imp)5 b(ortance)60 b(in)i(quan)-5 b(tum)-201 6571 y(dynamics)66 b([2,)h(3].)101 b(Indeed,)68 b(p)5 b(oin)-5 b(t)68 b(measures)e(ha)-5 b(ving)67 b(Hausdor\013)h(and)g(pac)-5 b(king)68 b(dimension)e(zero,)j (it)f(is)-201 6797 y(of)k(in)-5 b(terest)72 b(to)g(kno)-5 b(w)72 b(that)g(nev)-5 b(ertheless)72 b(the)f(generalized)i(fractal)f (dimensions)f Fm(D)5792 6731 y Fz(\006)5787 6842 y Fl(\026)5910 6797 y Fo(\()p Fm(q)7 b Fo(\))71 b(ma)-5 b(y)71 b(b)5 b(e)72 b(non)-201 7023 y(trivial)62 b(for)e Fm(q)e(<)50 b Fo(1)60 b(\(whic)-5 b(h)61 b(w)-5 b(ould)61 b(lead)g(to)f(non)h (trivial)h(di\013usion)f(in)g(quan)-5 b(tum)59 b(dynamics\).)81 7249 y(Other)43 b(examples)f(are)h(useful)g(illustrations)h(concerning) f(the)g(results)g(of)g(Sections)g(2-4.)74 b(In)43 b(particular,)-201 7475 y(Example)48 b(8)h(sho)-5 b(w)50 b(that)e(the)h(discrete)g(and)h (con)-5 b(tin)g(uous)49 b(v)-5 b(ersion)49 b(of)g(the)g(generalized)h (fractal)f(dimensions)-201 7701 y Fm(D)-45 7635 y Fz(\006)-50 7745 y Fl(\026)73 7701 y Fo(\()p Fm(q)7 b Fo(\))60 b(are)g(not)g(equiv) -10 b(alen)-5 b(t)61 b(for)f(negativ)-5 b(e)60 b Fm(q)7 b Fo('s.)81 8024 y(Finally)-15 b(,)86 b(in)81 b(the)e(App)5 b(endix,)86 b(w)-5 b(e)80 b(compare,)i(for)e Fm(q)90 b Fk(2)83 b Fo(\(0)p Fm(;)30 b Fo(1\),)84 b(the)c(dimensions)e Fm(D)5974 7958 y Fz(\006)5969 8069 y Fl(\026)6092 8024 y Fo(\()p Fm(q)7 b Fo(\))79 b(and)h(some)-201 8249 y(sligh)-5 b(tly)79 b(di\013eren)-5 b(t)78 b(dimensions,)i(where)e Fm(\026)p Fo(\()p Fm(x)51 b Fk(\000)g Fm(";)30 b(x)51 b Fo(+)g Fm(")p Fo(\))79 b(=)4233 8112 y Fg(R)4374 8249 y Fm(\037)4488 8286 y Ff([)p Fz(\000)p Ff(1)p Fl(;)p Ff(1])4865 8249 y Fo(\(\()p Fm(x)51 b Fk(\000)g Fm(y)7 b Fo(\))p Fm(=")p Fo(\)d)p Fm(y)84 b Fo(is)78 b(replaced)f(b)-5 b(y)-201 8502 y Fm(\026)-91 8436 y Ff(\()p Fl(R)q Ff(\))134 8502 y Fo(\()p Fm(x)24 b Fk(\000)g Fm(";)30 b(x)24 b Fo(+)g Fm(")p Fo(\))49 b(=)1356 8364 y Fg(R)1497 8502 y Fm(R)q Fo(\(\()p Fm(x)24 b Fk(\000)g Fm(y)7 b Fo(\))p Fm(=")p Fo(\)d)p Fm(y)g Fo(.)76 b(Here)53 b Fm(\037)3346 8539 y Ff([)p Fz(\000)p Ff(1)p Fl(;)p Ff(1])3775 8502 y Fo(is)f(the)h(c)-5 b(haracteristic)52 b(function)g(of)g(the)g(in)-5 b(terv)-10 b(al)-201 8728 y([)p Fk(\000)p Fo(1)p Fm(;)30 b Fo(1])61 b(and)i Fm(R)h Fo(is)f(an)-5 b(y)63 b(p)5 b(ositiv)-5 b(e)63 b(function)f(with)h(fast)f(deca)-5 b(y)62 b(at)h Fk(\0061)p Fo(.)86 b(W)-15 b(e)62 b(pro)-5 b(v)g(e)63 b(that)f Fm(\026)p Fo(\()p Fm(x)41 b Fk(\000)g Fm(";)30 b(x)42 b Fo(+)f Fm(")p Fo(\))-201 8953 y(and)62 b Fm(\026)264 8887 y Ff(\()p Fl(R)q Ff(\))488 8953 y Fo(\()p Fm(x)41 b Fk(\000)g Fm(";)30 b(x)40 b Fo(+)h Fm(")p Fo(\))61 b(giv)-5 b(e)62 b(rise)f(to)h(the)f(same)e(dimension)i (n)-5 b(um)g(b)5 b(ers,)59 b(namely)i Fm(D)5736 8887 y Fz(\006)5731 8998 y Fl(\026)5853 8953 y Fo(\()p Fm(q)7 b Fo(\).)83 b(That)61 b(w)-5 b(as)62 b(of)-201 9179 y(tec)-5 b(hnical)61 b(imp)5 b(ortance)60 b(in)h([2,)f(3])h(and)g([16)o(].)81 b(In)61 b([2)o(,)g(3])f(that)h(equiv)-10 b(alence)61 b(w)-5 b(as)61 b(pro)-5 b(v)g(ed)61 b(to)f(hold)i(in)f(some)-201 9405 y(particular)68 b(cases)d(\(lik)-5 b(e)68 b(compactly)d(supp)5 b(orted)66 b(measures\).)96 b(In)66 b(this)h(app)5 b(endix)67 b(w)-5 b(e)67 b(use)f(the)h(ideas)f(w)-5 b(e)-201 9631 y(dev)g(elopp)5 b(ed)61 b(in)g(Section)f(2)h(to)f(get)g(the)g(equiv)-10 b(alence)61 b(of)f(these)f(t)-5 b(w)g(o)61 b(dimensions)e(in)i(full)g (generalit)-5 b(y)-15 b(.)-201 10203 y Fp(2)270 b(De\014nitions)86 b(of)g(the)f(generalized)f(fractal)f(dimensions)k(and)f(equiv-)203 10502 y(alence)-201 10907 y Fo(The)58 b(main)f(result)h(of)g(this)g (section)g(is)g(concerned)f(with)i(the)e(equiv)-10 b(alen)-5 b(t)59 b(de\014nitions)f(of)f(the)h(generalized)-201 11133 y(fractal)66 b(dimensions)f Fm(D)1469 11067 y Fz(\006)1464 11178 y Fl(\026)1587 11133 y Fo(\()p Fm(q)7 b Fo(\).)96 b(As)67 b(already)g(men)-5 b(tionned)64 b(in)j(the)f(in)-5 b(tro)5 b(duction,)67 b(the)f(results)g(presen)-5 b(ted)3439 11631 y(5)p eop %%Page: 6 6 6 5 bop -201 1158 a Fo(here)61 b(are)f(already)h(kno)-5 b(wn)60 b(for)h Fm(q)c(>)50 b Fo(1)60 b(\(see)g(e.g.)79 b([26)o(]\).)h(Ho)-5 b(w)g(ev)g(er,)60 b(to)g(our)h(b)5 b(est)60 b(kno)-5 b(wledge,)60 b(for)g(general)-201 1384 y(probabilit)-5 b(y)65 b(measures)d(there)h(is)h(no)f(pro)5 b(of)64 b(in)g(the)f(case)g Fm(q)g Fk(\024)55 b Fo(1,)64 b(whic)-5 b(h)64 b(is)g(a)g(quite)f(di\013eren)-5 b(t)64 b(regime)f(as)-201 1610 y(explained)e(in)g(the)g(pro)5 b(of)60 b(of)g(Lemma)d(2.1.)81 1836 y(F)-15 b(rom)65 b(no)-5 b(w)66 b(on,)h(all)g(considered)f(measures,)f(denoted)g(b)-5 b(y)66 b Fm(\026)g Fo(or)g Fm(\027)12 b Fo(,)66 b(are)h(p)5 b(ositiv)-5 b(e)66 b(probabilit)-5 b(y)67 b(Borel)-201 2062 y(measures.)78 b(W)-15 b(e)60 b(\014rst)h(de\014ne)f(the)g(quan)-5 b(tities)60 b(that)g(are)h(of)f(in)-5 b(terest)60 b(for)g(us.)-201 2422 y Fc(De\014nition)71 b(2.1.)90 b Fn(The)64 b(functions)h Fm(\034)2454 2356 y Fz(\006)2433 2467 y Fl(\026)2572 2422 y Fn(.)-201 2648 y(F)-14 b(or)65 b Fm(q)57 b Fk(2)51 b Fj(R)100 b Fn(and)65 b Fm(")50 b Fk(2)h Fo(\(0)p Fm(;)30 b Fo(1\))p Fn(,)64 b(we)g(c)-9 b(onsider)64 b(the)h(fol)9 b(lowing)64 b(functions)g(with)h(values)f(in)h Fj(R)76 b Fk([)40 b(f)p Fo(+)p Fk(1g)1774 3104 y Fm(I)1854 3131 y Fl(\026)1948 3104 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))49 b(=)2585 2874 y Fg(Z)2677 3251 y Ff(supp)q Fl(\026)3091 3104 y Fm(\026)p Fo(\([)p Fm(x)39 b Fk(\000)h Fm(";)30 b(x)40 b Fo(+)g Fm(")p Fo(\)\))4367 3029 y Fl(q)5 b Fz(\000)p Ff(1)4623 3104 y Fo(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))64 b Fm(;)1601 b Fo(\(2.1\))-201 3584 y Fn(wher)-9 b(e)71 b Fo(supp)p Fm(\026)g Fn(is)i(the)f(supp)-9 b(ort)72 b(of)f(the)i(me)-9 b(asur)g(e)70 b Fm(\026)p Fn(.)104 b(We)72 b(then)g(de\014ne)g(the)g(two)h(functions)f(with)g(values)-201 3810 y(in)65 b Fj(R)76 b Fk([)41 b(f)p Fo(+)p Fk(1g)934 4296 y Fm(\034)1035 4221 y Fz(\000)1014 4341 y Fl(\026)1152 4296 y Fo(\()p Fm(q)7 b Fo(\))50 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b(In)-201 6531 y(the)72 b(sp)5 b(eci\014c)73 b(case)f Fm(q)77 b Fo(=)70 b(2)j(a)f(rigorous)h(pro)5 b(of)72 b(is)h(a)-5 b(v)-10 b(ailable)74 b(in)f([17)o(],)i([1)o(].)116 b(T)-15 b(o)73 b(our)g(kno)-5 b(wledge)73 b(the)f(b)5 b(est)-201 6757 y(kno)-5 b(wn)80 b(result)g(concerns)f(P)-5 b(oin)g(t)79 b(\(i\))h(of)f(Theorem)f(2.1)h(whic)-5 b(h)81 b(has)e(b)5 b(een)79 b(pro)-5 b(v)g(ed)80 b(to)f(hold)h(for)g Fm(q)89 b(>)81 b Fo(1)-201 6983 y(only)e(\(see)e(e.g.)132 b([26,)82 b Fk(x)p Fo(18])77 b(and)h(references)f(therein\).)133 b(But)78 b(as)f(w)-5 b(e)79 b(shall)g(discuss)e(it)i(in)f(the)g(pro)5 b(of)78 b(of)-201 7209 y(Theorem)50 b(2.1)g(the)h(regime)f Fm(q)57 b(>)50 b Fo(1)h(is)g(m)-5 b(uc)g(h)49 b(di\013eren)-5 b(t)51 b(\(and)g(simpler\))f(than)h(the)g(regime)e Fm(q)58 b Fk(2)50 b Fo(\(0)p Fm(;)30 b Fo(1])50 b(where)-201 7435 y(subtle)56 b(e\013ects)f(of)g(in)-5 b(terv)-10 b(als)57 b(of)e(small)g(w)-5 b(eigh)g(t)56 b(ma)-5 b(y)55 b(happ)5 b(en.)78 b(F)-15 b(rom)54 b(this)i(p)5 b(oin)-5 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b(0.)74 b(W)-15 b(e)42 b(sho)-5 b(w)42 b(that)g Fm(S)1999 10979 y Fz(f)p Fl(t)p Fz(g)1989 11096 y Fl(\026)2199 11075 y Fo(\(1)p Fm(;)30 b(")p Fo(\))49 b Fk(\024)h Fm(S)2951 11102 y Fl(\026)3044 11075 y Fo(\(1)p Fm(;)30 b(")p Fo(\))t(+)t(2.)72 b(De\014ne)42 b(b)-5 b(y)43 b Fm(f)61 b Fo(the)42 b(function)g Fm(f)20 b Fo(\()p Fm(x)p Fo(\))50 b(=)g Fm(x)30 b Fo(log)s(\(1)p Fm(=x)p Fo(\),)3439 11631 y(9)p eop %%Page: 10 10 10 9 bop -201 1171 a Fm(x)51 b(>)f Fo(0,)i(and)f Fm(f)20 b Fo(\(0\))49 b(=)h(0.)77 b(Hence)51 b Fm(S)2139 1075 y Fz(f)p Fl(t)p Fz(g)2129 1192 y Fl(\026)2339 1171 y Fo(\(1)p Fm(;)30 b(")p Fo(\))49 b(=)2979 1043 y Fg(P)3155 1217 y Fl(j)8 b Fz(2)p Fe(Z)3441 1171 y Fm(f)20 b Fo(\()p Fm(\026)p Fo(\()p Fm(t")h Fo(+)g Fm(I)4217 1198 y Fl(j)4287 1171 y Fo(\)\).)77 b(An)-5 b(y)51 b(in)-5 b(terv)-10 b(al)51 b Fm(t")21 b Fo(+)g Fm(I)5998 1198 y Fl(j)6121 1171 y Fo(in)-5 b(tersects)50 b(the)-201 1397 y(t)-5 b(w)g(o)50 b(in)-5 b(terv)-10 b(als)50 b Fm(I)930 1424 y Fl(j)1052 1397 y 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1979 y Fl(j)4376 1952 y Fo(\),)55 b(and)g Fm(\014)5014 1856 y Ff(\(2\))5004 2004 y Fl(j)5253 1952 y Fo(=)50 b Fm(\026)p Fo(\()p Fm(I)5706 1979 y Fl(j)5808 1952 y Fk(\\)30 b Fo(\()p Fm(t")g Fo(+)g Fm(I)6463 1979 y Fl(j)6533 1952 y Fo(\)\))p Fm(=\026)p Fo(\()p Fm(I)7027 1979 y Fl(j)7099 1952 y Fo(\))-201 2242 y(\(with)61 b(the)f(con)-5 b(v)g(en)g(tion)60 b(that)g Fm(\014)2013 2146 y Ff(\(1)p Fl(;)p Ff(2\))2003 2293 y Fl(j)2362 2242 y Fo(=)50 b(0)60 b(if)h Fm(\026)p Fo(\()p Fm(I)3133 2269 y Fl(j)3205 2242 y Fo(\))50 b(=)g(0\).)80 b(So)60 b(that)g(one)h(has)2880 2671 y Fm(\014)2993 2575 y Ff(\(1\))2983 2723 y Fl(j)3222 2671 y Fo(+)39 b Fm(\014)3516 2575 y Ff(\(2\))3506 2723 y Fl(j)3755 2671 y Fo(=)50 b(1)p Fm(;)-201 3037 y Fo(and)2016 3262 y Fm(\026)p Fo(\()p Fm(t")39 b Fo(+)h Fm(I)2649 3289 y Fl(j)2722 3262 y Fo(\))50 b(=)g Fm(\014)3148 3166 y Ff(\(2\))3138 3314 y Fl(j)3336 3262 y Fm(\026)p Fo(\()p Fm(I)3597 3289 y Fl(j)3669 3262 y Fo(\))40 b(+)g Fm(\014)4075 3166 y Ff(\(1\))4065 3314 y Fl(j)8 b Ff(+1)4318 3262 y Fm(\026)p Fo(\()p Fm(I)4579 3289 y Fl(j)g Ff(+1)4832 3262 y Fo(\))p Fm(:)-201 3575 y Fo(Finally)-15 b(,)50 b(note)c(that)g(for)h(an)-5 b(y)46 b Fm(a;)30 b(b)51 b(>)f Fo(0,)e(one)f(has)f Fm(f)20 b Fo(\()p Fm(a)13 b Fo(+)g Fm(b)p Fo(\))48 b Fk(\024)i Fm(f)20 b Fo(\()p Fm(a)p Fo(\))13 b(+)g Fm(f)20 b Fo(\()p Fm(b)p Fo(\).)73 b(Using)47 b(the)f(previous)h(notations)-201 3800 y(and)61 b(remarks,)d(and)j(the)f(fact)f(that)h Fm(f)20 b Fo(\()p Fm(x)p Fo(\))50 b Fk(\024)g Fo(1,)60 b(one)g(has)-177 4198 y Fm(S)-55 4123 y Fz(f)p Fl(t)p Fz(g)-65 4243 y Fl(\026)146 4198 y Fo(\(1)p Fm(;)30 b(")p Fo(\))165 b(=)1021 4036 y Fg(X)1017 4404 y Fl(j)8 b Fz(2)p Fe(Z)1295 4198 y Fm(f)20 b Fo(\()p Fm(\014)1588 4102 y Ff(\(2\))1578 4249 y Fl(j)1775 4198 y Fm(\026)p Fo(\()p Fm(I)2036 4225 y Fl(j)2109 4198 y Fo(\))40 b(+)g Fm(\014)2515 4102 y Ff(\(1\))2505 4249 y Fl(j)8 b Ff(+1)2758 4198 y Fm(\026)p Fo(\()p Fm(I)3019 4225 y Fl(j)g Ff(+1)3271 4198 y Fo(\)\))710 4857 y Fk(\024)1021 4696 y Fg(X)1017 5064 y Fl(j)g Fz(2)p Fe(Z)1295 4857 y Fm(\014)1408 4762 y Ff(\(2\))1398 4909 y Fl(j)1596 4857 y Fm(\026)p Fo(\()p Fm(I)1857 4884 y Fl(j)1929 4857 y Fo(\))2030 4520 y Fg(0)2030 4819 y(@)2176 4857 y Fo(log)2617 4734 y(1)p 2461 4815 404 7 v 2461 4982 a Fm(\026)p Fo(\()p Fm(I)2722 5009 y Fl(j)2794 4982 y Fo(\))2925 4857 y(+)40 b(log)3497 4734 y(1)p 3392 4815 302 7 v 3392 5031 a Fm(\014)3505 4935 y Ff(\(2\))3495 5082 y Fl(j)3713 4520 y Fg(1)3713 4819 y(A)3899 4857 y Fo(+)f Fm(\014)4193 4762 y Ff(\(1\))4183 4909 y Fl(j)8 b Ff(+1)4437 4857 y Fm(\026)p Fo(\()p Fm(I)4698 4884 y Fl(j)g Ff(+1)4950 4857 y Fo(\))5051 4520 y Fg(0)5051 4819 y(@)5197 4857 y Fo(log)5728 4734 y(1)p 5482 4815 585 7 v 5482 4982 a Fm(\026)p Fo(\()p Fm(I)5743 5009 y Fl(j)g Ff(+1)5995 4982 y Fo(\))6126 4857 y(+)40 b(log)6726 4734 y(1)p 6593 4815 357 7 v 6593 5031 a Fm(\014)6706 4935 y Ff(\(1\))6696 5082 y Fl(j)8 b Ff(+1)6969 4520 y Fg(1)6969 4819 y(A)710 5418 y Fk(\024)1021 5256 y Fg(X)1017 5624 y Fl(j)g Fz(2)p Fe(Z)1265 5418 y Fo(\()p Fm(\014)1449 5322 y Ff(\(1\))1439 5469 y Fl(j)1677 5418 y Fo(+)40 b Fm(\014)1972 5322 y Ff(\(2\))1962 5469 y Fl(j)2160 5418 y Fo(\))p Fm(\026)p Fo(\()p Fm(I)2492 5445 y Fl(j)2564 5418 y Fo(\))30 b(log)3106 5295 y(1)p 2950 5375 404 7 v 2950 5542 a Fm(\026)p Fo(\()p Fm(I)3211 5569 y Fl(j)3283 5542 y Fo(\))3414 5418 y(+)40 b(2)3721 5256 y Fg(X)3717 5624 y Fl(j)8 b Fz(2)p Fe(Z)3995 5418 y Fm(\026)p Fo(\()p Fm(I)4256 5445 y Fl(j)4328 5418 y Fo(\))710 5869 y(=)165 b Fm(S)1129 5896 y Fl(\026)1222 5869 y Fo(\(1)p Fm(;)30 b(")p Fo(\))39 b(+)h(2)30 b Fm(:)7028 6235 y Fb(\003)81 6461 y Fo(W)-15 b(e)84 b(turn)g(to)g(the)g(main)f(result)i(of)e(this)i (section,)k(that)83 b(is)i(Theorem)167 b(2.1,)89 b(whic)-5 b(h)85 b(relies)g(on)f(the)-201 6687 y(follo)-5 b(wing)64 b(three)f(lemmas.)83 b(They)63 b(constitute)f(the)h(pro)5 b(of)62 b(of)g(Theorem)g(2.1)g(\(for)g Fm(q)f Fk(\024)54 b Fo(0)62 b(the)h(announced)-201 6912 y(inequalities)f(are)e (trivial\).)-201 7252 y Fc(Lemma)68 b(2.1.)90 b Fn(F)-14 b(or)71 b(any)h Fm(q)d(>)61 b Fo(0)p Fn(,)73 b Fm(q)c Fk(6)p Fo(=)62 b(1)p Fn(,)73 b Fm(S)2917 7279 y Fl(\026)3009 7252 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))70 b Fn(and)i Fm(I)3915 7279 y Fl(\026)4008 7252 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))70 b Fn(ar)-9 b(e)71 b(either)f(b)-9 b(oth)72 b(c)-9 b(onver)g(ging)69 b(or)i(b)-9 b(oth)-201 7478 y(diver)g(ging,)63 b(and)i(ther)-9 b(e)64 b(exists)i(some)e(\014nite)i (c)-9 b(onstants)66 b Fm(C)3796 7505 y Ff(1)3875 7478 y Fo(\()p Fm(q)7 b Fo(\))p Fm(;)59 b(C)4345 7505 y Ff(2)4424 7478 y Fo(\()p Fm(q)7 b Fo(\))49 b Fm(>)h Fo(0)65 b Fn(so)g(that)h(for) f(al)9 b(l)64 b Fm(")50 b Fk(2)h Fo(\(0)p Fm(;)59 b Fo(1\))1895 7843 y Fm(C)2025 7870 y Ff(1)2104 7843 y Fo(\()p Fm(q)7 b Fo(\))p Fm(S)2446 7870 y Fl(\026)2538 7843 y Fo(\()p Fm(q)g(;)30 b(")p Fo(\))49 b Fk(\024)h Fm(I)3255 7870 y Fl(\026)3348 7843 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))49 b Fk(\024)h Fm(C)4115 7870 y Ff(2)4194 7843 y Fo(\()p Fm(q)7 b Fo(\))p Fm(S)4536 7870 y Fl(\026)4628 7843 y Fo(\()p Fm(q)g(;)30 b(")p Fo(\))p Fm(:)-201 8209 y Fn(In)66 b(p)-9 b(articular,)62 b Fm(S)1036 8236 y Fl(\026)1129 8209 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))63 b Fn(and)j Fm(I)2022 8236 y Fl(\026)2115 8209 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))64 b Fn(have)g(the)h(same)f(gr)-9 b(owth)64 b(exp)-9 b(onents)66 b Fm(D)5331 8143 y Fz(\006)5326 8254 y Fl(\026)5449 8209 y Fo(\()p Fm(q)7 b Fo(\))p Fn(,)63 b Fm(q)58 b(>)50 b Fo(0)p Fn(,)64 b Fm(q)57 b Fk(6)p Fo(=)50 b(1)p Fn(.)-201 8548 y Fc(Lemma)68 b(2.2.)90 b Fn(F)-14 b(or)64 b(any)i Fm(K)d(>)49 b Fo(2)p Fm(e)p Fn(,)65 b(one)g(has)-110 8989 y Fm(S)2 9016 y Fl(\026)95 8989 y Fo(\(1)p Fm(;)30 b(")p Fo(\))49 b Fk(\025)735 8759 y Fg(Z)827 9137 y Ff(supp)q Fl(\026)1242 8989 y Fo(log)2109 8866 y(1)p 1527 8947 1255 7 v 1527 9114 a Fm(\026)p Fo(\([)p Fm(x)39 b Fk(\000)h Fm(";)30 b(x)40 b Fo(+)g Fm(")p Fo(]\))2801 8989 y(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))50 b Fk(\025)3500 8751 y Fg(\022)3622 8989 y Fo(1)40 b Fk(\000)3993 8866 y Fo(2)p 3955 8947 168 7 v 3955 9114 a Fm(K)4142 8751 y Fg(\023)4295 8989 y Fm(S)4407 9016 y Fl(\026)4500 8989 y Fo(\(1)p Fm(;)30 b(")p Fo(\))51 b Fk(\000)5114 8866 y Fo(2)30 b(log)j Fm(K)p 5114 8947 554 7 v 5307 9114 a(K)5690 8989 y Fk(\000)r Fo(log)r(\()p Fm(K)15 b Fo(+)r(1\))90 b(\(2.13\))-201 9445 y Fn(Thus)65 b(the)g(fol)9 b(lowing)64 b(e)-9 b(qualities)62 b(hold)j(for)f(the)h(entr)-9 b(opy)65 b(dimensions)2192 9811 y Fm(h)2297 9736 y Fz(\000)2297 9856 y Fl(\026)2466 9811 y Fo(=)50 b Fm(R)q(h)2902 9736 y Fz(\000)2902 9856 y Fl(\026)3280 9811 y Fo(and)260 b Fm(h)3938 9736 y Ff(+)3938 9856 y Fl(\026)4106 9811 y Fo(=)50 b Fm(R)q(h)4542 9736 y Ff(+)4542 9856 y Fl(\026)4726 9811 y Fm(:)1928 b Fo(\(2.14\))-201 10176 y Fc(Lemma)68 b(2.3.)90 b Fn(F)-14 b(or)47 b(any)h Fm(q)57 b(>)50 b Fo(0)p Fn(,)h Fm(S)2250 10203 y Fl(\026)2343 10176 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))46 b Fn(and)i Fm(L)3245 10203 y Fl(\026)3338 10176 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))47 b Fn(ar)-9 b(e)47 b(either)f(b)-9 b(oth)48 b(c)-9 b(onver)g(ging)45 b(or)j(b)-9 b(oth)47 b(diver)-9 b(ging.)-201 10402 y(Mor)g(e)g(over,)63 b(for)h(any)h Fm(")51 b Fk(2)f Fo(\(0)p Fm(;)30 b Fo(1\))65 b Fn(we)f(have)2138 10768 y Fm(S)2250 10795 y Fl(\026)2342 10768 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))50 b Fk(\024)g Fm(L)3104 10795 y Fl(\026)3196 10768 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))50 b Fk(\024)g Fo(3)3925 10693 y Fl(q)5 b Ff(+1)4181 10768 y Fm(S)4293 10795 y Fl(\026)4386 10768 y Fo(\()p Fm(q)i(;)30 b(")p Fo(\))p Fm(:)-201 11133 y Fn(In)66 b(p)-9 b(articular,)62 b Fm(S)1036 11160 y Fl(\026)1129 11133 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))63 b Fn(and)j Fm(L)2066 11160 y Fl(\026)2159 11133 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))63 b Fn(have)h(the)i(same)e (gr)-9 b(owth)64 b(exp)-9 b(onents)66 b Fm(D)5375 11067 y Fz(\006)5370 11178 y Fl(\026)5492 11133 y Fo(\()p Fm(q)7 b Fo(\))p Fn(,)64 b Fm(q)57 b(>)50 b Fo(0)p Fn(,)65 b Fm(q)57 b Fk(6)p Fo(=)50 b(1)p Fn(.)3393 11631 y Fo(10)p eop %%Page: 11 11 11 10 bop 81 1158 a Fo(F)-15 b(or)70 b(the)g(pro)5 b(of)69 b(of)h(the)g(ab)5 b(o)-5 b(v)g(e)69 b(lemmas,)g(w)-5 b(e)70 b(adopt)g(the)f(follo)-5 b(wing)71 b(notations:)99 b(for)70 b(simplicit)-5 b(y)-15 b(,)71 b(w)-5 b(e)-201 1406 y(drop)44 b(the)g(indices)g(\()p Fm(")p Fo(\))g(from)e Fm(I)1862 1310 y Ff(\()p Fl(")p Ff(\))1848 1458 y Fl(j)2046 1406 y Fo(;)49 b(furthermore,)44 b(w)-5 b(e)45 b(de\014ne)e Fm(a)4045 1433 y Fl(j)4169 1406 y Fo(=)50 b Fm(\026)4501 1218 y Fg(\020)4600 1406 y Fm(I)4694 1310 y Ff(\()p Fl(")p Ff(\))4680 1458 y Fl(j)4877 1218 y Fg(\021)5021 1406 y Fo(and)44 b Fm(b)5436 1433 y Fl(j)5559 1406 y Fo(=)50 b Fm(a)5847 1433 y Fl(j)8 b Fz(\000)p Ff(1)6108 1406 y Fo(+)f Fm(a)6353 1433 y Fl(j)h Ff(+1)6606 1406 y Fo(,)47 b(where)-201 1663 y Fm(a)-105 1690 y Fl(j)-32 1663 y Fm(;)30 b(b)127 1690 y Fl(j)251 1663 y Fk(2)50 b Fo(\(0)p Fm(;)30 b Fo(1\))45 b(for)g(an)-5 b(y)46 b Fm(j)55 b Fo(and)1933 1534 y Fg(P)2108 1709 y Fl(j)2212 1663 y Fm(a)2308 1690 y Fl(j)2431 1663 y Fo(=)50 b(1.)75 b(The)46 b(basic)f(estimate)f(that)h(allo)-5 b(ws)46 b(one)f(to)g(compare)f(in) -5 b(tegrals)-201 1889 y(and)47 b(discrete)f(sums)f(comes)g(from)g(the) h(follo)-5 b(wing)48 b(simple)e(observ)-10 b(ation.)75 b(F)-15 b(or)47 b(an)-5 b(y)47 b Fm(x)j Fk(2)h Fm(I)5862 1916 y Fl(j)5985 1889 y Fo(=)f([)p Fm(j)10 b(";)30 b Fo(\()p Fm(j)23 b Fo(+)13 b(1\))p Fm(")p Fo(\),)1986 2297 y Fm(I)2066 2324 y Fl(j)2190 2297 y Fk(\032)49 b Fo([)p Fm(x)40 b Fk(\000)g Fm(";)30 b(x)40 b Fo(+)g Fm(")p Fo(\))50 b Fk(\032)g Fm(I)3728 2324 y Fl(j)8 b Fz(\000)p Ff(1)4021 2297 y Fk([)41 b Fm(I)4263 2324 y Fl(j)4376 2297 y Fk([)g Fm(I)4618 2324 y Fl(j)8 b Ff(+1)4932 2297 y Fm(;)-201 2706 y Fo(therefore)1416 2931 y Fm(a)1512 2958 y Fl(j)1635 2931 y Fk(\024)50 b Fm(\026)p Fo(\([)p Fm(x)39 b Fk(\000)h Fm(";)30 b(x)40 b Fo(+)g Fm(")p Fo(\)\))49 b Fk(\024)h Fm(a)3440 2958 y Fl(j)8 b Fz(\000)p Ff(1)3734 2931 y Fo(+)40 b Fm(a)4012 2958 y Fl(j)4126 2931 y Fo(+)f Fm(a)4403 2958 y Fl(j)8 b Ff(+1)4707 2931 y Fo(=)50 b Fm(a)4995 2958 y Fl(j)5109 2931 y Fo(+)40 b Fm(b)5369 2958 y Fl(j)5502 2931 y Fm(:)1152 b Fo(\(2.15\))-201 3365 y Fc(Pro)6 b(of)69 b(of)g(Lemma)e(2.1.)-201 3591 y Fo(The)61 b(in)-5 b(tegral)61 b Fm(I)915 3618 y Fl(\026)1008 3591 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))60 b(can)g(b)5 b(e)60 b(written)h(as)-110 4073 y Fm(I)-30 4100 y Fl(\026)63 4073 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))49 b(=)704 3911 y Fg(X)700 4279 y Fl(j)8 b Fz(2)p Fe(Z)978 3843 y Fg(Z)1070 4220 y Fl(I)1131 4240 y Fd(j)1197 4220 y Fz(\\)23 b Ff(supp)q Fl(\026)1729 4073 y Fm(\026)p Fo([)p Fm(x)h Fk(\000)g Fm(";)30 b(x)24 b Fo(+)g Fm(")p Fo(\))2800 3998 y Fl(q)5 b Fz(\000)p Ff(1)3084 4073 y Fo(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))49 b(=)3977 3911 y Fg(X)3782 4280 y Fl(j)8 b Fz(2)p Fe(Z)p Fl(;)29 b(a)4167 4300 y Fd(j)4232 4280 y Fz(6)p Ff(=0)4443 3843 y Fg(Z)4535 4220 y Fl(I)4596 4240 y Fd(j)4700 4073 y Fm(\026)p Fo([)p Fm(x)24 b Fk(\000)g Fm(";)30 b(x)24 b Fo(+)g Fm(")p Fo(\))5771 3998 y Fl(q)5 b Fz(\000)p Ff(1)6055 4073 y Fo(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))51 b Fm(:)91 b Fo(\(2.16\))-201 4772 y(Consider)61 b(\014rst)g(the)f(case)g Fm(q)d(>)50 b Fo(1.)80 b(F)-15 b(rom)59 b(\(2.16\))g(and)h(\(2.15\))g(w)-5 b(e)60 b(get)735 5180 y Fm(S)847 5207 y Fl(\026)940 5180 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))49 b(=)1772 5018 y Fg(X)1577 5387 y Fl(j)8 b Fz(2)p Fe(Z)p Fl(;)29 b(a)1962 5407 y Fd(j)2027 5387 y Fz(6)p Ff(=0)2237 5180 y Fm(a)2333 5092 y Fl(q)2333 5232 y(j)2460 5180 y Fk(\024)50 b Fm(I)2732 5207 y Fl(\026)2825 5180 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))165 b Fk(\024)3889 5018 y Fg(X)3693 5387 y Fl(j)8 b Fz(2)p Fe(Z)p Fl(;)29 b(a)4078 5407 y Fd(j)4143 5387 y Fz(6)p Ff(=0)4354 5180 y Fm(a)4450 5207 y Fl(j)4523 5180 y Fo(\()p Fm(a)4690 5207 y Fl(j)8 b Fz(\000)p Ff(1)4984 5180 y Fo(+)40 b Fm(a)5262 5207 y Fl(j)5375 5180 y Fo(+)g Fm(a)5653 5207 y Fl(j)8 b Ff(+1)5907 5180 y Fo(\))5978 5105 y Fl(q)d Fz(\000)p Ff(1)6705 5180 y Fo(\(2.17\))3386 5688 y Fk(\024)3889 5527 y Fg(X)3693 5896 y Fl(j)j Fz(2)p Fe(Z)p 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Fd(j)4498 10153 y Ff(\))4553 10115 y Fa(1)p Fi(\000)p Fd(q)4875 10086 y Fo(w)-5 b(ould)77 b(b)5 b(e)77 b(transformed)d(in)k Fm(a)7043 9998 y Fl(q)7043 10138 y(j)7119 10086 y Fo(;)-201 10383 y(otherwise)70 b(one)f(w)-5 b(ould)70 b(end)f(up)g(with)h Fm(a)2641 10410 y Fl(j)2714 10383 y Fm(=b)2883 10294 y Ff(1)p Fz(\000)p Fl(q)2883 10435 y(j)3140 10383 y Fo(,)h(whic)-5 b(h)70 b(ma)-5 b(y)67 b(b)5 b(e)70 b(signi\014can)-5 b(tly)69 b(smaller)g(than)g Fm(a)6635 10294 y Fl(q)6635 10435 y(j)6711 10383 y Fo(.)107 b(But)-201 10609 y(in)63 b(full)h(generalit) -5 b(y)63 b Fm(a)1271 10636 y Fl(j)1406 10609 y Fo(cannot)f(b)5 b(e)63 b(compared)e(to)h Fm(b)3387 10636 y Fl(j)3460 10609 y Fo(.)86 b(Of)62 b(course)g(if)h(one)f(imp)5 b(oses)61 b(that)h Fm(\026)g Fo(b)5 b(e)63 b(suc)-5 b(h)62 b(that)-201 10835 y Fm(a)-105 10862 y Fl(j)36 10835 y Fk(\025)k Fm(C)13 b Fo(\()p Fm(a)554 10862 y Fl(j)674 10835 y Fo(+)47 b Fm(b)941 10862 y Fl(j)1014 10835 y Fo(\))70 b(for)h(some)e(uniform)g (constan)-5 b(t)69 b Fm(C)84 b Fo(then)70 b(one)h(reco)-5 b(v)g(ers)70 b Fm(a)5127 10746 y Fl(q)5127 10886 y(j)5274 10835 y Fo(and)g(the)h(pro)5 b(of)70 b(functions)-201 11060 y(in)81 b(the)e(same)f(w)-5 b(a)g(y)80 b(as)f(b)5 b(efore)80 b(\()p Fm(q)89 b(>)82 b Fo(1\).)138 b(This)80 b(is)g(close)g(to)f(the)h(philosoph)-5 b(y)80 b(of)g(the)f(pro)5 b(of)79 b(of)h(Olsen)3393 11631 y(11)p eop %%Page: 12 12 12 11 bop -201 1158 a Fo([25],)79 b(where)e(the)f(doubling)h(condition) g(is)g(assumed)d(to)i(hold)h(\(with)g(a)g(di\013eren)-5 b(t)76 b(de\014nition)h(of)f(R)-5 b(\023)-86 b(en)-5 b(yi)-201 1384 y(dimension)60 b(though\).)81 1610 y(T)-15 b(o)42 b(get,)h(in)f(full)f(generalit)-5 b(y)-15 b(,)45 b(a)c(lo)-5 b(w)g(er)41 b(b)5 b(ound)41 b(in)h(terms)d(of)4012 1482 y Fg(P)4188 1656 y Fl(j)8 b Fz(2)p Fe(Z)4474 1610 y Fm(a)4570 1522 y Fl(q)4570 1662 y(j)4646 1610 y Fo(,)44 b(and)d(th)-5 b(us)41 b(pro)-5 b(v)g(e)40 b(the)h(equiv)-10 b(alence,)-201 1836 y(w)-5 b(e)64 b(shall)h(con)-5 b(trol)63 b(the)h(part)f(of)h(the)f(\\bad")g Fm(a)2988 1863 y Fl(j)3062 1836 y Fo('s,)g Fn(i.e.)89 b Fo(those)63 b(whic)-5 b(h)65 b(are)e(to)5 b(o)64 b(small)f(with)h(resp)5 b(ect)63 b(to)g Fm(b)7096 1863 y Fl(j)-201 2062 y Fo(\(see)d(the)g(set)g Fm(B)69 b Fo(b)5 b(elo)-5 b(w\).)81 b(This)61 b(will)i(b)5 b(e)60 b(ac)-5 b(hiev)g(ed)60 b(Line)h(\(2.21\).)81 2288 y(Fix)g Fm(K)i Fo(=)50 b(2)908 2222 y Ff(\()p Fl(q)5 b Ff(+2\))p Fl(=q)1473 2288 y Fo(and)61 b(de\014ne)f(the)g(sets)g Fm(A)h Fo(and)f Fm(B)70 b Fo(as)60 b(follo)-5 b(ws)916 2695 y Fm(A)51 b Fo(=)f Fk(f)p Fm(j)61 b Fk(2)50 b Fj(Z)78 b Fk(j)60 b Fo(0)51 b Fk(\024)f Fm(b)2398 2722 y Fl(j)2471 2695 y Fm(=K)62 b(<)50 b(a)3067 2722 y Fl(j)3141 2695 y Fk(g)60 b Fm(;)272 b(B)60 b Fo(=)50 b Fk(f)p Fm(j)60 b Fk(2)51 b Fj(Z)78 b Fk(j)60 b Fo(0)50 b Fm(<)g(a)5125 2722 y Fl(j)5249 2695 y Fk(\024)g Fm(b)5519 2722 y Fl(j)5592 2695 y Fm(=K)13 b Fk(g)60 b Fm(:)-201 3102 y Fo(Then)h(w)-5 b(e)61 b(ha)-5 b(v)g(e)1859 3348 y Fg(X)1664 3717 y Fl(j)8 b Fz(2)p 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y(j)5045 4672 y Fo(comes)78 b(from)g(the)i Fm(j)10 b Fo('s)80 b(in)g Fm(A)h Fo(.)-201 4932 y(Indeed)60 b(one)g(has,)g(b)-5 b(y)61 b(the)f(c)-5 b(hoice)60 b(of)g Fm(K)j Fo(=)50 b(2)2912 4866 y Ff(\()p Fl(q)5 b Ff(+2\))p Fl(=q)3417 4932 y Fo(,)1543 5177 y Fg(X)1527 5545 y Fl(j)j Fz(2)p Fl(B)1829 5339 y Fm(a)1925 5250 y Fl(q)1925 5391 y(j)2167 5339 y Fk(\024)166 b Fm(K)2643 5264 y Fz(\000)p Fl(q)2874 5177 y Fg(X)2858 5545 y Fl(j)8 b Fz(2)p Fl(B)3160 5339 y Fm(b)3238 5250 y Fl(q)3238 5391 y(j)3365 5339 y Fk(\024)50 b Fm(K)3725 5264 y Fz(\000)p Fl(q)3910 5339 y Fo(2)4001 5264 y Fl(q)4123 5177 y Fg(X)4107 5545 y Fl(j)8 b Fz(2)p Fl(B)4379 5339 y Fo(\()p Fm(a)4546 5250 y Fl(q)4546 5391 y(j)g Fz(\000)p Ff(1)4840 5339 y Fo(+)40 b Fm(a)5118 5250 y Fl(q)5118 5391 y(j)8 b Ff(+1)5371 5339 y Fo(\))2167 5894 y Fk(\024)166 b Fm(K)2643 5819 y Fz(\000)p Fl(q)2828 5894 y Fo(2)2919 5819 y Fl(q)5 b Ff(+1)3401 5732 y Fg(X)3205 6101 y Fl(j)j Fz(2)p Fe(Z)p Fl(;)30 b(a)3591 6121 y Fd(j)3656 6101 y Fz(6)p Ff(=0)3866 5894 y Fm(a)3962 5805 y Fl(q)3962 5946 y(j)4089 5894 y Fo(=)4301 5771 y(1)p 4301 5852 91 7 v 4301 6019 a(2)4637 5732 y Fg(X)4442 6101 y Fl(j)8 b Fz(2)p Fe(Z)p Fl(;)29 b(a)4827 6121 y Fd(j)4892 6101 y Fz(6)p Ff(=0)5103 5894 y Fm(a)5199 5805 y Fl(q)5199 5946 y(j)5275 5894 y Fm(:)1379 b Fo(\(2.21\))-201 6482 y(It)60 b(ob)-5 b(viously)61 b(implies)1537 6794 y Fg(X)1525 7161 y Fl(j)8 b Fz(2)p Fl(A)1820 6955 y Fm(a)1916 6867 y Fl(q)1916 7007 y(j)2042 6955 y Fo(=)2430 6794 y Fg(X)2234 7163 y Fl(j)g Fz(2)p Fe(Z)p Fl(;)30 b(a)2620 7183 y Fd(j)2685 7163 y Fz(6)p Ff(=0)2895 6955 y Fm(a)2991 6867 y Fl(q)2991 7007 y(j)3108 6955 y Fk(\000)3306 6794 y Fg(X)3290 7161 y Fl(j)8 b Fz(2)p Fl(B)3592 6955 y Fm(a)3688 6867 y Fl(q)3688 7007 y(j)3814 6955 y Fk(\025)4026 6832 y Fo(1)p 4026 6913 V 4026 7080 a(2)4363 6794 y Fg(X)4167 7163 y Fl(j)g Fz(2)p Fe(Z)p Fl(;)30 b(a)4553 7183 y Fd(j)4618 7163 y Fz(6)p Ff(=0)4828 6955 y Fm(a)4924 6867 y Fl(q)4924 7007 y(j)5061 6955 y Fm(:)1593 b Fo(\(2.22\))-201 7553 y(No)-5 b(w,)61 b(\(2.22\))e(together)h(with)h(\(2.20\))e(and)i(\(2.19\))e(implies)1643 7952 y(1)p 1643 8033 V 1643 8199 a(2)1785 7837 y Fg(\022)1907 8075 y Fo(1)40 b(+)g(2)2331 7921 y Fd(q)t Fa(+2)p 2331 7954 215 7 v 2408 8036 a Fd(q)2573 7837 y Fg(\023)2696 7883 y Fl(q)5 b Fz(\000)p Ff(1)2982 8075 y Fm(S)3094 8102 y Fl(\026)3187 8075 y Fo(\()p Fm(q)i(;)30 b(")p Fo(\))49 b Fk(\024)h Fm(I)3904 8102 y Fl(\026)3997 8075 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))50 b Fk(\024)f Fm(S)4746 8102 y Fl(\026)4839 8075 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))59 b Fm(:)-201 8553 y Fo(That)i(ends)f(the)g(pro)5 b(of)60 b(of)g(Lemma)d(2.1)7028 8779 y Fb(\003)-201 9104 y Fc(Pro)6 b(of)69 b(of)g(Lemma)e(2.2.)-201 9330 y Fo(Although)47 b(it)g(is)f(more)f(tec)-5 b(hnical,)49 b(the)d(spirit)h(of)f(the)g(pro) 5 b(of)46 b(of)g(Lemma)d(2.2)j(is)g(close)h(to)f(the)g(case)f Fm(q)58 b Fk(2)50 b Fo(\(0)p Fm(;)30 b Fo(1\))-201 9556 y(ab)5 b(o)-5 b(v)g(e.)86 b(It)62 b(is)g(b)5 b(ecause)62 b(of)g(the)g(conca)-5 b(vit)g(y)62 b(of)g(the)g(function)g Fm(x)54 b Fk(!)f Fm(x)30 b Fo(log)s(\(1)p Fm(=x)p Fo(\).)86 b(The)62 b(issue)h(is)f(still)i(to)e(get)-201 9781 y(a)f(con)-5 b(trol)60 b(on)h(the)f(con)-5 b(tribution)61 b(of)f(the)g(\\bad")g Fm(j)10 b Fo('s)61 b(for)f(whic)-5 b(h)61 b Fm(a)4382 9808 y Fl(j)4516 9781 y Fo(is)g(to)5 b(o)60 b(small)g(with)h(resp)5 b(ect)59 b(to)i Fm(b)6830 9808 y Fl(j)6903 9781 y Fo(.)-201 10007 y(Discretizing,)73 b(as)d(at)g(the)g(b)5 b(eginning)71 b(of)f(Lemma)d(2.1,)72 b(the)e(in)-5 b(tegral)71 b(that)f(en)-5 b(ters)69 b(in)-5 b(to)71 b(accoun)-5 b(t)69 b(in)i Fm(h)7051 9941 y Fz(\006)7051 10052 y Fl(\026)-201 10233 y Fo(leads)61 b(to)f(\(using)g(\(2.15\)\))270 10555 y Fg(X)75 10924 y Fl(j)8 b Fz(2)p Fe(Z)p Fl(;)29 b(a)460 10944 y Fd(j)525 10924 y Fz(6)p Ff(=0)735 10717 y Fm(a)831 10744 y Fl(j)935 10717 y Fo(log)1831 10594 y(1)p 1220 10675 1313 7 v 1220 10841 a Fm(a)1316 10868 y Fl(j)8 b Fz(\000)p Ff(1)1610 10841 y Fo(+)40 b Fm(a)1888 10868 y Fl(j)2001 10841 y Fo(+)g Fm(a)2279 10868 y Fl(j)8 b Ff(+1)2603 10717 y Fk(\024)2795 10487 y Fg(Z)2887 10864 y Ff(supp)q Fl(\026)3302 10717 y Fo(log)4168 10594 y(1)p 3587 10675 1255 7 v 3587 10841 a Fm(\026)p Fo(\([)p Fm(x)39 b Fk(\000)h Fm(";)30 b(x)39 b Fo(+)h Fm(")p Fo(]\))4861 10717 y(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))50 b Fk(\024)5755 10555 y Fg(X)5559 10924 y Fl(j)8 b Fz(2)p Fe(Z)p Fl(;)30 b(a)5945 10944 y Fd(j)6010 10924 y Fz(6)p Ff(=0)6220 10717 y Fm(a)6316 10744 y Fl(j)6420 10717 y Fo(log)6744 10594 y(1)p 6705 10675 170 7 v 6705 10841 a Fm(a)6801 10868 y Fl(j)6705 11133 y Fo(\(2.23\))3393 11631 y(12)p eop %%Page: 13 13 13 12 bop -201 1158 a Fo(As)41 b(previoulsy)-15 b(,)44 b(w)-5 b(e)41 b(should)f(obtain)h(a)f(lo)-5 b(w)g(er)41 b(b)5 b(ound)40 b(for)g(the)g(l.h.s.)73 b(of)40 b(\(2.23\))f(in)i (terms)d(of)5970 1030 y Fg(P)6146 1204 y Fl(a)6221 1224 y Fd(j)6286 1204 y Fz(6)p Ff(=0)6504 1158 y Fm(a)6600 1185 y Fl(j)6704 1158 y Fo(log)s(\(1)p Fm(=a)7288 1185 y Fl(j)7360 1158 y Fo(\).)-201 1384 y(Let)71 b Fm(K)80 b(>)68 b Fo(0)j(to)g(b)5 b(e)71 b(precised)g(later.)113 b(Consider)72 b Fm(A)c Fo(=)g Fk(f)p Fm(j)10 b Fk(j)68 b Fo(0)g Fk(\024)g Fm(b)4397 1411 y Fl(j)4470 1384 y Fm(=K)80 b(<)68 b(a)5102 1411 y Fl(j)5175 1384 y Fk(g)j Fo(and)h Fm(B)77 b Fo(=)68 b Fk(f)p Fm(j)10 b Fk(j)68 b Fo(0)g Fm(<)f(a)6886 1411 y Fl(j)7028 1384 y Fk(\024)-201 1610 y Fm(b)-123 1637 y Fl(j)-50 1610 y Fm(=K)13 b Fk(g)p Fo(.)80 b(Then)1002 1937 y Fg(X)806 2306 y Fl(j)8 b Fz(2)p Fe(Z)p Fl(;)30 b(a)1192 2326 y Fd(j)1257 2306 y Fz(6)p Ff(=0)1467 2098 y Fm(a)1563 2125 y Fl(j)1667 2098 y Fo(log)2177 1975 y(1)p 1952 2056 543 7 v 1952 2223 a Fm(a)2048 2250 y Fl(j)2161 2223 y Fo(+)40 b Fm(b)2421 2250 y Fl(j)2680 2098 y Fk(\025)3000 1937 y Fg(X)2988 2304 y Fl(j)8 b Fz(2)p Fl(A)3282 2098 y Fm(a)3378 2125 y Fl(j)3482 2098 y Fo(log)3993 1975 y(1)p 3767 2056 V 3767 2223 a Fm(a)3863 2250 y Fl(j)3976 2223 y Fo(+)40 b Fm(b)4236 2250 y Fl(j)4380 2098 y Fk(\025)4584 1937 y Fg(X)4572 2304 y Fl(j)8 b Fz(2)p Fl(A)4867 2098 y Fm(a)4963 2125 y Fl(j)5066 2098 y Fo(log)5701 1975 y(1)p 5351 2056 792 7 v 5351 2223 a(\()p Fm(K)53 b Fo(+)39 b(1\))p Fm(a)6069 2250 y Fl(j)2680 2668 y Fk(\025)3000 2507 y Fg(X)2988 2874 y Fl(j)8 b Fz(2)p Fl(A)3282 2668 y Fm(a)3378 2695 y Fl(j)3482 2668 y Fo(log)3806 2545 y(1)p 3767 2626 170 7 v 3767 2793 a Fm(a)3863 2820 y Fl(j)3996 2668 y Fk(\000)40 b Fo(log)s(\()p Fm(K)53 b Fo(+)39 b(1\))p Fm(:)1619 b Fo(\(2.24\))-201 3240 y(One)62 b(then)h(has)g(to)f(estimate)f(the)h(remaining)h(part)f (of)h(the)f(sum,)g(that)g(is)h Fk(\000)5164 3112 y Fg(P)5339 3286 y Fl(B)5491 3240 y Fm(a)5587 3267 y Fl(j)5690 3240 y Fo(log)33 b Fm(a)6051 3267 y Fl(j)6124 3240 y Fo(.)88 b(Notice)63 b(that)-201 3466 y(the)80 b(function)g Fm(x)i Fk(!)h Fm(x)30 b Fo(log)s(\(1)p Fm(=x)p Fo(\))80 b(\(de\014ned)f(to)h (b)5 b(e)80 b(zero)g(for)g Fm(x)j Fo(=)f(0\))e(increases)f(on)h(]0)p Fm(;)30 b Fo(1)p Fm(=e)p Fo([)79 b(and)h(then)-201 3692 y(decreases.)d(Hence,)56 b(b)-5 b(y)54 b(c)-5 b(ho)5 b(osing)55 b Fm(K)67 b Fo(so)55 b(that)f Fm(K)63 b(>)50 b Fo(2)p Fm(e)p Fo(,)55 b(one)f(mak)-5 b(es)53 b(sure)h(that)h(for)f (an)-5 b(y)55 b Fm(j;)84 b(b)6278 3719 y Fl(j)6352 3692 y Fm(=K)62 b Fk(\024)50 b Fo(1)p Fm(=e)p Fo(,)-201 3918 y(whic)-5 b(h)61 b(implies)g(that)f(for)g(an)-5 b(y)60 b Fm(j)h Fk(2)51 b Fm(B)1704 4442 y(a)1800 4469 y Fl(j)1903 4442 y Fo(log)2228 4319 y(1)p 2188 4400 V 2188 4567 a Fm(a)2284 4594 y Fl(j)2428 4442 y Fk(\024)2648 4319 y Fm(b)2726 4346 y Fl(j)p 2640 4400 168 7 v 2640 4567 a Fm(K)2858 4442 y Fo(log)3143 4319 y Fm(K)p 3143 4400 V 3151 4567 a(b)3229 4594 y Fl(j)3380 4442 y Fo(=)3600 4319 y Fm(b)3678 4346 y Fl(j)p 3592 4400 V 3592 4567 a Fm(K)3810 4154 y Fg( )3941 4442 y Fo(log)34 b Fm(K)52 b Fo(+)40 b(log)4911 4319 y(1)p 4881 4400 152 7 v 4881 4567 a Fm(b)4959 4594 y Fl(j)5052 4154 y Fg(!)5214 4442 y Fm(:)1440 b Fo(\(2.25\))-201 4967 y(As)61 b(a)f(second)g(remark,)f (note)h(that)g(for)g(an)-5 b(y)61 b Fm(a;)f(b)50 b(>)g Fo(0,)60 b(one)h(c)-5 b(hec)g(ks)59 b(that)2101 5438 y(\()p Fm(a)41 b Fo(+)e Fm(b)p Fo(\))30 b(log)3107 5315 y(1)p 2954 5396 397 7 v 2954 5562 a Fm(a)41 b Fo(+)f Fm(b)3421 5438 y Fk(\024)50 b Fm(a)30 b Fo(log)4027 5315 y(1)p 4025 5396 97 7 v 4025 5562 a Fm(a)4181 5438 y Fo(+)40 b Fm(b)30 b Fo(log)4756 5315 y(1)p 4756 5396 91 7 v 4762 5562 a Fm(b)6705 5438 y Fo(\(2.26\))-201 5909 y(Using)61 b(successiv)-5 b(ely)60 b(\(2.25\),)f(\(2.26\))g(and)i(the)f(fact)f (that)3818 5780 y Fg(P)3994 5955 y Fl(j)4097 5909 y Fm(a)4193 5936 y Fl(j)4317 5909 y Fo(=)50 b(1)60 b(leads)h(to)887 6308 y Fg(X)871 6676 y Fl(j)8 b Fz(2)p Fl(B)1174 6470 y Fm(a)1270 6497 y Fl(j)1373 6470 y Fo(log)1697 6347 y(1)p 1658 6428 170 7 v 1658 6595 a Fm(a)1754 6622 y Fl(j)2013 6470 y Fk(\024)2379 6347 y Fo(1)p 2341 6428 168 7 v 2341 6595 a Fm(K)2574 6308 y Fg(X)2558 6676 y Fl(j)g Fz(2)p Fl(B)2830 6470 y Fo(\()p Fm(a)2997 6497 y Fl(j)g Fz(\000)p Ff(1)3291 6470 y Fo(+)39 b Fm(a)3568 6497 y Fl(j)8 b Ff(+1)3822 6470 y Fo(\))3923 6182 y Fg( )4054 6470 y Fo(log)34 b Fm(K)52 b Fo(+)40 b(log)5409 6347 y(1)p 4994 6428 922 7 v 4994 6595 a Fm(a)5090 6622 y Fl(j)8 b Fz(\000)p Ff(1)5384 6595 y Fo(+)40 b Fm(a)5662 6622 y Fl(j)8 b Ff(+1)5935 6182 y Fg(!)2013 7034 y Fk(\024)2341 6911 y Fo(2)30 b(log)j Fm(K)p 2341 6992 554 7 v 2534 7159 a(K)2955 7034 y Fo(+)3195 6911 y(2)p 3156 6992 168 7 v 3156 7159 a Fm(K)3569 6872 y Fg(X)3374 7241 y Fl(j)8 b Fz(2)p Fe(Z)p Fl(;)29 b(a)3759 7261 y Fd(j)3824 7241 y Fz(6)p Ff(=0)4035 7034 y Fm(a)4131 7061 y Fl(j)4234 7034 y Fo(log)4558 6911 y(1)p 4519 6992 170 7 v 4519 7159 a Fm(a)4615 7186 y Fl(j)6705 7034 y Fo(\(2.27\))-201 7629 y(\(recall)65 b(w)-5 b(e)64 b(imp)5 b(osed)62 b(that)i Fm(a)1840 7656 y Fl(j)1969 7629 y Fo(=)55 b(0)64 b(implies)g Fm(a)3038 7656 y Fl(j)3141 7629 y Fo(log)s(\(1)p Fm(=a)3725 7656 y Fl(j)3798 7629 y Fo(\))56 b(=)f(0\).)90 b(Putting)64 b(together)f(\(2.24\))g(and)h(\(2.27\))-201 7855 y(yields)552 8268 y Fg(X)356 8637 y Fl(j)8 b Fz(2)p Fe(Z)p Fl(;)29 b(a)741 8657 y Fd(j)806 8637 y Fz(6)p Ff(=0)1017 8429 y Fm(a)1113 8456 y Fl(j)1216 8429 y Fo(log)1727 8306 y(1)p 1502 8387 543 7 v 1502 8554 a Fm(a)1598 8581 y Fl(j)1711 8554 y Fo(+)40 b Fm(b)1971 8581 y Fl(j)2230 8429 y Fk(\025)2733 8268 y Fg(X)2537 8637 y Fl(j)8 b Fz(2)p Fe(Z)p Fl(;)29 b(a)2922 8657 y Fd(j)2987 8637 y Fz(6)p Ff(=0)3198 8429 y Fm(a)3294 8456 y Fl(j)3398 8429 y Fo(log)3722 8306 y(1)p 3683 8387 170 7 v 3683 8554 a Fm(a)3779 8581 y Fl(j)3912 8429 y Fk(\000)4094 8092 y Fg(0)4094 8390 y(@)4255 8268 y(X)4239 8635 y Fl(j)8 b Fz(2)p Fl(B)4541 8429 y Fm(a)4637 8456 y Fl(j)4741 8429 y Fo(log)5065 8306 y(1)p 5026 8387 V 5026 8554 a Fm(a)5122 8581 y Fl(j)5255 8429 y Fo(+)40 b(log)s(\()p Fm(K)53 b Fo(+)39 b(1\))6294 8092 y Fg(1)6294 8390 y(A)2230 9008 y Fk(\025)2537 8770 y Fg(\022)2660 9008 y Fo(1)h Fk(\000)3031 8885 y Fo(2)p 2993 8966 168 7 v 2993 9133 a Fm(K)3180 8770 y Fg(\023)3528 8847 y(X)3333 9216 y Fl(j)8 b Fz(2)p Fe(Z)p Fl(;)29 b(a)3718 9236 y Fd(j)3783 9216 y Fz(6)p Ff(=0)3993 9008 y Fm(a)4089 9035 y Fl(j)4193 9008 y Fo(log)4517 8885 y(1)p 4478 8966 170 7 v 4478 9133 a Fm(a)4574 9160 y Fl(j)4707 9008 y Fk(\000)4909 8885 y Fo(2)h(log)j Fm(K)p 4909 8966 554 7 v 5102 9133 a(K)5523 9008 y Fk(\000)40 b Fo(log)s(\()p Fm(K)52 b Fo(+)40 b(1\))p Fm(:)92 b Fo(\(2.28\))-201 9603 y(Dividing)71 b(b)-5 b(y)69 b Fk(\000)30 b Fo(log)j Fm(")69 b Fo(and)h(taking)f(the)g (limit)g(\(inf)h(or)f(sup\),)i(\(2.23\))d(together)g(with)i(the)f(last) h(estimate)-201 9829 y(\(2.28\))59 b(implies)i(that,)e(for)h(an)-5 b(y)61 b Fm(K)i(>)50 b Fo(2)p Fm(e)p Fo(,)2345 10076 y Fg(\022)2468 10314 y Fo(1)40 b Fk(\000)2839 10191 y Fo(2)p 2801 10272 168 7 v 2801 10439 a Fm(K)2988 10076 y Fg(\023)3140 10314 y Fm(R)q(h)3384 10239 y Fz(\006)3384 10359 y Fl(\026)3553 10314 y Fk(\024)50 b Fm(h)3850 10239 y Fz(\006)3850 10359 y Fl(\026)4018 10314 y Fk(\024)g Fm(R)q(h)4454 10239 y Fz(\006)4454 10359 y Fl(\026)4573 10314 y Fm(:)-201 10809 y Fo(The)61 b(latter)f(clearly)i(implies)e (that)g Fm(R)q(h)2505 10743 y Fz(\006)2505 10854 y Fl(\026)2674 10809 y Fo(=)50 b Fm(h)2971 10743 y Fz(\006)2971 10854 y Fl(\026)3088 10809 y Fo(.)3889 b Fb(\003)-201 11133 y Fc(Pro)6 b(of)69 b(of)g(Lemma)e(2.3.)3393 11631 y Fo(13)p eop %%Page: 14 14 14 13 bop -201 1158 a Fo(The)61 b(in)-5 b(tegral)61 b Fm(L)959 1185 y Fl(\026)1052 1158 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))59 b(can)i(b)5 b(e)60 b(written)h(as)1092 1651 y Fm(L)1216 1678 y Fl(\026)1309 1651 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))50 b(=)1950 1489 y Fg(X)1947 1858 y Fl(j)8 b Fz(2)p Fe(Z)2224 1651 y Fm(L)2348 1678 y Fl(j)2421 1651 y Fo(\()p Fm(q)f(;)30 b(")p Fo(\))p Fm(;)150 b(L)3142 1678 y Fl(j)3214 1651 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))50 b(=)3871 1528 y(1)p 3871 1609 91 7 v 3874 1776 a Fm(")4013 1421 y Fg(Z)4105 1798 y Fl(I)4166 1818 y Fd(j)4270 1651 y Fm(\026)p Fo(\([)p Fm(x)39 b Fk(\000)h Fm(";)30 b(x)40 b Fo(+)f Fm(")p Fo(\)\))5545 1576 y Fl(q)5621 1651 y Fo(d)p Fm(x:)828 b Fo(\(2.29\))-201 2245 y(F)-15 b(or)60 b(an)-5 b(y)60 b Fm(q)d(>)50 b Fo(0)60 b(the)f(k)-5 b(ey)60 b(observ)-10 b(ation)60 b(\(2.15\))e(yields)j Fm(a)3686 2156 y Fl(q)3686 2297 y(j)3813 2245 y Fk(\024)50 b Fm(\026)p Fo(\([)p Fm(x)37 b Fk(\000)i Fm(";)30 b(x)39 b Fo(+)g Fm(")p Fo(\)\))5276 2179 y Fl(q)5401 2245 y Fk(\024)50 b Fo(\()p Fm(a)5760 2272 y Fl(j)8 b Fz(\000)p Ff(1)6053 2245 y Fo(+)38 b Fm(a)6329 2272 y Fl(j)6442 2245 y Fo(+)h Fm(a)6719 2272 y Fl(j)8 b Ff(+1)6972 2245 y Fo(\))7043 2179 y Fl(q)7119 2245 y Fo(,)-201 2471 y(for)60 b(an)-5 b(y)61 b Fm(x)50 b Fk(2)h Fm(I)827 2498 y Fl(j)900 2471 y Fo(.)80 b(In)-5 b(tegrating)60 b(this)h(inequalit)-5 b(y)61 b(o)-5 b(v)g(er)61 b Fm(I)3639 2498 y Fl(j)3712 2471 y Fo(,)f(w)-5 b(e)60 b(obtain)h(for)f(an)-5 b(y)61 b Fm(j)g Fk(2)50 b Fj(Z)8 b Fo(,)70 b(since)60 b Fk(j)p Fm(I)6394 2498 y Fl(j)6466 2471 y Fk(j)50 b Fo(=)g Fm(")p Fo(,)1818 2879 y Fm(a)1914 2791 y Fl(q)1914 2931 y(j)2041 2879 y Fk(\024)g Fm(L)2357 2906 y Fl(j)2430 2879 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))164 b Fk(\024)i Fo(\()o Fm(a)3464 2906 y Fl(j)8 b Fz(\000)p Ff(1)3758 2879 y Fo(+)40 b Fm(a)4036 2906 y Fl(j)4150 2879 y Fo(+)g Fm(a)4428 2906 y Fl(j)8 b Ff(+1)4681 2879 y Fo(\))4752 2795 y Fl(q)2990 3218 y Fk(\024)166 b Fo(3)3389 3143 y Fl(q)3495 3030 y Fg(\020)3594 3218 y Fm(a)3690 3130 y Fl(q)3690 3270 y(j)8 b Fz(\000)p Ff(1)3984 3218 y Fo(+)40 b Fm(a)4262 3130 y Fl(q)4262 3270 y(j)4378 3218 y Fo(+)g Fm(a)4656 3130 y Fl(q)4656 3270 y(j)8 b Ff(+1)4910 3030 y Fg(\021)5100 3218 y Fm(:)1554 b Fo(\(2.30\))-201 3658 y(The)61 b(result)g(of)f(the)g (Lemma)d(follo)-5 b(ws)61 b(directly)g(from)e(\(2.29\))g(-\(2.30\).) 2387 b Fb(\003)-201 4033 y Fc(Remark)69 b(2.4.)90 b Fo(The)79 b(result)g(is)g(not)g(true)g(if)g(one)f(in)-5 b(tegrates)79 b(not)f(o)-5 b(v)g(er)79 b Fj(R)114 b Fo(but)79 b(o)-5 b(v)g(er)79 b(supp)o(\()p Fm(\026)p Fo(\))f(in)i(the)-201 4259 y(de\014nition)70 b(of)e Fm(L)955 4286 y Fl(\026)1048 4259 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\).)104 b(F)-15 b(or)69 b(example,)g(for)g(a)f(measure)f Fm(\026)d Fo(=)g Fm(\016)4331 4286 y Ff(0)4479 4259 y Fo(that)k(w)-5 b(ould)70 b(giv)-5 b(e)69 b Fm(L)5926 4286 y Fl(\026)6019 4259 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))63 b(=)h(0,)71 b(but)-201 4485 y Fm(S)-89 4512 y Fl(\026)4 4485 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))58 b(=)h(1.)96 b(On)65 b(the)g(other)h(hand,)h(in)-5 b(tegrating)66 b(o)-5 b(v)g(er)65 b Fj(R)102 b Fo(for)65 b Fm(q)h(<)59 b Fo(0)66 b(yields)g Fm(L)5466 4512 y Fl(\026)5559 4485 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))58 b(=)h(+)p Fk(1)65 b Fo(for)g(an)-5 b(y)-201 4710 y(compactly)68 b(supp)5 b(orted)70 b(measure.)106 b(Therefore,)72 b(the)d(result)h(of) g(the)f(Lemma)e(can)j(hardly)g(b)5 b(e)70 b(extended)-201 4936 y(to)60 b(the)h(case)e Fm(q)f(<)50 b Fo(0)60 b(in)h(an)-5 b(y)61 b(reasonable)f(sense.)-201 5509 y Fp(3)270 b(Prop)7 b(erties)89 b(of)i(the)e(generalized)g(fractal)f(dimensions)-201 5915 y Fo(In)54 b(this)g(section)f(w)-5 b(e)54 b(list)h(and)e(pro)-5 b(v)g(e)54 b(some)e(basic)i(prop)5 b(erties)54 b(of)f(the)h(functions)f Fm(\034)5489 5849 y Fz(\006)5468 5960 y Fl(\026)5606 5915 y Fo(\()p Fm(q)7 b Fo(\))53 b(and)h(the)g(general-)-201 6141 y(ized)j(fractal)f(dimensions)f Fm(D)1819 6075 y Fz(\006)1814 6186 y Fl(\026)1937 6141 y Fo(\()p Fm(q)7 b Fo(\))56 b(in)-5 b(tro)5 b(duced)56 b(in)h(Section)f(2,)h (De\014nitions)g(2.1)f(and)g(2.2.)78 b(Some)55 b(of)h(them)-201 6367 y(are)71 b(w)-5 b(ell-kno)g(wn,)74 b(see)c(Theorem)g(3.1.)110 b(Some)69 b(others)i(are)f(part)h(of)g(a)f(\\common")e(kno)-5 b(wledge,)73 b(but,)g(as)-201 6592 y(far)60 b(as)h(w)-5 b(e)60 b(kno)-5 b(w,)61 b(with)g(no)f(pro)5 b(ofs.)-201 6818 y(The)61 b(results)f(con)-5 b(tained)61 b(in)g(this)f(section)g (use)h(t)-5 b(w)g(o)60 b(di\013eren)-5 b(t)61 b(crucial)g(to)5 b(ols,)60 b(whic)-5 b(h)61 b(leads)g(to)f(t)-5 b(w)g(o)61 b(series)-201 7044 y(of)68 b(prop)5 b(erties:)95 b(the)68 b(\\con)-5 b(v)g(exit)g(y")67 b(inequalit)-5 b(y)69 b(\(3.1\))e(and)h (the)f(equiv)-10 b(alence)69 b(pro)-5 b(v)g(ed)67 b(in)i(Section)f(2)g (that)-201 7270 y(allo)-5 b(ws)62 b(one)e(to)g(w)-5 b(ork)61 b(with)g(the)f(more)f(con)-5 b(v)g(enien)g(t)60 b(sums)3879 7141 y Fg(P)4054 7316 y Fl(j)4158 7270 y Fm(a)4254 7181 y Fl(q)4254 7322 y(j)4390 7270 y Fo(for)h Fm(q)c(>)50 b Fo(0.)-201 7645 y Fc(Theorem)68 b(3.1.)90 b Fn(L)-9 b(et)65 b Fm(\026)g Fn(b)-9 b(e)64 b(a)h(p)-9 b(ositive)64 b(pr)-9 b(ob)g(ability)63 b(Bor)-9 b(el)63 b(me)-9 b(asur)g(e.)-201 7871 y(\(i\))66 b Fm(\034)170 7805 y Fz(\000)149 7916 y Fl(\026)287 7871 y Fo(\()p Fm(q)7 b Fo(\))p Fn(,)64 b Fm(\034)738 7805 y Ff(+)717 7916 y Fl(\026)855 7871 y Fo(\()p Fm(q)7 b Fo(\))p Fn(,)64 b Fm(D)1361 7805 y Fz(\000)1356 7916 y Fl(\026)1478 7871 y Fo(\()p Fm(q)7 b Fo(\))65 b Fn(and)g Fm(D)2282 7805 y Ff(+)2277 7916 y Fl(\026)2399 7871 y Fo(\()p Fm(q)7 b Fo(\))65 b Fn(ar)-9 b(e)64 b(nonincr)-9 b(e)g(asing)63 b(functions)i(of)g Fm(q)57 b Fk(2)51 b Fj(R)26 b Fn(.)-201 8097 y(\(ii\))65 b Fm(\034)225 8031 y Fz(\006)204 8142 y Fl(\026)343 8097 y Fo(\(1\))49 b(=)h(0)p Fn(.)-201 8323 y(\(iii\))65 b Fm(\034)281 8257 y Ff(+)260 8368 y Fl(\026)398 8323 y Fo(\()p Fm(q)7 b Fo(\))65 b Fn(is)g(a)g(c)-9 b(onvex)64 b(function.)-201 8698 y Fc(Pro)6 b(of)53 b(of)h(Theorem)e(3.1.)76 b Fo(P)-5 b(art)47 b(\(i\))g(is)g(already)g(kno)-5 b(wn)48 b(\(see)e(e.g.)75 b([8)o(],)49 b([28]\),)f(and)f(is)h(a)f(straigh)-5 b(tforw)g(ard)-201 8924 y(consequence)59 b(of)i(Jensen)e(inequalities)i (\(for)f Fm(q)e Fo(=)50 b(1)60 b(see)g(Prop)5 b(osition)61 b(4.2\).)-201 9149 y(Prop)5 b(ert)-5 b(y)60 b(\(ii\))h(is)g(immediate)d (from)h(the)h(de\014nition)h(of)f Fm(\034)3799 9083 y Fz(\006)3778 9194 y Fl(\026)3916 9149 y Fo(\(1\).)-201 9375 y(P)-5 b(oin)g(t)77 b(\(iii\))i(go)5 b(es)76 b(as)h(follo)-5 b(ws.)131 b(Let)77 b Fm(t)g Fo(b)5 b(e)77 b(in)h(\(0)p Fm(;)30 b Fo(1\))76 b(and)i Fm(r)5 b Fo(,)81 b Fm(s)d Fo(b)5 b(e)77 b(t)-5 b(w)g(o)77 b(real)h(n)-5 b(um)g(b)5 b(ers.)129 b(Using)77 b(H\177)-91 b(older)-201 9601 y(inequalit)-5 b(y)62 b(with)f Fm(p)50 b Fo(=)g(1)p Fm(=t)60 b Fo(and)g Fm(p)2144 9535 y Fz(0)2241 9601 y Fo(=)50 b(1)p Fm(=)p Fo(\(1)40 b Fk(\000)f Fm(t)p Fo(\))60 b(giv)-5 b(es)787 10076 y Fm(I)867 10103 y Fl(\026)960 10076 y Fo(\()p Fm(tr)45 b Fo(+)40 b(\(1)g Fk(\000)g Fm(t)p Fo(\))p Fm(s;)30 b(")p Fo(\))164 b(=)2721 9846 y Fg(Z)2813 10223 y Ff(supp)q Fl(\026)3228 10076 y Fm(\026)p Fo(\([)p Fm(x)39 b Fk(\000)h Fm(";)30 b(x)39 b Fo(+)h Fm(")p Fo(\)\))4503 10001 y Fl(tr)t Ff(+\(1)p Fz(\000)p Fl(t)p Ff(\))p Fl(s)p Fz(\000)p Ff(1)5326 10076 y Fo(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))2413 10545 y(=)2721 10315 y Fg(Z)2813 10693 y Ff(supp)q Fl(\026)3228 10545 y Fm(\026)p Fo(\([)p Fm(x)f Fk(\000)h Fm(";)30 b(x)39 b Fo(+)h Fm(")p Fo(\)\))4503 10470 y Fl(t)p Ff(\()p Fl(r)t Fz(\000)p Ff(1\)+\(1)p Fz(\000)p Fl(t)p Ff(\)\()p Fl(s)p Fz(\000)p Ff(1\))5726 10545 y Fo(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))2413 10946 y Fk(\024)166 b Fo(\()p Fm(I)2872 10973 y Fl(\026)2965 10946 y Fo(\()p Fm(r)-5 b(;)30 b(")p Fo(\)\))3421 10870 y Fl(t)3479 10946 y Fo(\()p Fm(I)3630 10973 y Fl(\026)3723 10946 y Fo(\()p Fm(s;)g(")p Fo(\)\))4187 10870 y Ff(1)p Fz(\000)p Fl(t)4486 10946 y Fm(;)2259 b Fo(\(3.1\))3393 11631 y(14)p eop %%Page: 15 15 15 14 bop -201 1158 a Fo(and)83 b(th)-5 b(us)83 b(\(iii\))i(holds)e (true.)148 b(A)84 b(similar)f(result)h(w)-5 b(as)83 b(obtained)g(in)h ([28)o(])f(using)h(the)f(same)e(con)-5 b(v)g(exit)g(y)-201 1384 y(inequalit)g(y)62 b(\(3.1\).)5959 b Fb(\003)81 1710 y Fo(As)84 b(it)g(follo)-5 b(ws)84 b(directly)g(from)e(the)h (de\014nition)h(of)f Fm(\034)3834 1644 y Fz(\006)3813 1755 y Fl(\026)3951 1710 y Fo(\()p Fm(q)7 b Fo(\),)88 b(w)-5 b(e)84 b(ha)-5 b(v)g(e)83 b Fm(\034)5154 1644 y Fz(\006)5133 1755 y Fl(\026)5271 1710 y Fo(\()p Fm(q)7 b Fo(\))88 b Fk(\025)g Fo(0)c(for)f Fm(q)96 b(<)88 b Fo(1)83 b(and)-201 1936 y Fm(\034)-100 1870 y Fz(\006)-121 1980 y Fl(\026)17 1936 y Fo(\()p Fm(q)7 b Fo(\))74 b Fk(\024)f Fo(0)i(for)f Fm(q)81 b(>)73 b Fo(1.)123 b(As)75 b(to)f Fm(D)2311 1870 y Fz(\006)2306 1980 y Fl(\026)2429 1936 y Fo(\()p Fm(q)7 b Fo(\),)77 b(they)d(are)h(de\014ned)f(so)h(that) f(alw)-5 b(a)g(ys)75 b Fm(D)5566 1870 y Fz(\006)5561 1980 y Fl(\026)5683 1936 y Fo(\()p Fm(q)7 b Fo(\))74 b Fk(\025)f Fo(0.)123 b(The)75 b(\014rst)-201 2161 y(problem)58 b(w)-5 b(e)59 b(shall)g(consider)g(is)g(the)f(c)-5 b(haracterisation)58 b(of)g(the)h(set)f(of)g Fm(q)7 b Fo('s)57 b(where)i Fm(\034)5608 2095 y Fz(\006)5587 2206 y Fl(\026)5725 2161 y Fo(\()p Fm(q)7 b Fo(\))58 b(and)h Fm(D)6521 2095 y Fz(\006)6516 2206 y Fl(\026)6639 2161 y Fo(\()p Fm(q)7 b Fo(\))57 b(are)-201 2387 y(\014nite.)111 b(W)-15 b(e)70 b(shall)i(discuss)e (also)h(the)f(v)-10 b(alidit)-5 b(y)72 b(of)e(the)h(common)c(b)5 b(elief)71 b(that)f Fm(D)5436 2321 y Fz(\006)5431 2432 y Fl(\026)5554 2387 y Fo(\()p Fm(q)7 b Fo(\))70 b(are)h(smaller)f(than) -201 2613 y(1.)118 b(The)73 b(result)g(of)f(the)h(follo)-5 b(wing)74 b(prop)5 b(osition)74 b(is)f(w)-5 b(ell-kno)g(wn,)76 b Fn(e.g.)117 b Fo([1],)75 b([28)o(].)117 b(Ho)-5 b(w)g(ev)g(er,)76 b(the)d(pro)5 b(of)-201 2839 y(w)-5 b(e)59 b(pro)-5 b(vide)60 b(here)f(is)g(new)g(and)g(quite)g(elemen)-5 b(tary)-15 b(.)78 b(In)59 b(particular,)g(it)g(do)5 b(es)59 b(not)g(require)g(the) g(use)f(of)h(the)-201 3065 y(lo)5 b(cal)62 b(exp)5 b(onen)-5 b(ts)59 b Fm(\015)1187 2999 y Fz(\006)1177 3110 y Fl(\026)1305 3065 y Fo(\()p Fm(x)p Fo(\).)80 b(This)61 b(result)g(implies)f(that)g Fm(\034)3716 2999 y Fz(\006)3695 3110 y Fl(\026)3833 3065 y Fo(\()p Fm(q)7 b Fo(\))50 b Fm(>)f Fk(\0001)60 b Fo(for)h(an)-5 b(y)60 b Fm(q)d Fk(2)51 b Fj(R)26 b Fo(.)-201 3440 y Fc(Prop)6 b(osition)69 b(3.1.)90 b Fn(L)-9 b(et)65 b Fm(\026)f Fn(b)-9 b(e)64 b(a)h(p)-9 b(ositive)64 b(pr)-9 b(ob)g(ability)63 b(Bor)-9 b(el)64 b(me)-9 b(asur)g(e.)81 b(F)-14 b(or)64 b(any)i Fm(q)57 b(>)50 b Fo(1)p Fn(,)1103 3848 y Fo(1)41 b Fk(\000)e Fm(q)58 b Fk(\024)50 b Fm(\034)1848 3773 y Fz(\000)1827 3893 y Fl(\026)1965 3848 y Fo(\()p Fm(q)7 b Fo(\))49 b Fk(\024)h Fm(\034)2537 3773 y Ff(+)2516 3893 y Fl(\026)2654 3848 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(\024)g Fo(0)195 b(and)g(0)50 b Fk(\024)g Fm(D)4389 3773 y Fz(\000)4384 3893 y Fl(\026)4507 3848 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(\024)g Fm(D)5135 3773 y Ff(+)5130 3893 y Fl(\026)5252 3848 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(\024)g Fo(1)p Fm(:)-201 4257 y Fc(Pro)6 b(of)72 b(of)h(Prop)6 b(osition)72 b(3.1)p Fo(.)88 b(Let)63 b Fm(I)2572 4284 y Fl(j)2700 4257 y Fo(=)55 b([)p Fm(j)10 b(";)30 b Fo(\()p Fm(j)52 b Fo(+)42 b(1\))p Fm(")p Fo(\))p Fm(;)30 b(")54 b(<)h Fo(1)p Fm(=)p Fo(2)p Fm(;)93 b(j)66 b Fk(2)55 b Fj(Z)81 b Fo(and)64 b Fm(a)5688 4284 y Fl(j)5816 4257 y Fo(=)55 b Fm(\026)p Fo(\()p Fm(I)6274 4284 y Fl(j)6346 4257 y Fo(\).)89 b(F)-15 b(or)63 b(an)-5 b(y)-201 4483 y Fm(k)56 b Fk(2)51 b Fj(Z)8 b Fo(,)69 b(w)-5 b(e)61 b(ha)-5 b(v)g(e)2130 4820 y([)p Fm(k)6 b(;)30 b(k)44 b Fo(+)c(1\))50 b Fk(\032)3177 4608 y Fl(r)3119 4658 y Fg([)3088 5023 y Fl(j)8 b Ff(=)p Fl(p)3364 4820 y Fm(I)3444 4847 y Fl(j)3567 4820 y Fk(\032)50 b Fo([)p Fm(k)c Fk(\000)39 b Fo(1)p Fm(;)30 b(k)46 b Fo(+)40 b(2\))p Fm(;)1956 b Fo(\(3.2\))-201 5354 y(where)58 b Fm(p)50 b Fo(=)g([)727 5282 y Fl(k)p 727 5311 78 7 v 733 5416 a(")823 5354 y Fo(])34 b Fk(\000)g Fo(1)p Fm(;)88 b(r)55 b Fo(=)50 b([)1714 5282 y Fl(k)t Ff(+1)p 1714 5311 258 7 v 1810 5416 a Fl(")1991 5354 y Fo(])34 b(+)g(1)58 b(\([)p Fm(x)p Fo(])e(denotes)h(the)g(in)-5 b(teger)58 b(part)f(of)g Fm(x)p Fo(\).)79 b(Remark)56 b(that)h Fm(r)40 b Fk(\000)34 b Fm(p)50 b(<)g Fo(2)p Fm(=")-201 5579 y Fo(for)60 b Fm(")h Fo(small)f(enough.)80 b(It)60 b(follo)-5 b(ws)61 b(from)d(\(3.2\))i(that)2421 6109 y Fm(d)2516 6139 y Fl(k)2651 6109 y Fk(\021)50 b Fm(\026)p Fo(\([)p Fm(k)6 b(;)30 b(k)44 b Fo(+)c(1\)\))49 b Fk(\024)4141 5898 y Fl(r)4055 5948 y Fg(X)4052 6312 y Fl(j)8 b Ff(=)p Fl(p)4328 6109 y Fm(a)4424 6136 y Fl(j)4497 6109 y Fm(:)-201 6680 y Fo(With)61 b(H\177)-91 b(older)61 b(inequalit)-5 b(y)62 b(w)-5 b(e)60 b(estimate)f(for)h(an)-5 b(y)60 b Fm(q)e(>)50 b Fo(1)p Fm(;)90 b(k)56 b Fk(2)50 b Fj(Z)9 b Fo(:)1088 7306 y Fm(d)1183 7336 y Fl(k)1319 7306 y Fk(\024)1510 6968 y Fg(0)1510 7267 y(@)1745 7095 y Fl(r)1659 7144 y Fg(X)1656 7509 y Fl(j)f Ff(=)p Fl(p)1932 7306 y Fm(a)2028 7218 y Fl(q)2028 7358 y(j)2104 6968 y Fg(1)2104 7267 y(A)2249 7014 y Ff(1)p Fl(=q)2497 6968 y Fg(0)2497 7267 y(@)2731 7095 y Fl(r)2645 7144 y Fg(X)2642 7509 y Fl(j)g Ff(=)p Fl(p)2918 7306 y Fo(1)3009 6968 y Fg(1)3009 7267 y(A)3154 7014 y Ff(\()p Fl(q)d Fz(\000)p Ff(1\))p Fl(=q)3709 7306 y Fk(\024)50 b Fo(\()p Fm("=)p Fo(2\))4310 7231 y Ff(\(1)p Fz(\000)p Fl(q)5 b Ff(\))p Fl(=q)4844 6968 y Fg(0)4844 7267 y(@)5078 7095 y Fl(r)4992 7144 y Fg(X)4989 7509 y Fl(j)j Ff(=)p Fl(p)5265 7306 y Fm(a)5361 7218 y Fl(q)5361 7358 y(j)5437 6968 y Fg(1)5437 7267 y(A)5582 7014 y Ff(1)p Fl(=q)5830 7306 y Fm(:)-201 7885 y Fo(Therefore,)1925 8222 y Fm(S)2037 8249 y Fl(\026)2130 8222 y Fo(\()p Fm(q)f(;)30 b(")p Fo(\))49 b Fk(\021)2771 8061 y Fg(X)2767 8429 y Fl(j)8 b Fz(2)p Fe(Z)3045 8222 y Fm(a)3141 8134 y Fl(q)3141 8274 y(j)3268 8222 y Fk(\025)3548 8011 y Fl(r)3462 8061 y Fg(X)3459 8425 y Fl(j)g Ff(=)p Fl(p)3735 8222 y Fm(a)3831 8134 y Fl(q)3831 8274 y(j)3958 8222 y Fk(\025)50 b Fm(C)13 b Fo(\()p Fm(q)7 b Fo(\))p Fm(")4608 8147 y Fl(q)e Fz(\000)p Ff(1)4863 8222 y Fm(d)4958 8134 y Fl(q)4958 8279 y(k)6796 8222 y Fo(\(3.3\))-201 8732 y(with)66 b(some)c(p)5 b(ositiv)-5 b(e)65 b(constan)-5 b(t)64 b Fm(C)13 b Fo(\()p Fm(q)7 b Fo(\).)92 b(As)65 b Fm(\026)p Fo(\()p Fj(R)26 b Fo(\))66 b(=)3520 8604 y Fg(P)3695 8778 y Fl(k)t Fz(2)p Fe(Z)3994 8732 y Fm(d)4089 8762 y Fl(k)4231 8732 y Fo(=)57 b(1,)66 b(there)e(exists)g Fm(k)71 b Fo(suc)-5 b(h)64 b(that)g Fm(d)6686 8762 y Fl(k)6829 8732 y Fm(>)57 b Fo(0,)-201 8958 y(so)66 b(\(3.3\))f(yields)i Fm(S)1097 8985 y Fl(\026)1189 8958 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))59 b Fk(\025)g Fm(C)13 b(")2073 8892 y Fl(q)5 b Fz(\000)p Ff(1)2329 8958 y Fo(.)97 b(The)67 b(result)f(follo)-5 b(ws)66 b(no)-5 b(w)67 b(from)d(Lemma)e(2.1)k(and)g(the)g(de\014nition) -201 9184 y(of)60 b Fm(\034)107 9118 y Fz(\006)86 9229 y Fl(\026)224 9184 y Fo(\()p Fm(q)7 b Fo(\))p Fm(;)30 b(D)691 9118 y Fz(\006)686 9229 y Fl(\026)808 9184 y Fo(\()p Fm(q)7 b Fo(\).)5939 b Fb(\003)81 9509 y Fo(In)61 b(the)f(next)g(prop)5 b(osition,)61 b(w)-5 b(e)60 b(study)g(the)g(case) g Fm(q)d Fk(2)51 b Fo(\(0)p Fm(;)30 b Fo(1\).)-201 9885 y Fc(Prop)6 b(osition)69 b(3.2.)90 b Fn(L)-9 b(et)65 b Fm(d)1755 9915 y Fl(k)1890 9885 y Fo(=)50 b Fm(\026)p Fo(\([)p Fm(k)6 b(;)30 b(k)44 b Fo(+)c(1\)\))p Fn(,)64 b Fm(k)56 b Fk(2)50 b Fj(Z)9 b Fn(.)-201 10110 y(\(i\))66 b(If)260 9982 y Fg(P)435 10156 y Fl(k)551 10110 y Fm(d)646 10022 y Fl(q)646 10167 y(k)781 10110 y Fo(=)50 b(+)p Fk(1)64 b Fn(for)h(some)f Fm(q)57 b Fk(2)51 b Fo(\(0)p Fm(;)30 b Fo(1\))64 b Fn(then)h Fm(\034)3400 10044 y Fz(\006)3379 10155 y Fl(\026)3518 10110 y Fo(\()p Fm(q)7 b Fo(\))49 b(=)h(+)p Fk(1)64 b Fn(\(and)i(so)f Fm(D)5193 10044 y Fz(\006)5188 10155 y Fl(\026)5311 10110 y Fo(\()p Fm(q)7 b Fo(\))49 b(=)h(+)p Fk(1)p Fn(\).)-201 10336 y(\(ii\))73 b(If)330 10208 y Fg(P)506 10382 y Fl(k)621 10336 y Fm(d)716 10248 y Fl(q)716 10392 y(k)865 10336 y Fm(<)64 b Fo(+)p Fk(1)71 b Fn(for)h(some)h Fm(q)d Fk(2)65 b Fo(\(0)p Fm(;)30 b Fo(1\))71 b Fn(then)i Fo(0)64 b Fk(\024)f Fm(\034)3923 10270 y Fz(\000)3902 10381 y Fl(\026)4040 10336 y Fo(\()p Fm(q)7 b Fo(\))64 b Fk(\024)f Fm(\034)4640 10270 y Ff(+)4619 10381 y Fl(\026)4757 10336 y Fo(\()p Fm(q)7 b Fo(\))63 b Fk(\024)h Fo(1)46 b Fk(\000)f Fm(q)79 b Fn(and)73 b Fo(0)64 b Fk(\024)f Fm(D)6617 10270 y Fz(\000)6612 10381 y Fl(\026)6735 10336 y Fo(\()p Fm(q)7 b Fo(\))63 b Fk(\024)-201 10562 y Fm(D)-45 10496 y Ff(+)-50 10607 y Fl(\026)73 10562 y Fo(\()p Fm(q)7 b Fo(\))49 b Fk(\024)h Fo(1)p Fn(.)3393 11631 y Fo(15)p eop %%Page: 16 16 16 15 bop -201 1158 a Fc(Pro)6 b(of)69 b(of)g(Prop)6 b(osition)68 b(3.2.)-201 1384 y Fo(With)61 b(the)f(same)f(notations)h (as)g(in)h(the)f(pro)5 b(of)60 b(of)g(Prop)5 b(osition)61 b(3.1,)f(\(3.2\))f(implies)1597 1914 y Fm(d)1692 1944 y Fl(k)1827 1914 y Fk(\021)50 b Fm(\026)p Fo(\([)p Fm(k)6 b(;)30 b(k)44 b Fo(+)c(1\)\))49 b Fk(\024)3317 1703 y Fl(r)3231 1753 y Fg(X)3228 2117 y Fl(j)8 b Ff(=)p Fl(p)3504 1914 y Fm(a)3600 1941 y Fl(j)3724 1914 y Fk(\024)50 b Fm(d)4011 1944 y Fl(k)t Fz(\000)p Ff(1)4316 1914 y Fo(+)40 b Fm(d)4593 1944 y Fl(k)4719 1914 y Fo(+)f Fm(d)4995 1944 y Fl(k)t Ff(+1)5321 1914 y Fm(:)1424 b Fo(\(3.4\))-201 2504 y(Using)61 b(the)f(elemen)-5 b(tary)59 b(inequalit)-5 b(y)62 b(\()2468 2376 y Fg(P)2673 2504 y Fm(a)2769 2531 y Fl(j)2842 2504 y Fo(\))2913 2438 y Fl(q)3039 2504 y Fk(\024)3231 2376 y Fg(P)3437 2504 y Fm(a)3533 2416 y Fl(q)3533 2556 y(j)3609 2504 y Fo(,)e(where)h(0)50 b Fm(<)g(q)57 b(<)50 b Fo(1)61 b(and)f Fm(a)5510 2531 y Fl(j)5634 2504 y Fk(\025)50 b Fo(0,)60 b(w)-5 b(e)61 b(obtain)3049 3067 y Fm(d)3144 2978 y Fl(q)3144 3123 y(k)3280 3067 y Fk(\024)3560 2855 y Fl(r)3474 2905 y Fg(X)3471 3269 y Fl(j)8 b Ff(=)p Fl(p)3747 3067 y Fm(a)3843 2978 y Fl(q)3843 3118 y(j)-201 3637 y Fo(Therefore,)1871 3813 y Fg(X)1861 4182 y Fl(k)t Fz(2)p Fe(Z)2151 3975 y Fm(d)2246 3886 y Fl(q)2246 4031 y(k)2381 3975 y Fk(\024)2583 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Fm(C)13 b Fo(\()p Fm(q)7 b Fo(\))p Fm(")3248 6199 y Fl(q)e Fz(\000)p Ff(1)3503 6274 y Fo(\()p Fm(d)3669 6304 y Fl(k)t Fz(\000)p Ff(1)3974 6274 y Fo(+)40 b Fm(d)4251 6304 y Fl(k)4376 6274 y Fo(+)g Fm(d)4653 6304 y Fl(k)t Ff(+1)4919 6274 y Fo(\))4990 6199 y Fl(q)2482 6550 y Fk(\024)166 b Fm(C)13 b Fo(\()p Fm(q)7 b Fo(\))p Fm(")3248 6474 y Fl(q)e Fz(\000)p Ff(1)3503 6550 y Fo(\()p Fm(d)3669 6461 y Fl(q)3669 6606 y(k)t Fz(\000)p Ff(1)3974 6550 y Fo(+)40 b Fm(d)4251 6461 y Fl(q)4251 6606 y(k)4376 6550 y Fo(+)g Fm(d)4653 6461 y Fl(q)4653 6606 y(k)t Ff(+1)4919 6550 y Fo(\))60 b Fm(;)-201 6958 y Fo(where)h(w)-5 b(e)61 b(ha)-5 b(v)g(e)60 b(used)g(the)g(fact)g (that)g Fm(p)50 b Fo(=)g([)p Fm(k)6 b(=")p Fo(])38 b Fk(\000)i Fo(1,)60 b Fm(r)55 b Fo(=)50 b([\()p Fm(k)45 b Fo(+)40 b(1\))p Fm(=")p Fo(])f(+)h(1,)60 b Fm(r)46 b Fk(\000)40 b Fm(p)50 b(<)f Fo(2)p Fm(=")p Fo(.)80 b(Therefore,)1550 7488 y Fm(S)1662 7515 y Fl(\026)1754 7488 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))49 b(=)2395 7326 y Fg(X)2391 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10168 y Fo(=)50 b(0)65 b Fn(i\013)f Fm(q)1248 10102 y Fz(\003)1241 10213 y Fl(\026)1385 10168 y Fk(\024)50 b Fo(0)p Fn(.)81 10394 y(\(ii\))66 b(If)f Fm(s)683 10328 y Fz(\003)683 10439 y Fl(\026)827 10394 y Fm(>)50 b Fo(0)p Fn(,)64 b(then)h Fm(q)1722 10328 y Fz(\003)1715 10439 y Fl(\026)1859 10394 y Fo(=)50 b Fm(s)2136 10328 y Fz(\003)2136 10439 y Fl(\026)2230 10394 y Fn(.)81 10620 y(\(iii\))65 b(If)g Fo(0)51 b Fk(\024)f Fm(s)1072 10554 y Fz(\003)1072 10665 y Fl(\026)1216 10620 y Fm(<)g Fo(1)p Fn(,)64 b(then)i(for)e(any)h Fm(q)58 b Fk(2)50 b Fo(\()p Fm(s)3128 10554 y Fz(\003)3128 10665 y Fl(\026)3221 10620 y Fm(;)30 b Fo(1\))p Fn(,)65 b(we)f(have)2402 11045 y Fo(0)51 b Fk(\024)f Fm(\034)2837 10970 y Fz(\000)2816 11090 y Fl(\026)2954 11045 y Fo(\()p Fm(q)7 b Fo(\))49 b Fk(\024)h Fm(\034)3526 10970 y Ff(+)3505 11090 y Fl(\026)3643 11045 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(\024)g Fo(1)40 b Fk(\000)g Fm(q)7 b(;)3393 11631 y Fo(16)p eop %%Page: 17 17 17 16 bop -201 1158 a Fn(and)65 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b(on)-201 10562 y Fo(\()p Fm(q)-42 10496 y Fz(\003)-49 10607 y Fl(\026)44 10562 y Fm(;)30 b Fo(+)p Fk(1)p Fo(\))p Fn(.)-201 10814 y(ii\))60 b(F)-14 b(urthermor)-9 b(e)1144 10742 y Ff(d)p 1111 10771 147 7 v 1111 10876 a(d)p Fl(q)1277 10814 y Fm(\034)1378 10748 y Ff(+)1357 10859 y Fl(\026)1495 10814 y Fo(\()p Fm(q)7 b Fo(\))59 b Fn(exists)i(everywher)-9 b(e)57 b(on)k Fo(\()p Fm(q)3616 10748 y Fz(\003)3609 10859 y Fl(\026)3702 10814 y Fm(;)30 b Fo(+)p Fk(1)p Fo(\))59 b Fn(exc)-9 b(ept,)60 b(at)g(most,)h(on)g(a)f (c)-9 b(ountable)58 b(set)j(of)-201 11083 y(p)-9 b(oints,)65 b(and)796 11011 y Ff(d)p 762 11040 V 762 11145 a(d)p Fl(q)928 11083 y Fm(\034)1029 11017 y Fz(\000)1008 11127 y Fl(\026)1147 11083 y Fo(\()p Fm(q)7 b Fo(\))64 b Fn(exists)i(L)-9 b(eb)g(esgue)62 b(almost)j(everywher)-9 b(e)62 b(on)k Fo(\()p Fm(q)4619 11017 y Fz(\003)4612 11127 y Fl(\026)4705 11083 y Fm(;)30 b Fo(+)p Fk(1)p Fo(\))p Fn(.)3393 11631 y Fo(17)p eop %%Page: 18 18 18 17 bop -201 1158 a Fn(iii\))57 b(If)336 1087 y Ff(d)p 302 1116 147 7 v 302 1221 a(d)p Fl(q)468 1158 y Fm(\034)569 1092 y Ff(+)548 1203 y Fl(\026)686 1158 y Fo(\()p Fm(q)7 b Fo(\))57 b Fn(exists)h(for)f(some)g Fm(q)g(>)50 b(q)2621 1092 y Fz(\003)2614 1203 y Fl(\026)2708 1158 y Fn(,)58 b(then)g(for)f(al)9 b(l)57 b Fm(s)g Fn(such)g(that)i Fm(\034)4769 1092 y Ff(+)4748 1203 y Fl(\026)4886 1158 y Fo(\()p Fm(s)p Fo(\))50 b Fm(<)g Fo(+)p Fk(1)57 b Fn(and)g Fm(s)51 b(<)f(q)64 b Fn(we)57 b(have)2322 1463 y Fg(\014)2322 1563 y(\014)2322 1662 y(\014)2322 1762 y(\014)2441 1585 y Fo(d)p 2398 1666 189 7 v 2398 1833 a(d)p Fm(q)2606 1708 y(\034)2707 1633 y Ff(+)2686 1753 y Fl(\026)2824 1708 y Fo(\()p Fm(q)7 b Fo(\))3054 1463 y Fg(\014)3054 1563 y(\014)3054 1662 y(\014)3054 1762 y(\014)3159 1708 y Fk(\024)3351 1413 y Fg(\014)3351 1513 y(\014)3351 1613 y(\014)3351 1712 y(\014)3351 1812 y(\014)3426 1568 y Fm(\034)3527 1502 y Ff(+)3506 1613 y Fl(\026)3645 1568 y Fo(\()p Fm(s)p Fo(\))40 b Fk(\000)f Fm(\034)4194 1502 y Ff(+)4173 1613 y Fl(\026)4312 1568 y Fo(\()p Fm(q)7 b Fo(\))p 3426 1666 1115 7 v 3786 1833 a Fm(s)41 b Fk(\000)e Fm(q)4561 1413 y Fg(\014)4561 1513 y(\014)4561 1613 y(\014)4561 1712 y(\014)4561 1812 y(\014)6796 1708 y Fo(\(3.9\))-201 2242 y Fn(and)65 b(if)383 2170 y Ff(d)p 349 2199 147 7 v 349 2305 a(d)p Fl(q)515 2242 y Fm(\034)616 2176 y Fz(\000)595 2287 y Fl(\026)733 2242 y Fo(\()p Fm(q)7 b Fo(\))64 b Fn(exists)i(for)f(some)f Fm(q)57 b(>)50 b(q)2698 2176 y Fz(\003)2691 2287 y Fl(\026)2785 2242 y Fn(,)64 b(then)i(for)e(al)9 b(l)64 b Fm(s)i Fn(such)e(that)i Fm(\034)4897 2176 y Ff(+)4876 2287 y Fl(\026)5014 2242 y Fo(\()p Fm(s)p Fo(\))50 b Fm(<)g Fo(+)p Fk(1)64 b Fn(and)h Fm(s)51 b(<)f(q)7 b Fn(,)2264 2547 y Fg(\014)2264 2646 y(\014)2264 2746 y(\014)2264 2846 y(\014)2384 2668 y Fo(d)p 2340 2749 189 7 v 2340 2916 a(d)p Fm(q)2548 2791 y(\034)2649 2716 y Fz(\000)2628 2836 y Fl(\026)2767 2791 y Fo(\()p Fm(q)g Fo(\))2997 2547 y Fg(\014)2997 2646 y(\014)2997 2746 y(\014)2997 2846 y(\014)3101 2791 y Fk(\024)3293 2497 y 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4131 11104 a(r)46 b Fk(\000)40 b Fm(s)4546 10979 y(\034)4647 10904 y Ff(+)4626 11024 y Fl(\026)4764 10979 y Fo(\()p Fm(r)5 b Fo(\))40 b(+)5235 10856 y Fm(r)46 b Fk(\000)39 b Fm(q)p 5235 10937 397 7 v 5236 11104 a(r)46 b Fk(\000)40 b Fm(s)5652 10979 y(\034)5753 10904 y Fz(\000)5732 11024 y Fl(\026)5870 10979 y Fo(\()p Fm(s)p Fo(\))6097 10741 y Fg(\023)6314 10979 y Fm(:)340 b Fo(\(3.14\))3393 11631 y(18)p eop %%Page: 19 19 19 18 bop -201 1158 a Fc(Pro)6 b(of)69 b(of)g(Lemma)e(3.1.)-201 1384 y Fo(T)-15 b(aking)54 b Fm(t)c Fo(=)726 1303 y Fl(q)5 b Fz(\000)p Fl(s)p 726 1342 243 7 v 726 1447 a(r)t Fz(\000)p Fl(s)1043 1384 y Fo(in)54 b(inequalit)-5 b(y)54 b(\(3.1\))e(and)i (taking)g(the)f(log)j(on)e(b)5 b(oth)53 b(sides)g(giv)-5 b(es)54 b(for)f Fm(f)5902 1411 y Fl(")5975 1384 y Fo(\()p Fm(q)7 b Fo(\))50 b(:=)6517 1293 y Ff(log)24 b Fl(I)6784 1310 y Fd(\026)6867 1293 y Ff(\()p Fl(q)5 b(;")p Ff(\))p 6517 1342 632 7 v 6631 1447 a Fz(\000)24 b Ff(log)f Fl(")2175 1858 y Fm(f)2264 1885 y Fl(")2337 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10545 y Fl(\026)2189 10500 y Fk(\024)50 b Fm(s)h(<)f(q)57 b(<)50 b(r)2027 10726 y Fg(\014)2027 10826 y(\014)2027 10926 y(\014)2027 11025 y(\014)2027 11125 y(\014)2102 10881 y Fm(\034)2203 10816 y Ff(+)2182 10926 y Fl(\026)2320 10881 y Fo(\()p Fm(q)7 b Fo(\))40 b Fk(\000)g Fm(\034)2873 10816 y Ff(+)2852 10926 y Fl(\026)2990 10881 y Fo(\()p Fm(r)5 b Fo(\))p 2102 10979 1117 7 v 2462 11146 a Fm(q)47 b Fk(\000)40 b Fm(r)3238 10726 y Fg(\014)3238 10826 y(\014)3238 10926 y(\014)3238 11025 y(\014)3238 11125 y(\014)3344 11021 y Fk(\024)3536 10726 y Fg(\014)3536 10826 y(\014)3536 10926 y(\014)3536 11025 y(\014)3536 11125 y(\014)3611 10881 y Fm(\034)3712 10816 y Ff(+)3691 10926 y Fl(\026)3829 10881 y Fo(\()p Fm(r)5 b Fo(\))40 b Fk(\000)g Fm(\034)4381 10816 y Ff(+)4360 10926 y Fl(\026)4498 10881 y Fo(\()p Fm(s)p Fo(\))p 3611 10979 1114 7 v 3971 11146 a Fm(r)45 b Fk(\000)40 b Fm(s)4745 10726 y Fg(\014)4745 10826 y(\014)4745 10926 y(\014)4745 11025 y(\014)4745 11125 y(\014)4891 11021 y Fm(:)3393 11631 y Fo(19)p eop %%Page: 20 20 20 19 bop -201 1158 a Fo(T)-15 b(aking)66 b(the)f(limit)g(when)g Fm(r)f Fk(!)58 b Fm(q)7 b(;)95 b(r)63 b(>)58 b(q)7 b Fo(,)66 b(assuming)e(that)4063 1087 y Ff(d)p 4029 1116 147 7 v 4029 1221 a(d)p Fl(q)4195 1158 y Fm(\034)4296 1092 y Ff(+)4275 1203 y Fl(\026)4413 1158 y Fo(\()p Fm(q)7 b Fo(\))65 b(exists)g(and)g(using)g(the)g(con)-5 b(tin)g(uit)g(y)-201 1410 y(of)60 b Fm(\034)107 1344 y Ff(+)86 1455 y Fl(\026)285 1410 y Fo(at)g(the)g(p)5 b(oin)-5 b(t)61 b Fm(q)7 b Fo(,)60 b(w)-5 b(e)61 b(get)f(\(3.9\).)79 b(Inequalit)-5 b(y)60 b(\(3.10\))g(is)g(deriv)-5 b(ed)61 b(similarly)g(b)-5 b(y)61 b(using)f(\(3.14\).)266 b Fb(\003)81 1817 y Fo(W)-15 b(e)58 b(end)g(this)g(section)g(with)g(a)g(series)g(of)g(prop)5 b(erties)58 b(that)f(precise)h(the)g(b)5 b(eha)-5 b(viour)58 b(of)f(the)h(functions)-201 2043 y Fm(\034)-100 1977 y Fz(\006)-121 2088 y Fl(\026)17 2043 y Fo(\()p Fm(q)7 b Fo(\))59 b(in)h(the)f(t)-5 b(w)g(o)59 b(regions)g Fm(q)e Fk(\024)50 b Fo(0,)59 b Fm(q)e(>)50 b Fo(1,)59 b(and)h(in)f(particular) i(their)e(b)5 b(eha)-5 b(viour)60 b(at)f Fk(\0061)p Fo(.)79 b(First,)59 b(as)g(the)-201 2269 y(functions)h Fm(D)739 2203 y Fz(\006)734 2314 y Fl(\026)857 2269 y Fo(\()p Fm(q)7 b Fo(\))49 b Fk(2)i Fo([0)p Fm(;)30 b Fo(1])59 b(are)i(decreasing)f(in)h Fm(q)7 b Fo(,)60 b(the)g(limits)g(at)g Fk(\0061)g Fo(alw)-5 b(a)g(ys)61 b(exist:)1413 2782 y Fm(D)1569 2707 y Fz(\006)1564 2827 y Fl(\026)1687 2782 y Fo(\()p Fk(\0001)p Fo(\))49 b(:=)153 b(lim)2444 2890 y Fl(q)5 b Fz(!\0001)2934 2782 y Fm(D)3090 2707 y Fz(\006)3085 2827 y Fl(\026)3208 2782 y Fo(\()p Fm(q)i Fo(\))49 b(=)154 b(lim)3679 2890 y Fl(q)5 b Fz(!\0001)4189 2659 y Fm(\034)4290 2593 y Fz(\006)4408 2659 y Fo(\()p Fm(q)i Fo(\))p 4189 2740 448 7 v 4299 2906 a Fk(\000)p Fm(q)4707 2782 y Fk(2)51 b Fo([0)p Fm(;)30 b Fo(+)p Fk(1)p Fo(])g Fm(;)-201 3241 y Fo(and)1529 3527 y Fm(D)1685 3452 y Fz(\006)1680 3572 y Fl(\026)1803 3527 y Fo(\(+)p Fk(1)p Fo(\))49 b(:=)153 b(lim)2560 3635 y Fl(q)5 b Fz(!)p 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Fl(\026)4412 4582 y Fo(\()p Fk(\0001)p Fo(\))p Fm(:)1776 b Fo(\(3.20\))-201 4964 y Fn(\(ii\))65 b(Assume)g(that)h Fm(D)1331 4898 y Ff(+)1326 5009 y Fl(\026)1449 4964 y Fo(\()p Fk(\0001)p Fo(\))49 b Fm(<)h Fo(+)p Fk(1)64 b Fn(\(so)i(that)f Fm(q)3310 4898 y Fz(\003)3303 5009 y Fl(\026)3447 4964 y Fo(=)50 b Fk(\0001)p Fn(\).)83 b(Then)65 b(for)f(any)i Fm(q)57 b(<)50 b Fo(0)65 b Fn(we)g(have)2191 5363 y Fo(0)50 b Fk(\024)g Fm(\034)2625 5288 y Fz(\006)2604 5408 y Fl(\026)2742 5363 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(\024)g Fm(\034)3315 5288 y Ff(+)3294 5408 y Fl(\026)3432 5363 y Fo(\(0\))40 b Fk(\000)g Fm(q)7 b(D)4131 5288 y Fz(\006)4126 5408 y Fl(\026)4248 5363 y Fo(\()p Fk(\0001)p Fo(\))1991 b(\(3.21\))-201 5745 y Fn(As)66 b(a)f(c)-9 b(onse)g(quenc)g(e,)62 b(for)i(any)i Fm(q)57 b(<)50 b Fo(0)p Fn(,)1692 6127 y Fo(\(1)40 b Fk(\000)g Fm(q)7 b Fo(\))p Fm(D)2391 6052 y Fz(\006)2386 6172 y Fl(\026)2508 6127 y Fo(\(0\))49 b Fk(\024)h Fm(\034)3083 6052 y Fz(\006)3201 6127 y Fo(\()p Fm(q)7 b Fo(\))49 b Fk(\024)h Fm(D)3828 6052 y Ff(+)3823 6172 y Fl(\026)3946 6127 y Fo(\(0\))39 b Fk(\000)h Fm(q)7 b(D)4644 6052 y Fz(\006)4639 6172 y Fl(\026)4762 6127 y Fo(\()p Fk(\0001)p Fo(\))p Fm(:)1426 b Fo(\(3.22\))-201 6509 y Fn(\(iii\))65 b(F)-14 b(or)65 b(any)g Fm(q)57 b(>)50 b Fo(1)p Fn(,)1894 6735 y Fk(\000)p Fm(q)7 b(D)2280 6660 y Fz(\006)2275 6780 y Fl(\026)2397 6735 y Fo(\(+)p Fk(1)p Fo(\))49 b Fk(\024)h Fm(\034)3205 6660 y Fz(\007)3184 6780 y Fl(\026)3322 6735 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(\024)g Fo(\(1)40 b Fk(\000)g Fm(q)7 b Fo(\))p Fm(D)4493 6660 y Fz(\006)4488 6780 y Fl(\026)4610 6735 y Fo(\(+)p Fk(1)p Fo(\))1629 b(\(3.23\))-201 7055 y Fn(and)65 b(in)g(p)-9 b(articular,)63 b Fm(D)1419 6989 y Fz(\006)1414 7100 y Fl(\026)1536 7055 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(\024)g Fm(q)7 b(=)p Fo(\()p Fm(q)46 b Fk(\000)40 b Fo(1\))p Fm(D)2885 6989 y Fz(\006)2880 7100 y Fl(\026)3003 7055 y Fo(\(+)p Fk(1)p Fo(\))81 7443 y(If)55 b(one)f(considers)h(the)f(Example)g(1)h(of)f(Section)h(6)g(for)f Fm(\025)d(<)f(\013)30 b Fo(+)f(1)54 b(and)h Fm(q)i(<)50 b Fo(0,)55 b(w)-5 b(e)55 b(get)g Fm(\034)6160 7377 y Fz(\006)6139 7488 y Fl(\026)6277 7443 y Fo(\()p Fm(q)7 b Fo(\))50 b(=)6768 7362 y Ff(1)p Fz(\000)p Fl(\025q)p 6768 7401 331 7 v 6798 7506 a(\013)p Ff(+1)7119 7443 y Fo(,)-201 7700 y Fm(D)-45 7634 y Fz(\006)-50 7745 y Fl(\026)73 7700 y Fo(\(0\))g(=)h Fm(\034)650 7634 y Fz(\006)629 7745 y Fl(\026)767 7700 y Fo(\(0\))g(=)1364 7629 y Ff(1)p 1263 7658 271 7 v 1263 7763 a Fl(\013)p Ff(+1)1615 7700 y Fo(and)61 b Fm(D)2125 7634 y Fz(\006)2120 7745 y Fl(\026)2243 7700 y Fo(\()p Fk(\0001)p Fo(\))50 b(=)3066 7629 y Fl(\025)p 2972 7658 V 2972 7763 a(\013)p Ff(+1)3263 7700 y Fo(.)81 b(Therefore,)61 b(the)f(second)g(inequalit)-5 b(y)62 b(in)g(\(3.22\))e(turns)-201 7926 y(in)-5 b(to)61 b(equalitit)-5 b(y)61 b(for)f(all)i Fm(q)57 b Fk(\024)50 b Fo(0.)-201 8152 y(As)59 b(to)f(the)h(\014rst)f(inequalit)-5 b(y)60 b(in)f(\(3.23\),)e(it)i(turns)g(in)-5 b(to)59 b(equalit)-5 b(y)59 b(for)f(the)h(measure)d(of)j(the)f(Example)f(5)i(of)-201 8378 y(Section)i(6)f(for)g(an)-5 b(y)61 b Fm(q)c(>)50 b Fo(1)p Fm(=a)p Fo(.)-201 8604 y(W)-15 b(e)58 b(state)g(separately)g (the)h(follo)-5 b(wing)59 b(lemma)d(that)i(w)-5 b(e)59 b(need)f(to)h(pro)-5 b(v)g(e)58 b(P)-5 b(oin)g(t)59 b(\(iii\))g(of)g (Prop)5 b(osition)58 b(3.3,)-201 8829 y(but)j(it)f(ma)-5 b(y)59 b(b)5 b(e)61 b(of)f(indep)5 b(enden)-5 b(t)60 b(in)-5 b(terest.)-201 9183 y Fc(Lemma)68 b(3.2.)90 b Fn(F)-14 b(or)64 b(any)i Fo(0)50 b Fm(<)g(q)57 b(<)50 b(r)5 b Fn(,)65 b(we)f(have)2871 9537 y Fm(\034)2972 9471 y Fz(\006)2951 9582 y Fl(\026)3089 9537 y Fo(\()p Fm(r)5 b Fo(\))p 2871 9634 447 7 v 3051 9801 a Fm(r)3388 9677 y Fk(\024)3600 9537 y Fm(\034)3701 9471 y Fz(\006)3680 9582 y Fl(\026)3818 9537 y Fo(\()p Fm(q)i Fo(\))p 3600 9634 448 7 v 3780 9801 a Fm(q)4067 9677 y(:)-201 10136 y Fc(Pro)f(of)69 b(of)g(Lemma)e(3.2)-201 10362 y Fo(Since)61 b Fm(q)7 b(=r)55 b(<)50 b Fo(1,)60 b(one)g(has)1725 10971 y Fm(S)1837 10998 y Fl(\026)1929 10971 y Fo(\()p Fm(r)-5 b(;)30 b(")p Fo(\))2314 10896 y Fl(q)5 b(=r)2578 10971 y Fo(=)2770 10633 y Fg(0)2770 10932 y(@)2915 10809 y(X)3003 11174 y Fl(j)3186 10971 y Fm(a)3282 10896 y Fl(r)3282 11016 y(j)3358 10633 y Fg(1)3358 10932 y(A)3503 10679 y Fl(q)g(=r)3768 10971 y Fk(\024)3960 10809 y Fg(X)4047 11174 y Fl(j)4230 10971 y Fm(a)4326 10883 y Fl(q)4326 11023 y(j)4452 10971 y Fo(=)50 b Fm(S)4756 10998 y Fl(\026)4849 10971 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))3393 11631 y(20)p eop %%Page: 21 21 21 20 bop -201 1158 a Fo(W)-15 b(e)64 b(get)g Fm(S)535 1185 y Fl(\026)628 1158 y Fo(\()p Fm(r)-5 b(;)30 b(")p Fo(\))1013 1092 y Ff(1)p Fl(=r)1286 1158 y Fk(\024)57 b Fo(\()p Fm(S)1668 1185 y Fl(\026)1760 1158 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\)\))2227 1092 y Ff(1)p Fl(=q)2443 1158 y Fo(,)65 b(and)f(the)g(result)h(follo)-5 b(ws,)65 b(thanks)f(to)g(the)g(equiv)-10 b(alence)65 b(giv)-5 b(en)64 b(b)-5 b(y)-201 1384 y(Lemma)57 b(2.1.)7028 1610 y Fb(\003)-201 1836 y Fc(Pro)6 b(of)69 b(of)g(Prop)6 b(osition)68 b(3.3)81 2062 y Fo(The)c(\014rst)f(inequalit)-5 b(y)64 b(in)g(\(3.22\))e(and)h(the)g(second)f(ineqalit)-5 b(y)64 b(in)g(\(3.23\))e(are)h(trivially)i(true)e(b)5 b(ecause)-201 2288 y Fm(D)-45 2222 y Fz(\006)-50 2332 y Fl(\026)73 2288 y Fo(\()p Fm(q)i Fo(\))68 b(are)h(non-increasing.)106 b(They)69 b(turn)g(in)-5 b(to)69 b(equalities,)i(for)e(example,)g(if)h Fm(D)5508 2222 y Fz(\006)5503 2332 y Fl(\026)5625 2288 y Fo(\()p Fm(q)7 b Fo(\))69 b(are)g(constan)-5 b(t)67 b(on)-201 2513 y(\()p Fk(\0001)p Fm(;)30 b Fo(0])59 b(and)i(on)f(\(1)p Fm(;)30 b Fo(+)p Fk(1)p Fo(\))59 b(resp)5 b(ectiv)-5 b(ely)-15 b(.)81 2739 y(T)g(o)65 b(pro)-5 b(v)g(e)64 b(the)g(inequalit)-5 b(y)66 b(for)e Fm(\034)2395 2673 y Ff(+)2374 2784 y Fl(\026)2512 2739 y Fo(\()p Fm(q)7 b Fo(\))64 b(in)h(\(ii\),)g(tak)-5 b(e)64 b(inequalit)-5 b(y)65 b(\(3.13\))e(for)i(\014xed)f Fm(q)g(<)56 b Fo(0)64 b(and)h Fm(r)d Fo(=)56 b(0)-201 2965 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b(or)i(in\014m)-5 b(um)67 b(and)j(not)f Fm(\026)46 b Fk(\000)g Fo(ess)p Fm(:)p Fo(inf)83 b(or)69 b Fm(\026)46 b Fk(\000)g Fo(ess)p Fm(:)p Fo(sup)68 b(whic)-5 b(h)70 b(are)g(tak)-5 b(en.)107 b(It)69 b(is)g(w)-5 b(ell)71 b(kno)-5 b(wn)-201 8974 y(that)60 b Fm(\015)297 8908 y Fz(\006)287 9019 y Fl(\026)415 8974 y Fo(\()p Fm(x)p Fo(\))50 b Fk(\024)g Fo(1)61 b(for)f Fm(\026)40 b Fk(\000)g Fo(a)p Fm(:)p Fo(e)p Fm(:)59 b(x)i Fo(\()p Fn(e.g.)79 b Fo([1)o(]\).)h(Therefore,)60 b Fm(\026)p Fo(\()p Fm(A)p Fo(\))49 b(=)h(1,)60 b(and)2372 9381 y Fm(L)2496 9305 y Fz(\006)2496 9425 y Fl(\026)2665 9381 y Fk(\024)49 b Fm(\026)40 b Fk(\000)g Fo(ess)p Fm(:)p Fo(inf)74 b Fm(\015)3849 9305 y Fz(\006)3839 9425 y Fl(\026)3967 9381 y Fo(\()p Fm(x)p Fo(\))50 b Fk(\024)g Fo(1)p Fm(;)2503 9787 y(U)2647 9712 y Fz(\006)2627 9832 y Fl(\026)2815 9787 y Fk(\025)g Fm(\026)40 b Fk(\000)g Fo(ess)p Fm(:)p Fo(sup)59 b Fm(\015)4052 9712 y Fz(\006)4042 9832 y Fl(\026)4170 9787 y Fo(\()p Fm(x)p Fo(\))p Fm(:)-201 10111 y Fo(It)i(is)h(clear)g (that)g(strict)f(inequalities)i(are)e(quite)h(p)5 b(ossible.)84 b(The)62 b(results)g(of)f(the)g(next)h(t)-5 b(w)g(o)61 b(prop)5 b(ositions)-201 10337 y(should)61 b(b)5 b(e)60 b(compared)f(with)i(the)g(inequalities)2097 10727 y Fm(D)2253 10652 y Fz(\006)2248 10772 y Fl(\026)2371 10727 y Fo(\()p Fm(q)7 b Fo(\))49 b Fk(\024)h Fm(\026)40 b Fk(\000)g Fo(ess)p Fm(:)p Fo(inf)73 b Fm(\015)3834 10652 y Fz(\006)3824 10772 y Fl(\026)3953 10727 y Fo(\()p Fm(x)p Fo(\))p Fm(;)150 b(q)57 b(>)50 b Fo(1)p Fm(;)2071 11133 y(D)2227 11058 y Fz(\006)2222 11178 y Fl(\026)2344 11133 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(\025)g Fm(\026)40 b Fk(\000)g Fo(ess)p Fm(:)p Fo(sup)59 b Fm(\015)3861 11058 y Fz(\006)3851 11178 y Fl(\026)3979 11133 y Fo(\()p Fm(x)p Fo(\))p Fm(;)150 b(q)58 b(<)49 b Fo(1)p Fm(;)3393 11631 y Fo(21)p eop %%Page: 22 22 22 21 bop -201 1158 a Fo(of)60 b(Prop)5 b(osition)61 b(4.1)f(of)g(the)g(next)g(Section.)81 1610 y(W)-15 b(e)61 b(b)5 b(egin)61 b(with)g(the)f(case)g Fm(q)d(<)50 b Fo(0.)-201 1942 y Fc(Prop)6 b(osition)69 b(3.4.)90 b Fn(The)65 b(fol)9 b(lowing)63 b(statements)j(hold:)-201 2168 y(\(i\))2863 2394 y Fm(D)3019 2319 y Fz(\006)3014 2439 y Fl(\026)3137 2394 y Fo(\()p Fk(\0001)p Fo(\))49 b Fk(\025)h Fm(U)3988 2319 y Fz(\006)3968 2439 y Fl(\026)6705 2394 y Fo(\(3.24\))-201 2701 y Fn(\(ii\))65 b(F)-14 b(or)65 b(any)g Fm(q)58 b(<)49 b Fo(0)p Fn(,)2735 2965 y Fm(D)2891 2890 y Ff(+)2886 3010 y Fl(\026)3009 2965 y Fo(\()p Fm(q)7 b Fo(\))49 b Fk(\025)3657 2842 y Fm(q)p 3500 2923 401 7 v 3500 3089 a(q)e Fk(\000)40 b Fo(1)3921 2965 y Fm(U)4065 2890 y Ff(+)4045 3010 y Fl(\026)4183 2965 y Fm(:)2471 b Fo(\(3.25\))-201 3369 y Fn(\(iii\))65 b(If)g Fm(\015)475 3303 y Ff(+)465 3414 y Fl(\026)593 3369 y Fo(\()p Fm(x)p Fo(\))50 b(=)g Fm(\015)1185 3303 y Fz(\000)1175 3414 y Fl(\026)1303 3369 y Fo(\()p Fm(x)p Fo(\))65 b Fn(for)f(any)i Fm(x)50 b Fk(2)h Fm(A)p Fn(,)65 b(so)g(that)h Fm(U)3584 3303 y Ff(+)3564 3414 y Fl(\026)3753 3369 y Fo(=)50 b Fm(U)4089 3303 y Fz(\000)4069 3414 y Fl(\026)4257 3369 y Fo(:=)g Fm(U)4624 3396 y Fl(\026)4717 3369 y Fn(,)65 b(then)g(for)f(any)i Fm(q)57 b(<)50 b Fo(0)2757 3789 y Fm(D)2913 3714 y Fz(\000)2908 3834 y Fl(\026)3031 3789 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(\025)3679 3666 y Fm(q)p 3523 3747 V 3523 3913 a(q)d Fk(\000)39 b Fo(1)3943 3789 y Fm(U)4067 3816 y Fl(\026)4161 3789 y Fm(:)-201 4232 y Fc(Pro)6 b(of)92 b(of)f(Prop)6 b(osition)92 b(3.4)79 b Fo(.)140 b(Let)80 b Fm(x)2802 4259 y Ff(0)2964 4232 y Fk(2)k Fm(A;)30 b(q)90 b(<)83 b Fo(0)p Fm(;)30 b(")83 b Fk(2)h Fo(\(0)p Fm(;)30 b Fo(1\).)139 b(As)80 b(it)h(w)-5 b(as)80 b(men)-5 b(tioned)79 b(in)i(the)-201 4458 y(In)-5 b(tro)5 b(duction,)75 b(one)d(can)g(tak)-5 b(e)72 b Fm(\026)p Fo(\(\()p Fm(x)47 b Fk(\000)h Fm(";)30 b(x)48 b Fo(+)g Fm(")p Fo(\)\))71 b(instead)i(of)f Fm(\026)p Fo(\([)p Fm(x)47 b Fk(\000)g Fm(";)30 b(x)48 b Fo(+)g Fm(")p Fo(\)\))72 b(in)h(the)f(de\014nition)h(of)-201 4684 y Fm(I)14 b Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))60 b(without)g(c)-5 b(hanging)61 b(the)f(dimensions)g Fm(D)3189 4618 y Fz(\006)3184 4729 y Fl(\026)3306 4684 y Fo(\()p Fm(q)7 b Fo(\).)80 b(One)59 b(can)i(estimate)-201 5132 y Fm(I)-121 5159 y Fl(\026)-28 5132 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))50 b Fk(\025)609 4902 y Fg(Z)702 5279 y Ff(\()p Fl(x)836 5297 y Fa(0)905 5279 y Fz(\000)p Fl(";x)1198 5297 y Fa(0)1268 5279 y Ff(+)p Fl(")p Ff(\))p Fz(\\)p Ff(supp)q Fl(\026)2006 5132 y Fm(\026)p Fo(\(\()p Fm(x)q Fk(\000)q Fm(";)30 b(x)q Fo(+)q Fm(")p Fo(\)\))3147 5057 y Fl(q)5 b Fz(\000)p Ff(1)3400 5132 y Fm(d\026)p Fo(\()p Fm(x)p Fo(\))49 b Fk(\025)h Fm(\026)p Fo(\(\()p Fm(x)4448 5159 y Ff(0)4526 5132 y Fk(\000)q Fo(2)p Fm(";)30 b(x)5030 5159 y Ff(0)5108 5132 y Fo(+)q(2)p Fm(")p Fo(\)\))5569 5057 y Fl(q)5 b Fz(\000)p Ff(1)5824 5132 y Fm(\026)p Fo(\(\()p Fm(x)6180 5159 y Ff(0)6259 5132 y Fk(\000)q Fm(";)30 b(x)6672 5159 y Ff(0)6750 5132 y Fo(+)q Fm(")p Fo(\)\))p Fm(:)-201 5586 y Fo(Therefore,)741 5905 y(log)j Fm(I)1086 5932 y Fl(\026)1179 5905 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))p 696 5986 923 7 v 696 6153 a(\()p Fm(q)47 b Fk(\000)40 b Fo(1\))30 b(log)j Fm(")1689 6028 y Fk(\025)1901 5905 y Fo(log)g Fm(\026)p Fo(\(\()p Fm(x)2522 5932 y Ff(0)2640 5905 y Fk(\000)40 b Fo(2)p Fm(";)30 b(x)3183 5932 y Ff(0)3302 5905 y Fo(+)40 b(2)p Fm(")p Fo(\)\))p 1901 5986 1900 7 v 2676 6153 a(log)33 b Fm(")3861 6028 y Fo(+)4218 5905 y(1)p 4063 5986 401 7 v 4063 6153 a Fm(q)47 b Fk(\000)40 b Fo(1)4504 5905 y(log)33 b Fm(\026)p Fo(\(\()p Fm(x)5125 5932 y Ff(0)5243 5905 y Fk(\000)40 b Fm(";)30 b(x)5695 5932 y Ff(0)5814 5905 y Fo(+)39 b Fm(")p Fo(\)\))p 4504 5986 1718 7 v 5188 6153 a(log)33 b Fm(")6242 6028 y(:)-201 6492 y Fo(T)-15 b(aking)61 b(the)f(liminf)h(or)f(limsup,)g(w)-5 b(e)60 b(obtain)h(\(remem)-5 b(b)5 b(er)57 b(that)j Fm(\015)4350 6426 y Ff(+)4340 6536 y Fl(\026)4468 6492 y Fo(\()p Fm(x)4643 6519 y Ff(0)4722 6492 y Fo(\))50 b Fm(<)g Fo(+)p Fk(1)p Fo(\))2193 6950 y Fm(D)2349 6875 y Fz(\006)2344 6995 y Fl(\026)2467 6950 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(\025)f Fm(\015)3042 6875 y Fz(\006)3032 6995 y Fl(\026)3161 6950 y Fo(\()p Fm(x)3336 6977 y Ff(0)3414 6950 y Fo(\))40 b(+)3882 6827 y(1)p 3727 6908 401 7 v 3727 7074 a Fm(q)47 b Fk(\000)40 b Fo(1)4148 6950 y Fm(\015)4252 6875 y Ff(+)4242 6995 y Fl(\026)4370 6950 y Fo(\()p Fm(x)4545 6977 y Ff(0)4624 6950 y Fo(\))30 b Fm(:)1929 b Fo(\(3.26\))-201 7393 y(T)-15 b(aking)61 b(the)f(limit)h Fm(q)c Fk(!)50 b(\0001)60 b Fo(in)h(\(3.26\))e(yields)2695 7750 y Fm(D)2851 7675 y Fz(\006)2846 7795 y Fl(\026)2969 7750 y Fo(\()p Fk(\0001)p Fo(\))49 b Fk(\025)h Fm(\015)3780 7675 y Fz(\006)3770 7795 y Fl(\026)3898 7750 y Fo(\()p Fm(x)4073 7777 y Ff(0)4152 7750 y Fo(\))p Fm(:)2431 b Fo(\(3.27\))-201 8106 y(The)61 b(statemen)-5 b(ts)57 b(of)j(the)g(Prop)5 b(osition)61 b(follo)-5 b(w)61 b(directly)g(from)e(\(3.26\),)g(\(3.27\).)1730 b Fb(\003)81 8414 y Fo(In)72 b(the)f(\014rst)g(Example)f(of)h(Section)h (6,)i Fm(\026)47 b Fk(\000)g Fo(ess)p Fm(:)p Fo(sup)70 b Fm(\015)3948 8348 y Fz(\006)3938 8459 y Fl(\026)4067 8414 y Fo(\()p Fm(x)p Fo(\))e(=)g(0,)74 b(as)d(the)g(measure)f(is)i (pure)g(p)5 b(oin)-5 b(t.)-201 8665 y(Ho)g(w)g(ev)g(er,)61 b Fm(U)727 8599 y Fz(\006)707 8710 y Fl(\026)895 8665 y Fo(=)50 b Fm(\015)1191 8599 y Fz(\006)1181 8710 y Fl(\026)1309 8665 y Fo(\(0\))g(=)1804 8594 y Fl(\025)p Fz(\000)p Ff(1)p 1804 8623 263 7 v 1890 8728 a Fl(\013)2137 8665 y Fm(>)g Fo(0.)80 b(In)61 b(the)f(case)f Fm(\025)51 b(<)f Fo(1)40 b(+)g Fm(\013)q Fo(,)2481 9133 y Fm(D)2637 9058 y Fz(\006)2632 9178 y Fl(\026)2755 9133 y Fo(\()p Fk(\0001)p Fo(\))49 b(=)3644 9010 y Fm(\025)p 3482 9091 431 7 v 3482 9257 a Fo(1)40 b(+)g Fm(\013)3983 9133 y(>)50 b(U)4319 9058 y Fz(\006)4299 9178 y Fl(\026)4437 9133 y Fm(;)-201 9546 y Fo(and)61 b(in)g(the)f(case)g Fm(\025)50 b(>)g Fo(1)41 b(+)e Fm(\013)q Fo(,)1811 9930 y Fm(D)1967 9855 y Fz(\006)1962 9975 y Fl(\026)2084 9930 y Fo(\()p Fk(\0001)p Fo(\))49 b(=)h Fm(U)2935 9855 y Fz(\006)2915 9975 y Fl(\026)3235 9930 y Fo(and)182 b Fm(D)3866 9855 y Fz(\006)3861 9975 y Fl(\026)3984 9930 y Fo(\()p Fm(q)7 b Fo(\))49 b(=)4632 9807 y Fm(q)p 4475 9888 401 7 v 4475 10055 a(q)e Fk(\000)40 b Fo(1)4896 9930 y Fm(U)5040 9855 y Fz(\006)5020 9975 y Fl(\026)-201 10374 y Fo(for)62 b(an)-5 b(y)61 b Fm(q)e(<)52 b Fk(\000)1105 10302 y Fl(\013)p 919 10331 464 7 v 919 10436 a(\025)p Fz(\000)p Fl(\013)p Fz(\000)p Ff(1)1401 10374 y Fo(.)84 b(So,)62 b(the)f(inequalities)i(\(3.24\))d(and)i (\(3.25\))e(ma)-5 b(y)60 b(turn)i(in)-5 b(to)62 b(equalities)g(for)f (some)-201 10599 y Fm(\026)56 b Fo(and)h Fm(q)7 b Fo(.)78 b(One)56 b(can)g(note)g(also)h(that)f(a)g(single)h(p)5 b(oin)-5 b(t)57 b(\()p Fm(x)50 b Fo(=)g(0\))56 b(from)f(supp)o Fm(\026)i Fo(suc)-5 b(h)56 b(that)f Fm(\026)p Fo(\()p Fk(f)p Fm(x)p Fk(g)p Fo(\))50 b(=)g(0)56 b(ma)-5 b(y)-201 10825 y(determine)59 b(the)h(b)5 b(eha)-5 b(viour)61 b(of)f Fm(D)2168 10759 y Fz(\006)2163 10870 y Fl(\026)2286 10825 y Fo(\()p Fm(q)7 b Fo(\))60 b(as)g Fm(q)d Fk(!)50 b(\0001)p Fo(.)81 11133 y(A)61 b(similar)g(result)f(can)h(b)5 b(e)60 b(pro)-5 b(v)g(ed)61 b(for)f Fm(q)d Fk(!)51 b Fo(+)p Fk(1)p Fo(.)3393 11631 y(22)p eop %%Page: 23 23 23 22 bop -201 1158 a Fc(Prop)6 b(osition)69 b(3.5.)90 b Fn(F)-14 b(or)64 b(any)i Fm(q)57 b(>)50 b Fo(1)p Fn(,)2848 1493 y Fm(D)3004 1418 y Fz(\006)2999 1538 y Fl(\026)3121 1493 y Fo(\(+)p Fk(1)p Fo(\))g Fk(\024)f Fm(L)3952 1418 y Fz(\006)3952 1538 y Fl(\026)4070 1493 y Fm(:)2584 b Fo(\(3.28\))-201 1827 y Fc(Pro)6 b(of)77 b(of)h(Prop)6 b(osition)77 b(3.5)68 b Fo(.)103 b(As)68 b(in)h(the)e(pro)5 b(of)68 b(of)g(Prop)5 b(osition)68 b(3.4,)h(w)-5 b(e)68 b(easily)h(see)e(that)h(for)f(an)-5 b(y)-201 2053 y Fm(x)-97 2080 y Ff(0)32 2053 y Fk(2)51 b Fo(supp)p Fm(\026;)30 b(q)57 b(>)49 b Fo(1,)333 2444 y Fm(I)413 2471 y Fl(\026)506 2444 y Fo(\()p Fm(q)7 b(;)30 b Fo(2)p Fm(")p Fo(\))49 b Fk(\025)1234 2214 y Fg(Z)1326 2591 y Ff(\()p Fl(x)1460 2609 y Fa(0)1530 2591 y Fz(\000)p Fl(";x)1823 2609 y Fa(0)1893 2591 y Ff(+)p Fl(")p Ff(\))p Fz(\\)p Ff(supp)p Fl(\026)2631 2444 y Fm(\026)p Fo(\(\()p Fm(x)39 b Fk(\000)h Fo(2)p Fm(";)30 b(x)40 b Fo(+)g(2)p Fm(")p Fo(\)\))4109 2369 y Fl(q)5 b Fz(\000)p Ff(1)4364 2444 y Fm(d\026)p Fo(\()p Fm(x)p Fo(\))49 b Fk(\025)h Fm(\026)p Fo(\(\()p Fm(x)5412 2471 y Ff(0)5530 2444 y Fk(\000)40 b Fm(";)30 b(x)5982 2471 y Ff(0)6101 2444 y Fo(+)40 b Fm(")p Fo(\)\))6510 2369 y Fl(q)6585 2444 y Fm(:)-201 2876 y Fo(Therefore,)2593 3129 y Fm(D)2749 3054 y Fz(\006)2744 3174 y Fl(\026)2866 3129 y Fo(\()p Fm(q)7 b Fo(\))50 b Fk(\024)3514 3006 y Fm(q)p 3358 3087 401 7 v 3358 3254 a(q)d Fk(\000)40 b Fo(1)3779 3129 y Fm(\015)3883 3054 y Fz(\006)3873 3174 y Fl(\026)4001 3129 y Fo(\()p Fm(x)4176 3156 y Ff(0)4254 3129 y Fo(\))p Fm(:)-201 3503 y Fo(T)-15 b(aking)61 b(the)f(limit)h Fm(q)c Fk(!)50 b Fo(+)p Fk(1)60 b Fo(and)h(then)f(sup)g(o)-5 b(v)g(er)61 b Fm(x)3503 3530 y Ff(0)3581 3503 y Fo(,)f(w)-5 b(e)61 b(obtain)g(the)f(result)g(of)h(the)f(Prop)5 b(osition.)193 b Fb(\003)81 3804 y Fo(In)67 b(the)f(Example)e(5)j(of)f(Section)g(6,)h Fm(\026)44 b Fk(\000)g Fo(ess)p 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y(23)p eop %%Page: 24 24 24 23 bop -201 1158 a Fc(Pro)6 b(of)91 b(of)g(Prop)6 b(osition)90 b(4.1:)118 b Fo(In)79 b(\(4.1\))g(and)g(\(4.2\),)k(the)c (second)g(inequalities)h(are)g(ob)-5 b(vious.)137 b(The)-201 1384 y(equalities)78 b(are)g(w)-5 b(ell)79 b(kno)-5 b(wn)77 b([20])g([11)o(].)131 b(The)78 b(last)f(inequalities)i(in)f(\(4.1\))f (and)g(\(4.2\))g(w)-5 b(ere)77 b(pro)-5 b(v)g(ed)78 b(in)-201 1610 y([3].)118 b(The)73 b(pro)5 b(of)73 b(of)g(the)g(\014rst)g(t)-5 b(w)g(o)73 b(inequalities)h(can)f(b)5 b(e)73 b(done)g(in)h(the)f(same)e (spirit.)119 b(F)-15 b(or)73 b(the)g(sak)-5 b(e)73 b(of)-201 1836 y(completeness,)58 b(w)-5 b(e)60 b(shall)i(pro)-5 b(vide)61 b(b)5 b(elo)-5 b(w)61 b(the)f(pro)5 b(ofs)60 b(of)g(all)h(of)f(them.)81 2062 y(F)-15 b(or)61 b(an)-5 b(y)60 b Fm(\027)i Fk(2)51 b Fo(\(0)p Fm(;)30 b Fo(1\))59 b(consider)i(the)f(sets)1832 2464 y Fm(A)1968 2491 y Fl(\027)2055 2464 y Fo(\()p Fm(")p Fo(\))49 b(=)h Fk(f)p Fm(x)h Fk(2)f Fj(R)96 b Fk(j)60 b Fm(\026)p Fo(\()p Fm(x)40 b 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Fm(A)1335 1567 y Fl(\027)1473 1540 y Fo(:=)49 b(lim)30 b(sup)1814 1695 y Fl(k)t Fz(!1)2302 1540 y Fm(A)2438 1567 y Fl(\027)2525 1540 y Fo(\()p Fm(")2681 1570 y Fl(k)2766 1540 y Fo(\))50 b Fk(\033)g Fm(B)3217 1567 y Fl(\027)3353 1540 y Fo(:=)g Fk(f)p Fm(x)g Fk(2)h Fj(R)96 b Fk(j)60 b Fm(\015)4420 1465 y Fz(\000)4410 1585 y Fl(\026)4538 1540 y Fo(\()p Fm(x)p Fo(\))50 b Fk(\024)g Fm(a)41 b Fo(+)e Fm(\027)12 b(=)p Fo(2)p Fk(g)p Fm(:)-201 2021 y Fo(Again,)78 b(w)-5 b(e)75 b(ha)-5 b(v)g(e)74 b Fm(\026)p Fo(\()p Fm(A)1429 2048 y Fl(\027)1515 2021 y Fo(\))f Fk(\025)g Fm(\026)p Fo(\()p Fm(B)2193 2048 y Fl(\027)2279 2021 y Fo(\))g Fm(>)g Fo(0)h(for)g(an)-5 b(y)75 b Fm(\027)85 b(>)73 b Fo(0)h(as)g(a)g(direct)h(consequence)e(of)h(de\014nition)h(of) -201 2246 y Fm(\026)30 b Fo(ess)p Fm(:)p Fo(inf)13 b(.)93 b(Using)65 b(the)f(Borel-Can)-5 b(telli)66 b(Lemma)61 b(\(as)j(done)g(in)h([15]\),)f(w)-5 b(e)65 b(obtain)f(that)5894 2118 y Fg(P)6070 2292 y Fl(k)6185 2246 y Fm(\026)p Fo(\()p Fm(A)6502 2273 y 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5329 y Fo(\()p Fm(")2108 5359 y Fl(k)2193 5329 y Fo(\))50 b(=)g Fk(f)p Fm(x)g Fk(2)h Fj(R)96 b Fk(j)60 b Fm(\026)p Fo(\()p Fm(x)39 b Fk(\000)h Fm(")3817 5359 y Fl(k)3902 5329 y Fm(;)30 b(x)41 b Fo(+)e Fm(")4394 5359 y Fl(k)4479 5329 y Fo(\))51 b Fk(\024)f Fm(")4878 5253 y Fl(a)p Fz(\000)p Fl(\027)4878 5385 y(k)5148 5329 y Fk(g)-201 5711 y Fo(with)61 b(the)f(same)f Fm(")1067 5741 y Fl(k)1213 5711 y Fo(and)h Fm(a)51 b Fo(=)f Fm(\026)40 b Fk(\000)70 b Fo(ess)p Fm(:)p Fo(sup)59 b Fm(\015)2980 5645 y Ff(+)2970 5756 y Fl(\026)3098 5711 y Fo(\()p Fm(x)p Fo(\),)h(one)g(can)g(pro)-5 b(v)g(e)61 b(the)f(last)g(inequalit)-5 b(y)62 b(in)f(\(4.2\).)277 b Fb(\003)81 6027 y Fo(Quite)91 b(often,)97 b(the)90 b(\014rst)g(and)h(last)g(inequalities)h(in)f (\(4.1\))f(and)g(\(4.2\))g(turn)h(in)-5 b(to)91 b(equalities)g(\(for) -201 6253 y(example,)56 b(this)g(is)h(the)g(case)e(if)i Fm(D)2097 6187 y Fz(\006)2092 6298 y Fl(\026)2215 6253 y Fo(\()p Fm(q)7 b Fo(\))49 b(=)h(const)56 b(or)h(in)g(the)g(examples)d (1)j(and)g(5)f(of)g(Section)h(6\).)79 b(Ho)-5 b(w)g(ev)g(er,)-201 6479 y(as)60 b(sho)-5 b(w)61 b(the)f(examples)f(2)h(and)h(6,)f(strict)g (inequalities)i(ma)-5 b(y)59 b(o)5 b(ccur.)81 6705 y(As)80 b(it)f(follo)-5 b(ws)79 b(from)f(the)g(ab)5 b(o)-5 b(v)g(e)79 b(results,)k(if)c Fm(\026)30 b Fo(ess)p Fm(:)p Fo(inf)13 b Fm(\015)4075 6639 y Fz(\000)4065 6750 y Fl(\026)4193 6705 y Fo(\()p Fm(x)p Fo(\))81 b Fm(<)f(\026)30 b Fo(ess)p Fm(:)p Fo(sup)o Fm(\015)5535 6639 y Fz(\000)5525 6750 y Fl(\026)5653 6705 y Fo(\()p Fm(x)p Fo(\),)83 b(then)78 b Fm(D)6621 6639 y Fz(\000)6616 6750 y Fl(\026)6739 6705 y Fo(\()p Fm(q)7 b Fo(\))78 b(is)-201 6931 y(discon)-5 b(tin)g(uous)61 b(at)f Fm(q)d Fo(=)50 b(1)60 b(\(the)g(same)f(for)h Fm(\015)2858 6865 y Ff(+)2848 6975 y Fl(\026)3037 6931 y Fo(and)h Fm(D)3547 6865 y Ff(+)3542 6975 y Fl(\026)3664 6931 y Fo(\).)81 7156 y(On)g(the)f(other)g(hand,)g(if)h Fm(D)2001 7090 y Fz(\006)1996 7201 y Fl(\026)2118 7156 y Fo(\(1)40 b Fk(\000)g Fo(0\))50 b(=)g Fm(D)3062 7090 y Fz(\006)3057 7201 y Fl(\026)3180 7156 y Fo(\(1)40 b(+)g(0\))50 b(=)g Fm(D)5 b Fo(,)59 b(then)i(\(4.1\))e(and)h(\(4.2\))g(imply)1991 7555 y Fm(\015)2095 7480 y Ff(+)2085 7600 y Fl(\026)2213 7555 y Fo(\()p Fm(x)p Fo(\))50 b(=)g Fm(\015)2805 7480 y Fz(\000)2795 7600 y Fl(\026)2923 7555 y Fo(\()p Fm(x)p Fo(\))g(=)g Fm(D)5 b(;)211 b Fo(for)50 b Fm(\026)40 b Fk(\000)g Fo(a)p Fm(:)p Fo(e)p Fm(:)60 b(x)g(;)1818 b Fo(\(4.5\))-201 7936 y Fn(i.e.)76 b Fo(the)47 b(measure)f Fm(\026)i Fo(is)g(\\exact)g(dimensional")f([26)o(].)76 b(One)47 b(should)i(stress)e(that)g(the)h(con)-5 b(v)g(erse)47 b(is)h(not)g(true:)-201 8162 y(there)56 b(are)g(measures)e(suc)-5 b(h)56 b(that)f(\(4.5\))g(holds,)i(but)f Fm(D)3568 8096 y Fz(\006)3563 8207 y Fl(\026)3686 8162 y Fo(\()p Fm(q)7 b Fo(\))49 b Fk(6)p Fo(=)h(const)56 b(\(most)d(of)j(examples)e(of)i (Section)g(6\).)-201 8388 y(This)70 b(is)g(not)e(in)i(general)g(true)f (ev)-5 b(en)68 b(for)h Fm(q)j(>)64 b Fo(1)70 b(\(Example)d(5\),)k (unlik)-5 b(e)70 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b(or)h(the)f(Example)f(5)h(with)h(di\013eren)-5 b(t)61 b Fm(a)p Fo('s)f(\(in)h(the)f(case)g Fm(q)d(>)50 b Fo(1\)\).)-201 10096 y Fc(Prop)6 b(osition)69 b(4.2.)90 b Fn(We)64 b(have)3178 10477 y Fm(\026)30 b Fo(ess)p Fm(:)p Fo(inf)43 b Fm(\015)3948 10402 y Fz(\000)3938 10522 y Fl(\026)4067 10477 y Fo(\()p Fm(x)p Fo(\))50 b Fk(\024)f Fm(h)4659 10402 y Fz(\000)4659 10522 y Fl(\026)4842 10477 y Fm(;)1903 b Fo(\(4.6\))3126 10770 y Fm(h)3231 10695 y Ff(+)3231 10815 y Fl(\026)3399 10770 y Fk(\024)50 b Fm(\026)30 b Fo(ess)p Fm(:)p Fo(sup)f Fm(\015)4414 10695 y Ff(+)4404 10815 y Fl(\026)4532 10770 y Fo(\()p Fm(x)p Fo(\))64 b Fm(;)1903 b Fo(\(4.7\))1713 11062 y Fm(D)1869 10987 y Fz(\006)1864 11107 y Fl(\026)1987 11062 y Fo(\(1)40 b(+)g(0\))50 b Fk(\024)g Fm(h)2880 10987 y Fz(\006)2880 11107 y Fl(\026)3048 11062 y Fo(=)g Fm(D)3396 10987 y Fz(\006)3391 11107 y Fl(\026)3514 11062 y Fo(\(1\))g Fk(\024)f Fm(D)4144 10987 y Fz(\006)4139 11107 y Fl(\026)4262 11062 y Fo(\(1)40 b Fk(\000)g Fo(0\))65 b Fm(:)1872 b Fo(\(4.8\))3393 11631 y(25)p eop %%Page: 26 26 26 25 bop -201 1158 a Fc(Remark)69 b(4.2.)90 b Fn(A)-9 b(gain,)90 b(due)85 b(to)h(our)f(e)-9 b(quivalenc)g(e)82 b(L)-9 b(emmas,)90 b(the)c(ine)-9 b(qualities)83 b(\(4.3\),)91 b(\(4.4\))86 b(of)g([20])-201 1384 y(imply)64 b(\(4.6\),)h(\(4.7\))h (and)f(two)g(of)g(four)f(ine)-9 b(qualities)63 b(in)i(\(4.8\).)-201 1731 y Fc(Pro)6 b(of)76 b(of)h(Prop)6 b(osition)76 b(4.2:)93 b Fo(As)67 b(it)h(w)-5 b(as)67 b(men)-5 b(tioned)65 b(ab)5 b(o)-5 b(v)g(e,)68 b(the)f(inequalit)-5 b(y)68 b(\(4.6\))e(as)h(w)-5 b(ell)68 b(as)f(the)-201 1957 y(inequalit)-5 b(y)67 b(\(4.7\))d(is)i(a) g(consequence)e(of)h(our)h(Lemma)c(2.2)j(and)g(Theorem)g(4.1)f(in)j ([20)o(].)95 b(A)66 b(direct)g(pro)5 b(of)-201 2182 y(that)60 b(do)h(not)f(use)g(the)g(R)-5 b(\023)-86 b(en)-5 b(yi)61 b(en)-5 b(tropies)60 b Fm(R)q(h)2932 2117 y Fz(\006)2932 2227 y Fl(\026)3111 2182 y Fo(can)g(easily)h(b)5 b(e)61 b(done)f(b)-5 b(y)61 b(using)f(F)-15 b(atou's)59 b(lemma.)81 2408 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b(b)5 b(ound)55 b(in)h(the)g(spirit)g(of)-201 5760 y(\(5.3\))k(and)g(\(5.4\))f(could)i(b)5 b(e)61 b(obtained)f(for)h (the)f(en)-5 b(trop)g(y)60 b(dimensions.)-201 6141 y Fc(Pro)6 b(of)83 b(of)g(Prop)6 b(osition)83 b(5.1.)117 b Fo(Since,)76 b(thanks)c(to)g(Theorem)g(2.1,)j(the)d(di\013eren)-5 b(t)73 b(de\014nitions)g(of)g(the)-201 6367 y(m)-5 b(ultifractal)70 b(dimensions)f(of)i(a)g(measure)d(turn)j(out)g(to)f(supply)h(the)g (same)d(n)-5 b(um)g(b)5 b(er,)72 b(w)-5 b(e)71 b(shall)g(use,)i(in)-201 6593 y(eac)-5 b(h)60 b(case)g(the)g(most)f(con)-5 b(v)g(enien)g(t)60 b(de\014nitions.)80 b(W)-15 b(e)60 b(\014rst)h(pro)-5 b(v)g(e)60 b(\(5.1\)-\(5.2\).)-201 6819 y(F)-15 b(or)60 b Fm(q)e Fk(6)p Fo(=)50 b(1)60 b(the)g(fastest)f(w)-5 b(a)g(y)61 b(consists)e(in)i(using)g(the)f(mean-q)f(dimensions.)79 b(If)60 b Fm(q)d(>)50 b Fo(1,)60 b(w)-5 b(e)60 b(ha)-5 b(v)g(e)1015 7213 y Fm(L)1139 7240 y Fl(\027)9 b Fz(\003)p Fl(\026)1381 7213 y Fo(\()p Fm(q)e(;)30 b(")p Fo(\))164 b(=)2269 7090 y(1)p 2269 7171 91 7 v 2272 7338 a Fm(")2410 6983 y Fg(Z)2502 7361 y Fe(R)2606 7213 y Fo(\()p Fm(\027)52 b Fk(\003)40 b Fm(\026)p Fo(\))3131 7138 y Fl(q)3206 7213 y Fo(\([)p Fm(u)g Fk(\000)g Fm(";)30 b(u)40 b Fo(+)g Fm(")p Fo(\)\))p Fm(du)1941 7677 y Fo(=)2269 7554 y(1)p 2269 7634 V 2272 7801 a Fm(")2410 7447 y Fg(Z)2502 7824 y Fe(R)2637 7438 y Fg(\022)2759 7447 y(Z)2851 7824 y Fe(R)2986 7677 y Fm(\026)p Fo(\([)p Fm(u)f Fk(\000)h Fm(x)g Fk(\000)g Fm(";)30 b(u)40 b Fk(\000)g Fm(x)g Fo(+)g Fm(")p Fo(\)\))p Fm(d\027)12 b Fo(\()p Fm(x)p Fo(\))5357 7438 y Fg(\023)5477 7484 y Fl(q)5583 7677 y Fm(du)1941 8141 y Fk(\024)2269 8018 y Fo(1)p 2269 8099 V 2272 8266 a Fm(")2410 7912 y Fg(Z)2502 8289 y Fe(R)2637 7903 y Fg(\022)2759 7912 y(Z)2851 8289 y Fe(R)2986 8141 y Fo(\()o Fm(\026)p Fo(\([)p Fm(u)40 b Fk(\000)f Fm(x)i Fk(\000)f Fm(";)30 b(u)40 b Fk(\000)f Fm(x)i Fo(+)f Fm(")p Fo(\)\)\))5055 8058 y Fl(q)5161 8141 y Fm(d\027)12 b Fo(\()p Fm(x)p Fo(\))5604 7903 y Fg(\023)5755 8141 y Fm(du)1941 8606 y Fo(=)2269 8483 y(1)p 2269 8564 V 2272 8731 a Fm(")2410 8376 y Fg(Z)2502 8754 y Fe(R)2637 8368 y Fg(\022)2759 8376 y(Z)2851 8754 y Fe(R)2986 8606 y Fo(\()o Fm(\026)p Fo(\([)p Fm(u)40 b Fk(\000)f Fm(x)i Fk(\000)f Fm(";)30 b(u)40 b Fk(\000)f Fm(x)i Fo(+)f Fm(")p Fo(\)\)\))5055 8523 y Fl(q)5161 8606 y Fm(du)5360 8368 y Fg(\023)5512 8606 y Fm(d\027)12 b Fo(\()p Fm(x)p Fo(\))1941 9067 y(=)2269 8944 y(1)p 2269 9025 V 2272 9191 a Fm(")2410 8837 y Fg(Z)2502 9214 y Fe(R)2637 9067 y Fo(\()o Fm(\026)p Fo(\([)p Fm(u)40 b Fk(\000)f Fm(";)30 b(u)40 b Fo(+)g Fm(")p Fo(\)\)\))4053 8983 y Fl(q)4160 9067 y Fm(du)50 b Fo(=)g Fm(L)4725 9094 y Fl(\026)4818 9067 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))59 b Fm(:)1472 b Fo(\(5.5\))-201 9480 y(This)58 b(giv)-5 b(es,)57 b(together)f(with)i(Lemma)53 b(2.3)j(that)h Fm(D)3310 9414 y Fz(\006)3305 9525 y Fl(\027)9 b Fz(\003)p Fl(\026)3546 9480 y Fo(\()p Fm(q)e Fo(\))50 b Fk(\025)g Fm(D)4174 9414 y Fz(\006)4169 9525 y Fl(\027)4291 9480 y Fo(\()p Fm(q)7 b Fo(\).)79 b(Rev)-5 b(ersing)56 b(the)h(role)g(of)g Fm(\027)68 b Fo(and)57 b Fm(\026)g Fo(in)-201 9705 y(the)j(ab)5 b(o)-5 b(v)g(e)60 b(inequalities)i(giv)-5 b(es)61 b(the)f(exp)5 b(ected)59 b(result)i(for)f Fm(q)67 b Fo(larger)62 b(than)e(1.)-201 9931 y(In)k(the)f(case)h Fm(q)e Fk(2)57 b Fo(\(0)p Fm(;)30 b Fo(1\),)63 b(the)h(inequalities)h(\(5.5\))d(are)i(rev)-5 b(ersed.)90 b(Going)64 b(bac)-5 b(k)64 b(to)f(the)h(de\014nition)g(of)g (the)-201 10157 y(generalized)53 b(fractal)e(dimensions)g(of)h(a)g (measure)e(for)i Fm(q)57 b Fk(2)51 b Fo(\(0)p Fm(;)30 b Fo(1\),)52 b(leads)h(also)f(to)g(inequalities)h(\(5.1\)-\(5.2\).)81 10456 y(W)-15 b(e)58 b(turn)f(to)g(the)g(pro)5 b(of)57 b(of)g(\(5.3\))f(and)i(\(5.4\),)e(and)i(w)-5 b(e)57 b(shall)h(use)f (the)h(de\014nition)f(of)g(the)g(m)-5 b(ultifractal)-201 10682 y(dimensions)78 b(that)h(in)-5 b(v)g(olv)g(es)79 b(the)g(sum)f(o)-5 b(v)g(er)79 b(the)g(in)-5 b(terv)-10 b(als)79 b Fm(I)4166 10709 y Fl(j)4239 10682 y Fo(.)137 b(W)-15 b(e)79 b(adopt)f(the)h(follo)-5 b(wing)81 b(simpli\014ed)-201 10907 y(notations:)1120 11133 y Fm(I)1214 11037 y Ff(\()p Fl(")p Ff(\))1200 11185 y Fl(j)1448 11133 y Fo(=)50 b Fm(I)1720 11160 y Fl(j)1843 11133 y Fo(=)g([)p Fm(j)10 b(";)30 b Fo(\()p Fm(j)50 b Fo(+)40 b(1\))p Fm(")p Fo(\))p Fm(;)271 b(a)3451 11160 y Fl(j)3575 11133 y Fo(=)50 b Fm(\026)p Fo(\()p Fm(I)4028 11160 y Fl(j)4100 11133 y Fo(\))242 b(and)g Fm(b)5026 11160 y Fl(j)5150 11133 y Fo(=)50 b Fm(\027)12 b Fo(\()p Fm(I)5595 11160 y Fl(j)5667 11133 y Fo(\))60 b Fm(:)3393 11631 y Fo(27)p eop %%Page: 28 28 28 27 bop 81 1158 a Fo(W)-15 b(e)61 b(\014rst)g(treat)f(the)h(case)f Fm(q)f Fk(2)51 b Fo(\(0)p Fm(;)30 b Fo(1\).)81 b(Note)61 b(that)f(for)h(an)-5 b(y)61 b Fm(x)52 b Fk(2)f Fm(I)4629 1188 y Fl(k)4714 1158 y Fo(,)61 b(the)g(set)f Fk(\000)p Fm(x)40 b Fo(+)h Fm(I)5973 1185 y Fl(j)6107 1158 y Fo(is)61 b(included)h(in)-201 1384 y Fm(I)-121 1414 y Fl(j)8 b Fz(\000)p Fl(k)t Fz(\000)p Ff(1)360 1384 y Fk([)40 b Fm(I)601 1414 y Fl(j)8 b Fz(\000)p Fl(k)861 1384 y Fo(.)80 b(Th)-5 b(us)1285 1869 y Fm(\026)40 b Fk(\003)g Fm(\027)12 b Fo(\()p Fm(I)1819 1896 y Fl(j)1891 1869 y Fo(\))50 b(=)2204 1639 y Fg(Z)2400 1869 y Fo(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))p Fm(\027)12 b Fo(\()p Fk(\000)p Fm(x)38 b Fo(+)i Fm(I)3576 1896 y Fl(j)3649 1869 y Fo(\))50 b Fk(\024)3962 1707 y Fg(X)4043 2076 y Fl(k)4232 1869 y Fm(a)4328 1899 y Fl(k)4414 1869 y Fo(\()p Fm(b)4563 1899 y Fl(j)8 b Fz(\000)p Fl(k)t Fz(\000)p Ff(1)5043 1869 y Fo(+)40 b Fm(b)5303 1899 y Fl(j)8 b Fz(\000)p Fl(k)5563 1869 y Fo(\))p Fm(:)1111 b Fo(\(5.6\))-201 2428 y(Therefore,)60 b(since)g Fm(q)e Fk(2)50 b Fo(\(0)p Fm(;)30 b Fo(1\))1510 2811 y Fg(X)1597 3175 y Fl(j)1780 2972 y Fo(\()p Fm(\026)40 b Fk(\003)g Fm(\027)12 b Fo(\()p Fm(I)2385 2999 y Fl(j)2457 2972 y Fo(\)\))2598 2888 y Fl(q)2841 2972 y Fk(\024)3148 2811 y Fg(X)3236 3175 y Fl(j)3418 2684 y Fg( )3550 2811 y(X)3631 3180 y Fl(k)3820 2972 y Fm(a)3916 3002 y Fl(k)4001 2972 y Fo(\()p Fm(b)4150 3002 y Fl(j)c Fz(\000)p Fl(k)t Fz(\000)p Ff(1)4630 2972 y Fo(+)40 b Fm(b)4890 3002 y Fl(j)8 b Fz(\000)p Fl(k)5150 2972 y Fo(\))5221 2684 y Fg(!)5352 2730 y Fl(q)2841 3462 y Fk(\024)3148 3301 y Fg(X)3236 3665 y Fl(j)3418 3301 y Fg(X)3500 3670 y Fl(k)3688 3462 y Fm(a)3784 3374 y Fl(q)3784 3518 y(k)3870 3462 y Fo(\()p Fm(b)4019 3374 y Fl(q)4019 3518 y(j)g Fz(\000)p Fl(k)t Fz(\000)p Ff(1)4499 3462 y Fo(+)40 b Fm(b)4759 3374 y Fl(q)4759 3518 y(j)8 b Fz(\000)p Fl(k)5019 3462 y Fo(\))p Fm(:)-201 4033 y Fo(It)60 b(follo)-5 b(ws)61 b(that)858 4415 y Fg(X)946 4780 y Fl(j)1128 4577 y Fo(\()p Fm(\026)40 b Fk(\003)g Fm(\027)12 b Fo(\()p Fm(I)1733 4604 y Fl(j)1805 4577 y Fo(\)\))1947 4493 y Fl(q)2073 4577 y Fk(\024)2265 4415 y Fg(X)2346 4784 y Fl(k)2535 4577 y Fm(a)2631 4488 y Fl(q)2631 4633 y(k)2747 4415 y Fg(X)2834 4780 y Fl(j)2987 4577 y Fo(\()p Fm(b)3136 4488 y Fl(q)3136 4633 y(j)c Fz(\000)p Fl(k)t Fz(\000)p Ff(1)3616 4577 y Fo(+)40 b Fm(b)3876 4488 y Fl(q)3876 4633 y(j)8 b Fz(\000)p Fl(k)4136 4577 y Fo(\))50 b Fk(\024)g Fo(2)4570 4289 y Fg( )4702 4415 y(X)4783 4784 y Fl(k)4972 4577 y Fm(a)5068 4488 y Fl(q)5068 4633 y(k)5153 4289 y Fg(!)5315 4239 y(0)5315 4538 y(@)5460 4415 y(X)5548 4780 y Fl(j)5730 4577 y Fm(b)5808 4488 y Fl(q)5808 4628 y(j)5884 4239 y Fg(1)5884 4538 y(A)6060 4577 y Fm(:)-201 5166 y Fo(Th)-5 b(us,)60 b(this)h(inequalit)-5 b(y)61 b(together)f(with)h(Lemma)c(2.1)j(giv)-5 b(es)60 b Fm(\034)4107 5193 y Fl(\026)p Fz(\003)p Fl(\027)4349 5166 y Fo(\()p Fm(q)7 b Fo(\))49 b Fk(\024)h Fm(\034)4900 5193 y Fl(\026)4993 5166 y Fo(\()p Fm(q)7 b Fo(\))40 b(+)f Fm(\034)5524 5193 y Fl(\026)5617 5166 y Fo(\()p Fm(q)7 b Fo(\).)-201 5491 y(W)-15 b(e)62 b(no)-5 b(w)62 b(turn)g(to)g(case)f Fm(q)f(>)52 b Fo(1.)85 b(W)-15 b(e)61 b(shall)i(need)f(here)g(t)-5 b(w)g(o)62 b(grids)g(of)g(t)-5 b(w)g(o)61 b(di\013eren)-5 b(t)63 b(sizes:)82 b Fm(")62 b Fo(and)g(2)p Fm(")p Fo(.)85 b(In)-201 5739 y(order)61 b(to)f(stress)g(the)g(dep)5 b(endency)60 b(of)g Fm(I)2586 5766 y Fl(j)2719 5739 y Fo(in)h Fm(")p Fo(,)f(w)-5 b(e)61 b(shall)g(use)f(the)h(notation)f Fm(I)5269 5643 y Ff(\()p Fl(")p Ff(\))5255 5790 y Fl(j)5503 5739 y Fo(=)50 b([)p Fm(j)10 b(";)30 b Fo(\()p Fm(j)49 b Fo(+)40 b(1\))p Fm(")p Fo(\).)-201 6028 y(Note)62 b(that)g(if)g Fm(x)54 b Fo(=)e(\()p Fm(k)47 b Fo(+)41 b(1\))p Fm(")62 b Fo(then)g Fk(\000)p Fm(x)41 b Fo(+)g Fm(I)2841 5932 y Ff(\(2)p Fl(")p Ff(\))2827 6080 y Fl(j)3148 6028 y Fo(=)53 b([\(2)p Fm(j)e Fk(\000)41 b Fm(k)47 b Fk(\000)41 b Fo(1\))p Fm(";)30 b Fo(\(2)p Fm(j)50 b Fk(\000)41 b Fm(k)47 b Fo(+)41 b(1\))p Fm(")p Fo(\))62 b(and)g(if)g Fm(x)53 b Fo(=)g Fm(k)6 b(")62 b Fo(then)-201 6318 y Fk(\000)p Fm(x)40 b Fo(+)g Fm(I)361 6222 y Ff(\(2)p Fl(")p Ff(\))347 6370 y Fl(j)665 6318 y Fo(=)50 b([\(2)p Fm(j)g Fk(\000)40 b Fm(k)6 b Fo(\))p Fm(";)30 b Fo(\(2)p Fm(j)49 b Fk(\000)40 b Fm(k)45 b Fo(+)40 b(2\))p Fm(")p Fo(\).)80 b(Therefore)60 b(one)g(c)-5 b(hec)g(ks)60 b(that)1008 6790 y Fk(8)30 b Fm(x)51 b Fk(2)f Fm(I)1559 6694 y Ff(\()p Fl(")p Ff(\))1545 6846 y Fl(k)1743 6790 y Fm(;)212 b(I)2100 6694 y Ff(\()p Fl(")p Ff(\))2086 6846 y(2)p Fl(j)8 b Fz(\000)p Fl(k)2466 6790 y Fo(=)50 b([\(2)p Fm(j)g Fk(\000)40 b Fm(k)6 b Fo(\))p Fm(";)30 b Fo(\(2)p Fm(j)49 b Fk(\000)40 b Fm(k)45 b Fo(+)40 b(1\))p Fm(")p Fo(\))50 b Fk(\032)4866 6602 y Fg(\020)4965 6790 y Fk(\000)p Fm(x)40 b Fo(+)g Fm(I)5527 6694 y Ff(\(2)p Fl(")p Ff(\))5513 6842 y Fl(j)5781 6602 y Fg(\021)5910 6790 y Fm(:)-201 7220 y Fo(It)60 b(follo)-5 b(ws)61 b(that)1163 7649 y Fm(\026)40 b Fk(\003)h Fm(\027)1576 7461 y Fg(\020)1675 7649 y Fm(I)1769 7553 y Ff(\(2)p Fl(")p Ff(\))1755 7701 y Fl(j)2023 7461 y Fg(\021)2173 7649 y Fo(=)2365 7487 y Fg(X)2446 7856 y Fl(k)2635 7419 y Fg(Z)2727 7796 y Fl(I)2798 7726 y Fa(\()p Fd(")p Fa(\))2788 7846 y Fd(k)3000 7649 y Fo(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))50 b Fm(\027)3638 7461 y Fg(\020)3738 7649 y Fk(\000)p Fm(x)39 b Fo(+)h Fm(I)4299 7553 y Ff(\(2)p Fl(")p Ff(\))4285 7701 y Fl(j)4553 7461 y Fg(\021)4703 7649 y Fk(\025)4895 7487 y Fg(X)4976 7856 y Fl(k)5165 7649 y Fm(a)5261 7679 y Fl(k)5346 7649 y Fm(b)5424 7679 y Ff(2)p Fl(j)8 b Fz(\000)p Fl(k)5755 7649 y Fm(:)-201 8268 y Fo(Where,)62 b(again,)h Fm(a)1067 8295 y Fl(j)1194 8268 y Fo(=)53 b Fm(\026)1529 8079 y Fg(\020)1628 8268 y Fm(I)1722 8172 y Ff(\()p Fl(")p Ff(\))1708 8319 y Fl(j)1906 8079 y Fg(\021)2067 8268 y Fo(and)63 b Fm(b)2501 8295 y Fl(j)2628 8268 y Fo(=)53 b Fm(\027)2954 8079 y Fg(\020)3054 8268 y Fm(I)3148 8172 y Ff(\()p Fl(")p Ff(\))3134 8319 y Fl(j)3331 8079 y Fg(\021)3493 8268 y Fo(\(to)62 b(alleviate)h(the)f (notations)g(w)-5 b(e)62 b(drop)h(the)f(dep)5 b(en-)-201 8524 y(dency)60 b(of)g Fm(a)622 8551 y Fl(j)756 8524 y Fo(and)h Fm(b)1188 8551 y Fl(j)1321 8524 y Fo(on)g(the)f(size)h Fm(")f Fo(of)g(the)g(grid\).)81 b(Then,)60 b(since)h Fm(q)c(>)50 b Fo(1,)1328 8914 y Fg(X)1416 9279 y Fl(j)1598 9076 y Fm(\026)40 b Fk(\003)h Fm(\027)2011 8888 y Fg(\020)2110 9076 y Fm(I)2204 8980 y Ff(\(2)p Fl(")p Ff(\))2190 9128 y Fl(j)2458 8888 y Fg(\021)2558 8933 y Fl(q)2800 9076 y Fk(\025)3107 8914 y Fg(X)3195 9279 y Fl(j)3377 8788 y Fg( )3509 8914 y(X)3590 9283 y Fl(k)3779 9076 y Fm(a)3875 9106 y Fl(k)3960 9076 y Fm(b)4038 9106 y Ff(2)p Fl(j)8 b Fz(\000)p Fl(k)4369 8788 y Fg(!)4500 8834 y Fl(q)2800 9566 y Fk(\025)3107 9404 y Fg(X)3195 9769 y Fl(j)3377 9404 y Fg(X)3459 9773 y Fl(k)3648 9566 y Fm(a)3744 9478 y Fl(q)3744 9622 y(k)3829 9566 y Fm(b)3907 9478 y Fl(q)3907 9622 y Ff(2)p Fl(j)g Fz(\000)p Fl(k)4288 9566 y Fo(=)4480 9404 y Fg(X)4561 9773 y Fl(k)4750 9566 y Fm(a)4846 9478 y Fl(q)4846 9622 y(k)4962 9404 y Fg(X)5049 9769 y Fl(j)5232 9566 y Fm(b)5310 9478 y Fl(q)5310 9622 y Ff(2)p Fl(j)g Fz(\000)p Fl(k)-201 10286 y Fo(The)61 b(latter)f(estimate)f(can)h(b)5 b(e)60 b(written)1092 10533 y Fg(X)1180 10898 y Fl(j)1362 10695 y Fm(\026)40 b Fk(\003)g Fm(\027)1775 10506 y Fg(\020)1874 10695 y Fm(I)1968 10599 y Ff(\(2)p Fl(")p Ff(\))1954 10746 y Fl(j)2222 10506 y Fg(\021)2321 10552 y Fl(q)2448 10695 y Fk(\025)2640 10533 y Fg(X)2721 10902 y Fl(k)2910 10695 y Fm(a)3006 10606 y Fl(q)3006 10751 y Ff(2)p Fl(k)3192 10533 y Fg(X)3280 10898 y Fl(j)3463 10695 y Fm(b)3541 10606 y Fl(q)3541 10751 y Ff(2)p Fl(j)8 b Fz(\000)p Ff(2)p Fl(k)3982 10695 y Fo(+)4164 10533 y Fg(X)4245 10902 y Fl(k)4434 10695 y Fm(a)4530 10606 y Fl(q)4530 10751 y Ff(2)p Fl(k)t Ff(+1)4897 10533 y Fg(X)4984 10898 y Fl(j)5167 10695 y Fm(b)5245 10606 y Fl(q)5245 10751 y Ff(2)p Fl(j)g Fz(\000)p Ff(2)p Fl(k)t Fz(\000)p Ff(1)5826 10695 y Fm(:)919 b Fo(\(5.7\))3393 11631 y(28)p eop %%Page: 29 29 29 28 bop -201 1158 a Fo(Without)61 b(loss)f(of)g(generalit)-5 b(y)-15 b(,)60 b(assume)f(that)2981 1030 y Fg(P)3156 1204 y Fl(j)3259 1158 y Fm(b)3337 1070 y Fl(q)3337 1210 y Ff(2)p Fl(j)3531 1158 y Fk(\025)3723 1030 y Fg(P)3898 1204 y Fl(j)4002 1158 y Fm(b)4080 1070 y Fl(q)4080 1210 y Ff(2)p Fl(j)8 b Ff(+1)4404 1158 y Fo(,)60 b(that)g(is)5092 1030 y Fg(P)5267 1204 y Fl(j)5370 1158 y Fm(b)5448 1070 y Fl(q)5448 1210 y Ff(2)p Fl(j)5642 1158 y Fk(\025)5854 1087 y Ff(1)p 5854 1116 71 7 v 5854 1221 a(2)5975 1030 y Fg(P)6150 1204 y Fl(j)6254 1158 y Fm(b)6332 1070 y Fl(q)6332 1210 y(j)6408 1158 y Fo(.)80 b(Then)1824 1596 y Fg(X)1912 1960 y Fl(j)2094 1758 y Fm(\026)40 b Fk(\003)h Fm(\027)2507 1569 y Fg(\020)2607 1758 y Fm(I)2701 1662 y Ff(\(2)p Fl(")p Ff(\))2687 1809 y Fl(j)2955 1569 y Fg(\021)3054 1615 y Fl(q)3180 1758 y Fk(\025)3392 1635 y Fo(1)p 3392 1715 91 7 v 3392 1882 a(2)3533 1470 y Fg( )3665 1596 y(X)3746 1965 y Fl(k)3935 1758 y Fm(a)4031 1669 y Fl(q)4031 1814 y Ff(2)p Fl(k)4187 1470 y Fg(!)4349 1420 y(0)4349 1719 y(@)4494 1596 y(X)4581 1960 y Fl(j)4764 1758 y Fm(b)4842 1669 y Fl(q)4842 1809 y(j)4918 1420 y Fg(1)4918 1719 y(A)5094 1758 y Fm(:)-201 2334 y Fo(If)60 b(again)466 2205 y Fg(P)641 2380 y Fl(k)757 2334 y Fm(a)853 2245 y Fl(q)853 2390 y Ff(2)p Fl(k)1059 2334 y Fk(\025)1251 2205 y Fg(P)1426 2380 y Fl(k)1542 2334 y Fm(a)1638 2245 y Fl(q)1638 2390 y Ff(2)p Fl(k)t Ff(+1)2035 2334 y Fo(then)g(w)-5 b(e)61 b(are)f(done)h(since)1319 2925 y Fm(S)1431 2952 y Fl(\026)p Fz(\003)p Fl(\027)1672 2925 y Fo(\()p Fm(q)7 b(;)30 b Fo(2)p Fm(")p Fo(\))49 b(=)2400 2764 y Fg(X)2488 3128 y Fl(j)2670 2925 y Fm(\026)40 b Fk(\003)h Fm(\027)3083 2737 y Fg(\020)3182 2925 y Fm(I)3276 2830 y Ff(\(2)p Fl(")p Ff(\))3262 2977 y Fl(j)3530 2737 y Fg(\021)3630 2783 y Fl(q)3756 2925 y Fk(\025)3968 2802 y Fo(1)p 3968 2883 V 3968 3050 a(4)4109 2638 y Fg( )4241 2764 y(X)4322 3133 y Fl(k)4511 2925 y Fm(a)4607 2837 y Fl(q)4607 2982 y(k)4692 2638 y Fg(!)4854 2588 y(0)4854 2887 y(@)4999 2764 y(X)5087 3128 y Fl(j)5269 2925 y Fm(b)5347 2837 y Fl(q)5347 2977 y(j)5424 2588 y Fg(1)5424 2887 y(A)5599 2925 y Fm(:)1146 b Fo(\(5.8\))-201 3551 y(If)44 b(not,)j(that)e(is)g (if)1023 3423 y Fg(P)1198 3597 y Fl(k)1314 3551 y Fm(a)1410 3463 y Fl(q)1410 3608 y Ff(2)p Fl(k)t Ff(+1)1796 3551 y Fk(\025)1988 3423 y Fg(P)2164 3597 y Fl(k)2279 3551 y Fm(a)2375 3463 y Fl(q)2375 3608 y Ff(2)p Fl(k)2531 3551 y Fo(,)j(one)c(then)h(w)-5 b(orks)45 b(with)g(the)g(family)e(of)i (in)-5 b(terv)-10 b(als)6009 3506 y(~)5987 3551 y Fm(I)6081 3456 y Ff(\(2)p Fl(")p Ff(\))6067 3603 y Fl(j)6385 3551 y Fo(=)50 b Fm(")9 b Fo(+)g Fm(I)6916 3456 y Ff(\(2)p Fl(")p Ff(\))6902 3603 y Fl(j)-201 3841 y Fo(instead)60 b(of)h(the)f Fm(I)1041 3745 y Ff(\(2)p Fl(")p Ff(\))1027 3893 y Fl(j)1295 3841 y Fo('s.)79 b(Instead)60 b(of)g(\(5.7\),)f(this)h (leads)h(to)1092 4134 y Fg(X)1180 4499 y Fl(j)1362 4296 y Fm(\026)40 b Fk(\003)g Fm(\027)1775 4108 y Fg(\020)1896 4250 y Fo(~)1874 4296 y Fm(I)1968 4200 y Ff(\(2)p Fl(")p Ff(\))1954 4348 y Fl(j)2222 4108 y Fg(\021)2321 4153 y Fl(q)2448 4296 y Fk(\025)2640 4134 y Fg(X)2721 4503 y Fl(k)2910 4296 y Fm(a)3006 4208 y Fl(q)3006 4352 y Ff(2)p Fl(k)3192 4134 y Fg(X)3280 4499 y Fl(j)3463 4296 y Fm(b)3541 4208 y Fl(q)3541 4352 y Ff(2)p Fl(j)8 b Fz(\000)p Ff(2)p Fl(k)t Ff(+1)4162 4296 y Fo(+)4344 4134 y Fg(X)4425 4503 y Fl(k)4614 4296 y Fm(a)4710 4208 y Fl(q)4710 4352 y Ff(2)p Fl(k)t Ff(+1)5077 4134 y Fg(X)5164 4499 y Fl(j)5347 4296 y Fm(b)5425 4208 y Fl(q)5425 4352 y Ff(2)p Fl(j)g Fz(\000)p Ff(2)p Fl(k)5826 4296 y Fm(:)-201 4845 y Fo(And)61 b(th)-5 b(us)1079 5044 y Fg(X)1167 5409 y Fl(j)1349 5206 y Fm(\026)40 b Fk(\003)h Fm(\027)1762 5017 y Fg(\020)1883 5160 y Fo(~)1862 5206 y Fm(I)1956 5110 y Ff(\(2)p Fl(")p Ff(\))1942 5257 y Fl(j)2210 5017 y Fg(\021)2309 5063 y Fl(q)2435 5206 y Fk(\025)2647 5083 y Fo(1)p 2647 5164 V 2647 5330 a(2)2788 5044 y Fg(X)2870 5413 y Fl(k)3058 5206 y Fm(a)3154 5117 y Fl(q)3154 5262 y Ff(2)p Fl(k)t Ff(+1)3521 5044 y Fg(X)3609 5409 y Fl(j)3791 5206 y Fm(b)3869 5117 y Fl(q)3869 5257 y(j)3996 5206 y Fk(\025)4208 5083 y Fo(1)p 4208 5164 V 4208 5330 a(4)4349 4918 y Fg( )4480 5044 y(X)4562 5413 y Fl(k)4750 5206 y Fm(a)4846 5117 y Fl(q)4846 5262 y(k)4932 4918 y Fg(!)5094 4868 y(0)5094 5167 y(@)5239 5044 y(X)5326 5409 y Fl(j)5509 5206 y Fm(b)5587 5117 y Fl(q)5587 5257 y(j)5663 4868 y Fg(1)5663 5167 y(A)5839 5206 y Fm(:)906 b Fo(\(5.9\))-201 5773 y(But)42 b(b)-5 b(y)43 b(Theorem)e(2.2,)k Fm(S)1568 5800 y Fl(\026)p Fz(\003)p Fl(\027)1809 5773 y Fo(\()p Fm(q)7 b(;)30 b Fo(2)p Fm(")p Fo(\))49 b Fk(\025)h Fo(2)2628 5707 y Fz(\000)p Ff(\()p Fl(q)5 b Ff(+1\))3134 5644 y Fg(P)3310 5819 y Fl(j)3413 5773 y Fm(\026)t Fk(\003)t Fm(\027)3754 5584 y Fg(\020)3875 5727 y Fo(~)3853 5773 y Fm(I)3947 5677 y Ff(\(2)p Fl(")p Ff(\))3933 5824 y Fl(j)4201 5584 y Fg(\021)4300 5630 y Fl(q)4376 5773 y Fo(.)74 b(In)42 b(consequence)g(the)g(estimates)e(\(5.8\))-201 6030 y(and)61 b(\(5.9\))f(leads)g(to)h(\(5.3\))f(and)g(\(5.4\))g(of)g(the)h(Prop)5 b(osition.)81 b(This)61 b(ends)g(the)f(pro)5 b(of)61 b(of)f(the)h(Prop)5 b(osition.)-201 6255 y Fb(\003)-201 6825 y Fp(6)270 b(Examples)-201 7231 y Fc(Example)70 b(1.)-201 7457 y Fo(F)-15 b(or)70 b Fm(\025)65 b(>)h Fo(1)j(and)h Fm(\013)d(>)e Fo(0,)71 b(let)f Fm(a)2008 7484 y Fl(n)2168 7457 y Fo(=)65 b Fm(a=n)2671 7391 y Fl(\025)2762 7457 y Fo(,)71 b Fm(x)2988 7484 y Fl(n)3148 7457 y Fo(=)65 b(1)p Fm(=n)3646 7391 y Fl(\013)3745 7457 y Fo(,)72 b(where)e Fm(a)65 b(>)h Fo(0)j(is)h(a)g(normalization)f (constan)-5 b(t.)-201 7682 y(W)-15 b(e)62 b(de\014ne)f(the)g(pure)h(p)5 b(oin)-5 b(t)62 b(probabilit)-5 b(y)63 b(measure)d Fm(\026)52 b Fo(=)3843 7554 y Fg(P)4019 7606 y Fz(1)4019 7728 y Fl(n)p Ff(=1)4323 7682 y Fm(a)4419 7709 y Fl(n)4513 7682 y Fm(\016)4594 7709 y Fl(x)4673 7726 y Fd(n)4829 7682 y Fo(on)61 b Fj(R)27 b Fo(.)93 b(This)62 b(measure)e(has)i(b)5 b(een)-201 7908 y(sho)-5 b(w)g(ed)50 b(in)g([3)o(])f(to)g(ha)-5 b(v)g(e)50 b(non)f(trivial)i(dimensions,)f(whic)-5 b(h)50 b(w)-5 b(as)50 b(of)f(particular)i(in)-5 b(terest)49 b(in)h(the)f(setting)g([3)o(].)-201 8134 y(In)44 b(the)g(con)-5 b(text)42 b(of)i(this)g(pap)5 b(er,)47 b(this)d(example)e(is)j (particularly)g(in)-5 b(teresting)44 b(to)g(emphasize)e(the)h (di\013erence)-201 8360 y(b)5 b(et)-5 b(w)g(een)65 b(the)f(regime)g Fm(q)h(>)57 b Fo(1)66 b(and)f Fm(q)f(<)58 b Fo(1.)94 b(W)-15 b(e)64 b(pro)-5 b(vide)66 b(here)f(the)f(explicit)i(v)-10 b(alues)65 b(of)g(the)f Fm(D)6417 8294 y Fz(\006)6412 8405 y Fl(\026)6535 8360 y Fo(\()p Fm(q)7 b Fo(\)'s)64 b(for)-201 8586 y(that)i(atomic)e(measure.)95 b(W)-15 b(e)65 b(shall)i(not)f(supply)g(all)h(the)e(tec)-5 b(hnicals)66 b(details)g(but)g(restrict)g(ourselv)-5 b(es)66 b(to)-201 8811 y(pro)-5 b(vide)55 b(the)e(reader)h(the)g(main)f(signi\014can)-5 b(t)54 b(steps)f(of)g(the)h(calculations.)78 b(First,)55 b(as)e(the)h(measure)e(is)i(pure)-201 9037 y(p)5 b(oin)-5 b(t,)60 b Fm(D)480 8971 y Fz(\006)475 9082 y Fl(\026)598 9037 y Fo(\()p Fm(q)7 b Fo(\))50 b(=)g(0)60 b(for)g(an)-5 b(y)61 b Fm(q)c(>)50 b Fo(1.)80 b(One)60 b(can)g(also)h(note)f(that)g (supp)g Fm(\026)50 b Fo(=)g Fk(f)p Fo(0)p Fm(;)30 b(x)5429 9064 y Ff(1)5508 9037 y Fm(;)g(x)5693 9064 y Ff(2)5771 9037 y Fm(;)g(:::)p Fk(g)p Fo(.)-201 9263 y(When)60 b(analyzing,)f(for) h(a)f(\014xed)h Fm(")p Fo(,)f(the)g(b)5 b(eha)-5 b(viour)60 b(in)g Fm(x)3723 9290 y Fl(n)3877 9263 y Fo(of)f Fm(\026)p Fo(\()p Fm(x)4368 9290 y Fl(n)4500 9263 y Fk(\000)38 b Fm(";)30 b(x)4950 9290 y Fl(n)5081 9263 y Fo(+)38 b Fm(")p Fo(\),)59 b(three)g(regimes)g(clearly)-201 9489 y(app)5 b(ear.)80 b(One)60 b(shall)i(compute)1817 9875 y Fm(I)1897 9902 y Fl(\026)1990 9875 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))49 b(=)2640 9714 y Fg(X)2627 10079 y Fl(n)p Fz(\025)p Ff(1)2923 9875 y Fm(\026)p Fo(\()p Fk(f)p Fm(x)3299 9902 y Fl(n)3392 9875 y Fk(g)p Fo(\))p Fm(\026)3664 9800 y Fl(q)5 b Fz(\000)p Ff(1)3920 9875 y Fo(\()p Fm(x)4095 9902 y Fl(n)4229 9875 y Fk(\000)40 b Fm(";)30 b(x)4681 9902 y Fl(n)4814 9875 y Fo(+)40 b Fm(")p Fo(\))-201 10422 y(for)60 b Fm(q)e(<)50 b Fo(1)60 b(b)-5 b(y)61 b(cutting)f(it)h(in)g(three)f(corresp)5 b(onding)61 b(pieces)f Fm(I)4072 10449 y Ff(1)4151 10422 y Fm(;)30 b(I)4312 10449 y Ff(2)4390 10422 y Fm(;)g(I)4551 10449 y Ff(3)4630 10422 y Fo(.)-201 10648 y Fn(R)-9 b(e)g(gime)63 b(1)p Fo(:)730 10576 y Ff(1)p 678 10605 175 7 v 678 10710 a Fl(n)764 10672 y Fd(\013)922 10648 y Fm(<)50 b(")p Fo(,)60 b(or)h(equiv)-10 b(alen)-5 b(tly)61 b Fm(n)51 b(>)f Fo(\()2986 10576 y Ff(1)p 2986 10605 71 7 v 2989 10710 a Fl(")3076 10648 y Fo(\))3177 10522 y Fa(1)p 3166 10546 81 7 v 3166 10628 a Fd(\013)3275 10648 y Fo(.)-201 10873 y(Then)71 b Fm(x)388 10900 y Fl(n)553 10873 y Fo(is)g(close)g (enough)g(to)f(0)h(so)g(that)f Fm(\026)p Fo(\()p Fm(x)3146 10900 y Fl(n)3287 10873 y Fk(\000)47 b Fm(";)30 b(x)3746 10900 y Fl(n)3886 10873 y Fo(+)47 b Fm(")p Fo(\))67 b(=)h Fm(\026)p Fo(\(0)p Fm(;)30 b(x)4965 10900 y Fl(n)5105 10873 y Fo(+)47 b Fm(")p Fo(\),)72 b(and)f(hence)g Fm(\026)p Fo(\(0)p Fm(;)30 b(")p Fo(\))66 b Fk(\024)-201 11133 y Fm(\026)p Fo(\()p Fm(x)84 11160 y Fl(n)220 11133 y Fk(\000)41 b Fm(";)30 b(x)673 11160 y Fl(n)809 11133 y Fo(+)42 b Fm(")p Fo(\))55 b Fk(\024)g Fm(\026)p Fo(\(0)p Fm(;)30 b Fo(2)p Fm(")p Fo(\).)87 b(It)63 b(th)-5 b(us)63 b(comes)f(out)h(that)g Fm(\026)p Fo(\()p Fm(x)4290 11160 y Fl(n)4425 11133 y Fk(\000)42 b Fm(";)30 b(x)4879 11160 y Fl(n)5015 11133 y Fo(+)42 b Fm(")p Fo(\))55 b Fk(\030)f Fm(")5711 11005 y Fd(\025)p Fi(\000)p Fa(1)p 5711 11032 231 7 v 5786 11114 a Fd(\013)5970 11133 y Fo(.)89 b(In)63 b(particular,)3393 11631 y(29)p eop %%Page: 30 30 30 29 bop -201 1158 a Fm(\015)-97 1092 y Fz(\006)-107 1203 y Fl(\026)22 1158 y Fo(\(0\))57 b(=)531 1087 y Fl(\025)p Fz(\000)p Ff(1)p 531 1116 263 7 v 617 1221 a Fl(\013)878 1158 y Fo(\(and)65 b Fm(\015)1411 1092 y Fz(\006)1401 1203 y Fl(\026)1529 1158 y Fo(\()p Fm(x)p Fo(\))57 b(=)h(0)64 b(for)h(other)g(p)5 b(oin)-5 b(ts)65 b Fm(x)57 b Fo(=)g Fm(x)3965 1185 y Fl(n)4117 1158 y Fk(2)h Fo(supp)65 b Fm(\026)p Fo(\).)93 b(The)65 b(corresp)5 b(onding)65 b(part)f Fm(I)7090 1185 y Ff(1)-201 1460 y Fo(of)c Fm(I)86 1487 y Fl(\026)179 1460 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))60 b(is)h(equiv)-10 b(alen)-5 b(t)60 b(to)g Fm(")2008 1321 y Fd(q)t Fa(\()p Fd(\025)p Fi(\000)p Fa(1\))p 2008 1359 387 7 v 2161 1441 a Fd(\013)2423 1460 y Fo(.)-201 1734 y Fn(R)-9 b(e)g(gime)63 b(2)p Fo(:)80 b(\()749 1662 y Fl(\013)p 749 1691 91 7 v 762 1797 a(")859 1734 y Fo(\))1036 1594 y Fa(1)p 950 1618 235 7 v 950 1701 a Fd(\013)p Fa(+1)1263 1734 y Fk(\024)50 b Fm(n)g Fk(\024)g Fo(\()1897 1662 y Ff(1)p 1897 1691 71 7 v 1900 1797 a Fl(")1987 1734 y Fo(\))2088 1608 y Fa(1)p 2078 1632 81 7 v 2078 1714 a Fd(\013)2186 1734 y Fo(.)-201 1960 y(Then)k Fm(x)371 1987 y Fl(n)492 1960 y Fk(\000)27 b Fm(")50 b(>)g Fo(0)k(and)f Fm(x)1583 1987 y Fl(n)1704 1960 y Fk(\000)27 b Fm(x)1977 1987 y Fl(n)p Ff(+1)2301 1960 y Fm(<)50 b(")p Fo(.)78 b(That)54 b(means)e(that)h(\()p Fm(x)4267 1987 y Fl(n)4388 1960 y Fk(\000)27 b Fm(";)j(x)4827 1987 y Fl(n)4946 1960 y Fo(+)d Fm(")p Fo(\))53 b(con)-5 b(tains)54 b(more)e(than)i(one)-201 2186 y(p)5 b(oin)-5 b(t)54 b(of)g(the)f(supp)5 b(ort)54 b(but)g(do)5 b(es)53 b(not)h(reac)-5 b(h)53 b(the)h(edge)f(0.)78 b(One)53 b(then)h(gets)f Fm(\026)p Fo(\()p Fm(x)5333 2213 y Fl(n)5453 2186 y Fk(\000)27 b Fm(";)j(x)5892 2213 y Fl(n)6012 2186 y Fo(+)d Fm(")p Fo(\))50 b Fk(\030)6816 2114 y Fl(")p 6599 2143 501 7 v 6599 2257 a(n)6685 2219 y Fd(\025)p Fi(\000)p Fa(1)p Fi(\000)p Fd(\013)7119 2186 y Fo(.)-201 2411 y(If)60 b Fm(\025)51 b(<)f Fo(1)40 b(+)g Fm(\013)q Fo(,)60 b(then)g(for)g(an)-5 b(y)61 b Fm(q)c(<)50 b Fo(1)60 b(w)-5 b(e)61 b(ha)-5 b(v)g(e)3047 2863 y Fm(I)3127 2890 y Ff(2)3256 2863 y Fk(\030)50 b Fm(")3553 2715 y Fd(q)t(\025)p Fi(\000)p Fa(1)p 3553 2748 291 7 v 3581 2830 a(1+)p Fd(\013)3871 2863 y Fm(:)2874 b Fo(\(6.1\))-201 3272 y(If)60 b Fm(\025)51 b(>)f Fo(1)40 b(+)g Fm(\013)q Fo(,)60 b(then)g(\(6.1\))f(holds)i(for)f Fm(q)d(>)50 b(q)2884 3299 y Ff(0)3014 3272 y Fo(:=)f Fk(\000)3604 3200 y Fl(\013)p 3418 3229 464 7 v 3418 3334 a(\025)p Fz(\000)p Ff(1)p Fz(\000)p Fl(\013)3901 3272 y Fo(,)60 b(and)g(for)h Fm(q)c(<)50 b(q)5055 3299 y Ff(0)5134 3272 y Fo(,)2798 3765 y Fm(I)2878 3792 y Ff(2)3007 3765 y Fk(\030)g Fm(")3304 3622 y Fd(q)t Fa(\()p Fd(\025)p Fi(\000)p Fa(1\))p 3304 3660 387 7 v 3457 3742 a Fd(\013)3769 3765 y Fk(\030)g Fm(I)4041 3792 y Ff(1)4120 3765 y Fm(:)-201 4196 y Fn(R)-9 b(e)g(gime)63 b(3)p Fo(:)80 b Fm(n)51 b(<)e Fo(\()1100 4124 y Fl(\013)p 1100 4153 91 7 v 1113 4259 a(")1211 4196 y Fo(\))1388 4056 y Fa(1)p 1301 4080 235 7 v 1301 4162 a Fd(\013)p Fa(+1)1564 4196 y Fo(.)-201 4422 y(In)60 b(this)h(regime,)e(\()p Fm(x)1193 4449 y Fl(n)1327 4422 y Fk(\000)40 b Fm(";)30 b(x)1779 4449 y Fl(n)1913 4422 y Fo(+)39 b Fm(")p Fo(\))h Fk(\\)h Fo(supp)p Fm(\026)50 b Fo(=)g Fk(f)p Fm(x)3374 4449 y Fl(n)3467 4422 y Fk(g)p Fo(,)60 b(th)-5 b(us)60 b Fm(\026)p Fo(\()p Fm(x)4354 4449 y Fl(n)4488 4422 y Fk(\000)40 b Fm(";)30 b(x)4940 4449 y Fl(n)5074 4422 y Fo(+)39 b Fm(")p Fo(\))50 b(=)5725 4350 y Ff(1)p 5673 4379 175 7 v 5673 4484 a Fl(n)5759 4447 y Fd(\013)5867 4422 y Fo(.)81 b(This)61 b(yields)2641 4830 y Fm(I)2721 4857 y Ff(3)2850 4830 y Fk(\030)50 b Fo(const)o Fm(;)151 b(q)57 b Fk(\025)50 b Fo(1)p Fm(=\025;)-201 5239 y Fo(and)2637 5464 y Fm(I)2717 5491 y Ff(3)2846 5464 y Fk(\030)g Fm(")3143 5316 y Fd(q)t(\025)p Fi(\000)p Fa(1)p 3143 5349 291 7 v 3171 5431 a(1+)p Fd(\013)3461 5464 y Fm(;)151 b(q)58 b(<)49 b Fo(1)p Fm(=\025:)-201 5798 y Fo(F)-15 b(or)59 b(an)-5 b(y)59 b Fm(q)e(<)50 b Fo(1)59 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Fl(\026)961 10545 y Fo(\(0\).)-201 10871 y Fc(Example)c(2.)74 b Fo(With)41 b(the)e(same)g(notations)h(as)g(in)h(the)f(previous)g (example,)j(let)d(us)g(tak)-5 b(e)40 b Fm(a)5896 10898 y Fl(n)6040 10871 y Fo(=)6451 10799 y Fl(a)p 6252 10828 473 7 v 6252 10934 a(n)23 b Ff(log)6544 10882 y Fd(p)6639 10934 y Fl(n)6745 10871 y Fm(;)70 b(x)6970 10898 y Fl(n)7115 10871 y Fo(=)-70 11062 y Ff(1)p -181 11091 292 7 v -181 11196 a(log)24 b Fl(n)131 11133 y Fo(,)65 b(where)h Fm(n)58 b Fk(\025)f Fo(2)p Fm(;)95 b(p)58 b(>)f Fo(1.)94 b(One)65 b(can)f(do)i(the)e(calculations)i(similar)f(to)f(those)h(of)f(example)g (1.)94 b(First,)3393 11631 y(30)p eop %%Page: 31 31 31 30 bop -201 1158 a Fo(one)79 b(sho)-5 b(ws)78 b(that)g Fm(\015)1207 1092 y Fz(\006)1197 1203 y Fl(\026)1325 1158 y Fo(\(0\))j(=)f Fm(p)52 b Fk(\000)g Fo(1)78 b(\(and)h(at)f(the)g (same)f(time)g Fm(\026)52 b Fk(\000)g Fo(ess)p Fm(:)p Fo(sup)78 b Fm(\015)5376 1092 y Fz(\006)5366 1203 y Fl(\026)5494 1158 y Fo(\()p Fm(x)p Fo(\))i(=)g(0\).)135 b(As)78 b(to)h(the)-201 1384 y(dimensions)60 b Fm(D)887 1318 y Fz(\006)882 1429 y Fl(\026)1004 1384 y Fo(\()p Fm(q)7 b Fo(\),)59 b(the)i(result)f(is)h (the)f(follo)-5 b(wing.)81 1610 y(1\))61 b(The)f(case)g(1)51 b Fm(<)f(p)g Fk(\024)g Fo(2.)1752 2010 y Fm(D)1908 1935 y Fz(\006)1903 2055 y Fl(\026)2026 2010 y Fo(\()p Fm(q)7 b Fo(\))50 b(=)g(0)p Fm(;)151 b(q)57 b(>)50 b Fo(1)p Fm(;)30 b Fo(and)120 b Fm(D)3862 1935 y Fz(\006)3857 2055 y Fl(\026)3980 2010 y Fo(\()p Fm(q)7 b Fo(\))50 b(=)g(1)p Fm(;)151 b(q)57 b(<)50 b Fo(1)p Fm(:)81 2410 y Fo(2\))61 b(The)f(case)g Fm(p)50 b(>)g Fo(2.)683 2905 y Fm(D)839 2830 y Fz(\006)834 2950 y Fl(\026)956 2905 y Fo(\()p Fm(q)7 b Fo(\))50 b(=)g(0)p Fm(;)90 b(q)57 b(>)50 b Fo(1;)91 b Fm(D)2379 2830 y Fz(\006)2374 2950 y Fl(\026)2496 2905 y Fo(\()p Fm(q)7 b Fo(\))50 b(=)g(1)p Fm(;)90 b(q)57 b Fk(2)51 b Fo(\()p Fm(q)3662 2932 y Ff(0)3741 2905 y Fm(;)30 b Fo(1\);)90 b Fm(D)4281 2830 y Fz(\006)4276 2950 y Fl(\026)4398 2905 y Fo(\()p Fm(q)7 b Fo(\))50 b(=)4890 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y(in)-5 b(terv)-10 b(al)62 b(\(3)p Fm(;)e Fo(+)p Fk(1)p Fo(\).)81 b(F)-15 b(urthermore,)59 b(w)-5 b(e)61 b(assume)e(that)i(\()p Fm(a;)30 b(b)p Fo(\))51 b Fk(2)h Fo(\(\(1)p Fm(;)59 b Fo(+)p Fk(1)p Fo(\))40 b Fk(\002)g Fj(R)27 b Fo(\))49 b Fk([)41 b Fo(\()p Fk(f)p Fo(1)p Fk(g)g(\002)f Fo(\(1)p Fm(;)60 b Fo(+)p Fk(1)p Fo(\)\))g(so)-201 10358 y(that)66 b Fm(\026)f Fo(is)i(a)f(\014nite)g (measure.)95 b(Then)66 b(for)g(all)h Fm(q)f Fk(2)60 b Fj(R)101 b Fo(there)66 b(exists)f Fm(C)4641 10385 y Ff(1)4720 10358 y Fo(\()p Fm(q)7 b Fo(\))65 b(and)i Fm(C)5505 10385 y Ff(2)5583 10358 y Fo(\()p Fm(q)7 b Fo(\))66 b(suc)-5 b(h)65 b(that)h(for)f(all)-201 10584 y Fm(")50 b Fk(2)h Fo(\(0)p Fm(;)30 b Fo(1\),)371 11077 y Fm(C)501 11104 y Ff(1)580 11077 y Fm(")665 11002 y Fl(q)5 b Fz(\000)p Ff(1)951 10847 y Fg(Z)1117 10900 y Ff(+)p Fz(1)1044 11225 y Ff(3)1407 10889 y Fg(\020)1506 11077 y Fm(x)1610 11002 y Fz(\000)p Fl(a)1803 11077 y Fo(\(log)33 b Fm(x)p Fo(\))2314 11002 y Fz(\000)p Fl(b)2492 10889 y Fg(\021)2591 10935 y Fl(q)2697 11077 y Fo(d)p Fm(x)51 b Fk(\024)f Fm(I)3225 11104 y Fl(\026)3318 11077 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))49 b Fk(\024)h Fm(C)4085 11104 y Ff(2)4164 11077 y Fm(")4249 11002 y Fl(q)5 b Fz(\000)p Ff(1)4535 10847 y Fg(Z)4701 10900 y Ff(+)p Fz(1)4628 11225 y Ff(3)4991 10889 y Fg(\020)5090 11077 y Fm(x)5194 11002 y Fz(\000)p Fl(a)5387 11077 y Fo(\(log)33 b Fm(x)p Fo(\))5898 11002 y Fz(\000)p Fl(b)6076 10889 y Fg(\021)6175 10935 y Fl(q)6282 11077 y Fo(d)p Fm(x)60 b(:)3393 11631 y Fo(31)p eop %%Page: 32 32 32 31 bop -201 1158 a Fo(This)61 b(giv)-5 b(es)61 b Fm(q)748 1092 y Fz(\003)741 1203 y Fl(\026)885 1158 y Fo(=)1099 1087 y Ff(1)p 1097 1116 75 7 v 1097 1221 a Fl(a)1252 1158 y Fo(and)1416 1915 y Fm(D)1572 1840 y Fz(\006)1567 1960 y Fl(\026)1690 1915 y Fo(\()p Fm(q)7 b Fo(\))49 b(=)2161 1421 y Fg(8)2161 1570 y(>)2161 1620 y(>)2161 1670 y(>)2161 1720 y(<)2161 2019 y(>)2161 2068 y(>)2161 2118 y(>)2161 2168 y(:)2392 1549 y Fo(1)1663 b(for)61 b Fm(q)c(>)50 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Fd(n)98 3846 y Ff(\))-121 3992 y(2)-50 3954 y Fd(n)35 3992 y Fz(\000)p Ff(1)284 3942 y Fo(in)-5 b(tersects)59 b(the)i(supp)5 b(ort)60 b(of)g Fm(\026)p Fo(,)g(and)g(in)h(addition)g Fm(\026)3926 3754 y Fg(\020)4025 3942 y Fm(I)4119 3846 y Ff(\()p Fl(")4239 3863 y Fd(n)4325 3846 y Ff(\))4105 3992 y(2)4176 3954 y Fd(n)4261 3992 y Fz(\000)p Ff(1)4450 3754 y Fg(\021)4599 3942 y Fo(=)50 b(2)4882 3876 y Fz(\000)p Fl(n)5078 3829 y Fa(2)5155 3942 y Fo(.)80 b(Therefore)2276 4408 y Fm(S)10 b Fo(\()p Fm(q)d(;)30 b(")2723 4435 y Fl(n)2816 4408 y Fo(\))50 b Fk(\025)g Fm(\026)3269 4220 y Fg(\020)3368 4408 y Fm(I)3462 4312 y Ff(\()p Fl(")3582 4329 y Fd(n)3668 4312 y Ff(\))3448 4458 y(2)3519 4420 y Fd(n)3604 4458 y Fz(\000)p Ff(1)3792 4220 y Fg(\021)3892 4265 y Fl(q)4018 4408 y Fo(=)g(2)4301 4333 y Fz(\000)p Fl(q)5 b(n)4565 4286 y Fa(2)4642 4408 y Fm(:)-201 4801 y Fo(T)-15 b(aking)61 b(the)f(log,)h(dividing)h(b)-5 b(y)60 b(log)34 b Fm(")2367 4828 y Fl(n)2521 4801 y Fo(and)60 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Fp(A)269 b(App)7 b(endix)-201 6962 y Fo(Let)47 b Fm(R)h Fo(b)5 b(e)47 b(a)g(real)h(con)-5 b(tin)g(uous)46 b(function)h(suc)-5 b(h)47 b(that)f Fm(R)j Fo(deca)-5 b(ys)46 b(faster)g(than)h(an)-5 b(y)47 b(p)5 b(olynomials)47 b(at)f(in\014nit)-5 b(y)-15 b(.)-201 7188 y(W)g(e)61 b(supp)5 b(ose)60 b(also)h(that)f Fm(R)q Fo(\()p Fm(t)p Fo(\))50 b Fk(\025)g Fo(0)61 b(for)g(an)-5 b(y)60 b Fm(t)h Fo(and)g(inf)3621 7225 y Ff([)p Fz(\000)p Ff(1)p Fl(;)p Ff(1])4028 7188 y Fm(R)q Fo(\()p Fm(t)p Fo(\))50 b(=)h Fm(b)g(>)f Fo(0.)81 b(Giv)-5 b(en)61 b Fm(q)c Fk(2)p Fm(>)51 b Fo(0,)60 b Fm(q)e Fk(6)p Fo(=)50 b(1,)60 b(and)-201 7414 y Fm(")50 b Fk(2)h Fo(\(0)p Fm(;)30 b Fo(1\))60 b(w)-5 b(e)61 b(de\014ne)1803 7706 y Fm(K)1971 7631 y Ff(\()p Fl(R)q Ff(\))1958 7751 y Fl(\026)2195 7706 y Fo(\()p Fm(q)7 b(;)60 b(")p Fo(\))50 b(=)2863 7476 y Fg(Z)2955 7853 y Ff(supp)q Fl(\026)3369 7518 y Fg(\020)3469 7706 y Fm(b)3547 7631 y Ff(\()p Fl(R)q Ff(\))3772 7706 y Fo(\()p Fm(x;)30 b(")p Fo(\))4184 7518 y Fg(\021)4282 7564 y Fl(q)5 b Fz(\000)p Ff(1)4599 7706 y Fo(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))59 b Fm(;)-201 8111 y Fo(where)2005 8373 y Fm(b)2083 8297 y Ff(\()p Fl(R)q Ff(\))2308 8373 y Fo(\()p Fm(x;)30 b(")p Fo(\))50 b(=)2962 8143 y Fg(Z)3054 8520 y Ff(supp)q Fl(\026)3468 8373 y Fm(R)3638 8134 y Fg(\022)3780 8250 y Fm(x)40 b Fk(\000)g Fm(y)p 3780 8330 422 7 v 3949 8497 a(")4222 8134 y Fg(\023)4405 8373 y Fo(d)p Fm(\026)p Fo(\()p Fm(y)7 b Fo(\))59 b Fm(:)-201 8847 y Fo(The)87 b(in)-5 b(tegral)88 b Fm(K)1056 8751 y Ff(\()p Fl(R)q Ff(\))1043 8868 y Fl(\026)1280 8847 y Fo(\()p Fm(q)7 b(;)60 b(")p Fo(\))86 b(is)i(quite)f(close)f(to)h(the) f(in)-5 b(tegrals)88 b Fm(I)4378 8874 y Fl(\026)4471 8847 y Fo(\()p Fm(q)7 b(;)59 b(")p Fo(\))87 b(w)-5 b(e)87 b(studied)f(throughout)h(the)-201 9073 y(presen)-5 b(t)79 b(pap)5 b(er.)137 b(Indeed)79 b(note)g(that)g Fm(\026)p Fo(\()p Fm(x)53 b Fk(\000)f Fm(";)30 b(x)53 b Fo(+)f Fm(")p Fo(\))82 b(=)4045 8935 y Fg(R)4186 9073 y Fm(\037)4300 9110 y Ff([)p Fz(\000)p Ff(1)p Fl(;)p Ff(1])4677 9073 y Fo(\(\()p Fm(x)52 b Fk(\000)h Fm(y)7 b Fo(\))p Fm(=")p Fo(\)d)p Fm(y)g Fo(.)136 b(In)79 b(other)g(w)-5 b(ords)-201 9355 y Fm(K)-33 9259 y Ff(\()p Fl(R)q Ff(\))-46 9376 y Fl(\026)192 9355 y Fo(\()p Fm(q)7 b(;)59 b(")p Fo(\))h(is)g(a)g (generalization)h(of)f(the)f(in)-5 b(tegral)61 b Fm(I)3433 9382 y Fl(\026)3526 9355 y Fo(\()p Fm(q)7 b(;)60 b(")p Fo(\))f(where)i(the)e(c)-5 b(haracteristic)60 b(function)g Fm(\037)6793 9392 y Ff([)p Fz(\000)p Ff(1)p Fl(;)p Ff(1])-201 9658 y Fo(is)66 b(replaced)g(b)-5 b(y)65 b(a)h(function)f Fm(R)i Fo(with)f(fast)e(deca)-5 b(y)66 b(at)f Fk(\0061)p Fo(:)89 b Fm(I)4101 9685 y Fl(\026)4194 9658 y Fo(\()p Fm(q)7 b(;)60 b(")p Fo(\))e(=)g Fm(K)5046 9542 y Ff(\()p Fl(\037)5189 9571 y Fa([)p Fi(\000)p Fa(1)p Fd(;)p Fa(1])5522 9542 y Ff(\))5033 9679 y Fl(\026)5586 9658 y Fo(\()p Fm(q)7 b(;)59 b(")p Fo(\).)95 b(The)66 b(result)f(is)-201 9884 y(the)60 b(follo)-5 b(wing.)-201 10228 y Fc(Theorem)68 b(A.1.)91 b Fn(Pick)73 b Fm(q)f Fk(2)66 b Fo(\(0)p Fm(;)30 b Fo(1\))p Fn(.)108 b(Ther)-9 b(e)72 b(exists)i Fo(0)66 b Fm(<)f(c)p Fo(\()p Fm(q)7 b Fo(\))64 b Fm(<)h Fk(1)74 b Fn(such)f(that)h(for)e(al)9 b(l)73 b Fm(")66 b Fk(2)g Fo(\(0)p Fm(;)30 b Fo(1\))p Fn(,)74 b(we)-201 10454 y(have)2023 10588 y Fo(1)p 1915 10669 308 7 v 1915 10836 a Fm(c)p Fo(\()p Fm(q)7 b Fo(\))2242 10711 y Fm(I)2322 10738 y Fl(\026)2416 10711 y Fo(\()p Fm(q)g(;)30 b(")p Fo(\))49 b Fk(\024)h Fm(K)3221 10636 y Ff(\()p Fl(R)q Ff(\))3208 10756 y Fl(\026)3445 10711 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))50 b Fk(\024)f Fm(c)p Fo(\()p Fm(q)7 b Fo(\))p Fm(I)4471 10738 y Fl(\026)4563 10711 y Fo(\()p Fm(q)g(;)30 b(")p Fo(\))64 b Fm(:)1676 b Fo(\(A.1\))-201 11133 y Fn(In)66 b(p)-9 b(articular,)62 b Fm(I)1004 11160 y Fl(\026)1097 11133 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))64 b Fn(and)h Fm(K)2065 11160 y Fl(\026)2158 11133 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))63 b Fn(have)i(the)g(same)f(gr)-9 b(owth)64 b(exp)-9 b(onents:)84 b Fm(\034)5393 11067 y Fz(\006)5372 11178 y Fl(\026)5510 11133 y Fo(\()p Fm(q)7 b Fo(\))p Fn(.)3393 11631 y Fo(32)p eop %%Page: 33 33 33 32 bop 81 1158 a Fo(Theorem)72 b(A.1)g(is)h(of)f(particular)h(in)-5 b(terest)72 b(in)h(the)g(setting)f(of)g(quan)-5 b(tum)70 b(dynamics.)115 b(Indeed)72 b(as)g(it)-201 1384 y(is)d(sho)-5 b(wn)69 b(in)g([2,)f(3])g(and)h([16)o(],)h(a)f(lo)-5 b(w)g(er)69 b(b)5 b(ound)69 b(on)g(the)f(dynamics)f(can)i(b)5 b(e)69 b(obtained)f(in)i(terms)c(of)j(the)-201 1642 y(in)-5 b(tegral)64 b Fm(K)632 1546 y Ff(\()p Fl(R)q Ff(\))619 1663 y Fl(\026)857 1642 y Fo(\()p Fm(q)7 b(;)59 b(")p Fo(\))k(ab)5 b(o)-5 b(v)g(e,)63 b Fm(q)f Fk(2)56 b Fo(\(0)p Fm(;)30 b Fo(1\).)87 b(That)64 b(leads)f(to)g(a)g(di\013usion)i(co)5 b(e\016cien)-5 b(t)62 b(whic)-5 b(h)64 b(needs)f(not)g(to)-201 1868 y(b)5 b(e)61 b(equal)g(to)f Fm(D)905 1802 y Fz(\006)900 1913 y Fl(\026)1023 1868 y Fo(\()p Fm(q)7 b Fo(\))59 b Fn(a)66 b(priori)p Fo(.)79 b(Under)61 b(some)e(assumptions,)f(among)h (them)f Fm(\026)j Fo(with)g(compact)d(supp)5 b(ort,)-201 2093 y(it)59 b(has)f(b)5 b(een)58 b(pro)-5 b(v)g(ed)58 b(in)h([3)o(])f(that)f(b)5 b(oth)58 b Fm(I)2675 2120 y Fl(\026)2768 2093 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))57 b(and)i Fm(K)3728 2120 y Fl(\026)3820 2093 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))57 b(lead)i(to)f(the)g(same)e(dimension)h(n) -5 b(um)g(b)5 b(ers)-201 2319 y(for)63 b Fm(q)e Fk(2)54 b Fo(\(0)p Fm(;)30 b Fo(1\),)62 b(namely)f Fm(D)1704 2253 y Fz(\006)1699 2364 y Fl(\026)1822 2319 y Fo(\()p Fm(q)7 b Fo(\).)86 b(Nev)-5 b(ertheless)62 b(it)h(w)-5 b(as)63 b(not)g(clear)g(whether)f(the)h(equalit)-5 b(y)63 b(should)g(hold)-201 2545 y(in)e(full)h(generalit)-5 b(y)60 b(or)h(not.)80 b(Theorem)59 b(A.1)i(answ)-5 b(ers)60 b(p)5 b(ositiv)-5 b(ely)61 b(to)f(that)g(issue.)-201 2851 y Fc(Pro)6 b(of)69 b(of)g(Theorem)f(A.1:)-201 3077 y Fo(As)55 b(in)g(Section)g(2,)g(for)g(all)g Fm(k)h Fk(2)51 b Fj(Z)72 b Fo(and)54 b Fm(")d Fk(2)f Fo(\(0)p Fm(;)30 b Fo(1\),)55 b(w)-5 b(e)55 b(denote)f Fm(I)4201 3107 y Fl(k)4336 3077 y Fo(=)c([)p Fm("k)6 b(;)30 b(")p Fo(\()p Fm(k)i Fo(+)d(1\)\))54 b(and)g Fm(a)6030 3107 y Fl(k)6166 3077 y Fo(=)c Fm(\026)p Fo(\()p Fm(I)6619 3107 y Fl(k)6704 3077 y Fo(\).)78 b(F)-15 b(or)-201 3303 y Fm(n)51 b Fk(2)g Fj(Z)77 b Fo(let)614 3257 y(~)575 3303 y Fm(R)q Fo(\()p Fm(n)p Fo(\))51 b(=)e(sup)1481 3346 y Fl(t)p Fz(2)p Ff([)p Fl(n)p Fz(\000)p Ff(1)p Fl(;)23 b(n)p Ff(+1\))2354 3303 y Fm(R)q Fo(\()p Fm(t)p Fo(\).)79 b(No)-5 b(w)62 b(for)e(all)h Fm(k)6 b(;)30 b(j)60 b Fk(2)51 b Fj(Z)78 b Fo(and)60 b(for)h(all)g Fm(x)51 b Fk(2)f Fm(I)5737 3330 y Fl(j)5810 3303 y Fo(,)60 b(w)-5 b(e)61 b(ha)-5 b(v)g(e)1092 3536 y Fg(Z)1184 3913 y Fl(I)1245 3936 y Fd(k)1360 3766 y Fm(R)1530 3527 y Fg(\022)1672 3643 y Fm(x)40 b Fk(\000)g Fm(y)p 1672 3723 422 7 v 1841 3890 a(")2114 3527 y Fg(\023)2297 3766 y Fo(d)p Fm(\026)p Fo(\()p Fm(y)7 b Fo(\))164 b Fk(\024)3218 3536 y Fg(Z)3310 3913 y Fl(I)3371 3936 y Fd(k)3795 3766 y Fo(sup)3547 3928 y Ff(\()p Fl(x;y)5 b Ff(\))p Fz(2)p Fl(I)4005 3948 y Fd(j)4070 3928 y Fz(\002)p Fl(I)4241 3951 y Fd(k)4348 3766 y Fm(R)4517 3527 y Fg(\022)4659 3643 y Fm(x)41 b Fk(\000)f Fm(y)p 4659 3723 V 4828 3890 a(")5101 3527 y Fg(\023)5284 3766 y Fo(d)p Fm(\026)p Fo(\()p Fm(y)7 b Fo(\))2910 4281 y Fk(\024)3218 4051 y Fg(Z)3310 4428 y Fl(I)3371 4451 y Fd(k)3526 4235 y Fo(~)3486 4281 y Fm(R)r Fo(\()p Fm(j)50 b Fk(\000)40 b Fm(k)6 b Fo(\))30 b(d)p Fm(\026)p Fo(\()p Fm(y)7 b Fo(\))48 b(=)i Fm(a)4991 4311 y Fl(k)5116 4235 y Fo(~)5077 4281 y Fm(R)q Fo(\()p Fm(j)h Fk(\000)39 b Fm(k)6 b Fo(\))60 b Fm(:)873 b Fo(\(A.2\))-201 4760 y(Without)78 b(loss)f(of)g(generalit) -5 b(y)-15 b(,)82 b(w)-5 b(e)78 b(ma)-5 b(y)76 b(assume)f(that)3823 4714 y(~)3784 4760 y Fm(R)q Fo(\(0\))j(=)h(sup)4729 4804 y Ff([)p Fz(\000)p Ff(1)p Fl(;)p Ff(1])5136 4760 y Fm(R)q Fo(\()p Fm(t)p Fo(\))f Fk(\024)h Fo(1.)131 b(Therefore,)82 b(for)-201 4986 y Fm(q)57 b Fk(2)51 b Fo(\(0)p Fm(;)30 b Fo(1\),)1107 5500 y Fm(K)1275 5425 y Ff(\()p Fl(R)q Ff(\))1262 5545 y Fl(\026)1500 5500 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))165 b(=)2372 5338 y Fg(X)2368 5707 y Fl(j)8 b Fz(2)p Fe(Z)2646 5270 y Fg(Z)2738 5647 y Fl(I)2799 5667 y Fd(j)2903 5212 y Fg( )3034 5338 y(X)3116 5707 y Fl(k)3304 5270 y Fg(Z)3397 5647 y Fl(I)3458 5670 y Fd(k)3573 5500 y Fm(R)3742 5262 y Fg(\022)3885 5377 y Fm(x)40 b Fk(\000)g Fm(y)p 3885 5458 V 4053 5625 a(")4326 5262 y Fg(\023)4509 5500 y Fo(d)p Fm(\026)p Fo(\()p Fm(y)7 b Fo(\))4958 5212 y Fg(!)5088 5258 y Fl(q)e Fz(\000)p Ff(1)5405 5500 y Fo(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))2061 5994 y Fk(\025)2372 5832 y Fg(X)2368 6200 y Fl(j)j Fz(2)p Fe(Z)2646 5994 y Fm(a)2742 6021 y Fl(j)2845 5994 y Fo(\()p Fm(a)3012 6021 y Fl(j)3125 5994 y Fo(+)40 b Fm(b)3385 6021 y Fl(j)3458 5994 y Fo(\))3529 5910 y Fl(q)5 b Fz(\000)p Ff(1)3876 5994 y Fm(;)2823 b Fo(\(A.3\))-201 6551 y(where)81 b Fm(b)423 6578 y Fl(j)579 6551 y Fo(:=)854 6423 y Fg(P)1029 6597 y Fl(k)t Fz(6)p Ff(=)p Fl(j)1359 6505 y Fo(~)1319 6551 y Fm(R)r Fo(\()p Fm(j)63 b Fk(\000)53 b Fm(k)6 b Fo(\))p Fm(a)2131 6581 y Fl(k)2215 6551 y Fo(.)139 b(F)-15 b(or)80 b(\014xed)g Fm(N)103 b(>)83 b Fo(1,)h(let)c Fm(A)k Fo(=)e Fk(f)p Fm(j)94 b Fk(2)83 b Fj(Z)97 b Fk(j)80 b Fo(0)j Fk(\024)g Fm(b)5909 6578 y Fl(j)5982 6551 y Fm(=)-10 b(N)103 b(<)82 b(a)6632 6578 y Fl(j)6705 6551 y Fk(g)e Fo(and)-201 6777 y Fm(B)60 b Fo(=)50 b Fk(f)p Fm(j)60 b Fk(2)51 b Fj(Z)78 b Fk(j)60 b Fo(0)50 b Fm(<)g(a)1309 6804 y Fl(j)1433 6777 y Fk(\024)g Fm(b)1703 6804 y Fl(j)1776 6777 y Fm(=)-10 b(N)20 b Fk(g)p Fo(.)80 b(Then)61 b(from)d(\(A.3\),)i (w)-5 b(e)60 b(get)2061 7150 y Fm(K)2229 7075 y Ff(\()p Fl(R)q Ff(\))2216 7195 y Fl(\026)2453 7150 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))49 b Fk(\025)h Fo(\(1)40 b(+)g Fm(N)20 b Fo(\))3711 7075 y Fl(q)5 b Fz(\000)p Ff(1)4010 6988 y Fg(X)3998 7356 y Fl(j)j Fz(2)p Fl(A)4292 7150 y Fm(a)4388 7061 y Fl(q)4388 7201 y(j)4525 7150 y Fm(:)2174 b Fo(\(A.4\))-201 7665 y(On)60 b(the)g(other)h(hand,)f(w)-5 b(e)60 b(ha)-5 b(v)g(e)1910 8029 y Fg(X)1894 8397 y Fl(j)8 b Fz(2)p Fl(B)2196 8191 y Fm(a)2292 8102 y Fl(q)2292 8243 y(j)2534 8191 y Fk(\024)166 b Fm(N)3008 8116 y Fz(\000)p Fl(q)3240 8029 y Fg(X)3224 8397 y Fl(j)8 b Fz(2)p Fl(B)3526 7853 y Fg(0)3526 8152 y(@)3677 8029 y(X)3671 8398 y Fl(k)t Fz(6)p Ff(=)p Fl(j)3992 8145 y Fo(~)3953 8191 y Fm(R)q Fo(\()p Fm(j)51 b Fk(\000)40 b Fm(k)6 b Fo(\))p Fm(a)4739 8221 y Fl(k)4823 7853 y Fg(1)4823 8152 y(A)4968 7899 y Fl(q)2534 8710 y Fk(\024)166 b Fm(N)3008 8635 y Fz(\000)p Fl(q)3236 8549 y Fg(X)3224 8916 y Fl(j)8 b Fz(2)p Fl(Z)3524 8549 y Fg(X)3518 8918 y Fl(k)t Fz(6)p Ff(=)p Fl(j)3800 8522 y Fg(\020)3938 8664 y Fo(~)3899 8710 y Fm(R)q Fo(\()p Fm(j)51 b Fk(\000)39 b Fm(k)6 b Fo(\))4588 8522 y Fg(\021)4687 8568 y Fl(q)4793 8710 y Fm(a)4889 8622 y Fl(q)4889 8766 y(k)2534 9226 y Fk(\024)166 b Fm(N)3008 9151 y Fz(\000)p Fl(q)3242 9064 y Fg(X)3224 9433 y Fl(k)t Fz(2)p Fl(Z)3530 9226 y Fm(a)3626 9137 y Fl(q)3626 9282 y(k)3748 9064 y Fg(X)3742 9433 y Fl(j)8 b Fz(6)p Ff(=)p Fl(k)4024 9038 y Fg(\020)4162 9180 y Fo(~)4123 9226 y Fm(R)q Fo(\()p Fm(j)51 b Fk(\000)39 b Fm(k)6 b Fo(\))4812 9038 y Fg(\021)4911 9084 y Fl(q)2534 9720 y Fk(\024)166 b Fm(C)13 b Fo(\()p Fm(q)7 b Fo(\))p Fm(N)3381 9645 y Fz(\000)p Fl(q)3614 9559 y Fg(X)3596 9928 y Fl(k)t Fz(2)p Fl(Z)3902 9720 y Fm(a)3998 9632 y Fl(q)3998 9777 y(k)4144 9720 y Fm(;)2555 b Fo(\(A.5\))-201 10261 y(where)72 b Fm(C)13 b Fo(\()p Fm(q)7 b Fo(\))69 b(:=)1039 10133 y Fg(P)1214 10307 y Fl(k)t Fz(6)p Ff(=0)1549 10215 y Fo(~)1510 10261 y Fm(R)q Fo(\()p Fm(k)6 b Fo(\))1892 10195 y Fl(q)2037 10261 y Fm(<)68 b Fk(1)k Fo(since)g Fm(R)h Fo(deca)-5 b(ys)71 b(faster)g(than)g(an)-5 b(y)72 b(p)5 b(olynomials.)113 b(T)-15 b(aking)73 b Fm(N)89 b Fo(=)-201 10515 y(\(2)p Fm(C)13 b Fo(\()p Fm(q)7 b Fo(\)\))405 10449 y Ff(1)p Fl(=q)681 10515 y Fo(in)62 b(\(A.5\))d(giv)-5 b(es,)60 b(together)g(with)h(\(A.4\),)e(that)2207 10976 y Fm(K)2375 10901 y Ff(\()p Fl(R)q Ff(\))2362 11021 y Fl(\026)2600 10976 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))49 b Fk(\025)3257 10853 y Fo(\(1)40 b(+)g Fm(N)20 b Fo(\))3878 10787 y Fl(q)5 b Fz(\000)p Ff(1)p 3257 10934 877 7 v 3650 11101 a Fo(2)4196 10815 y Fg(X)4184 11182 y Fl(j)j Fz(2)p Fl(Z)4478 10976 y Fm(a)4574 10888 y Fl(q)4574 11028 y(j)4711 10976 y Fm(:)1988 b Fo(\(A.6\))3393 11631 y(33)p eop %%Page: 34 34 34 33 bop -201 1158 a Fo(On)60 b(the)g(other)h(side,)f(w)-5 b(e)60 b(also)h(ha)-5 b(v)g(e)60 b(the)g(follo)-5 b(wing)62 b(inequalit)-5 b(y)-15 b(,)61 b(for)f Fm(q)d(<)50 b Fo(1,)1515 1647 y Fm(K)1683 1572 y Ff(\()p Fl(R)q Ff(\))1670 1692 y Fl(\026)1908 1647 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))49 b(=)2545 1417 y Fg(Z)2637 1794 y Ff(supp)q Fl(\026)3052 1409 y Fg(\022)3174 1647 y Fm(R)3344 1409 y Fg(\022)3486 1524 y Fm(x)40 b Fk(\000)g Fm(y)p 3486 1604 422 7 v 3654 1771 a(")3928 1409 y Fg(\023)4110 1647 y Fo(d)p Fm(\026)p Fo(\()p Fm(y)7 b Fo(\))4559 1409 y Fg(\023)4680 1454 y Fl(q)e Fz(\000)p Ff(1)4997 1647 y Fo(d)p Fm(\026)p Fo(\()p Fm(x)p Fo(\))1344 2225 y Fk(\024)50 b Fm(b)1614 2150 y Fl(q)5 b Fz(\000)p Ff(1)1901 1995 y Fg(Z)1993 2372 y Ff(supp)q Fl(\026)2408 2225 y Fm(\026)p Fo(\([)p Fm(x)39 b Fk(\000)h Fm(";)30 b(x)40 b Fo(+)f Fm(")p Fo(]\))3663 2150 y Fl(q)5 b Fz(\000)p Ff(1)3919 2225 y Fm(d\026)p Fo(\()p Fm(x)p Fo(\))49 b(=)h Fm(b)4689 2150 y Fl(q)5 b Fz(\000)p Ff(1)4945 2225 y Fm(I)5025 2252 y Fl(\026)5118 2225 y Fo(\()p Fm(q)i(;)30 b(")p Fo(\))p Fm(;)1185 b Fo(\(A.7\))-201 2647 y(where)62 b Fm(b)54 b Fo(=)e(inf)873 2684 y Ff([)p Fz(\000)p Ff(1)p Fl(;)p Ff(1])1280 2647 y Fm(R)q Fo(\()p Fm(t)p Fo(\))h Fm(>)f Fo(0.)85 b(Inequalities)62 b(\(A.6\))g(and)g(\(A.7\))f(and)h(Lemma)d(2.1)i(conclude)h(the)g(pro)5 b(of)-201 2873 y(of)60 b(the)g(Theorem.)5949 b Fb(\003)-201 3228 y Fc(Remark)69 b(A.1.)91 b Fo(If)68 b Fm(q)73 b(>)65 b Fo(1)k(the)g(conclusions)h(of)f(Theorem)f(A.1)i(still)g(hold.)108 b(Lik)-5 b(e)69 b(for)h(Lemma)c(2.1)j(the)-201 3454 y(pro)5 b(of)75 b(is)g(ev)-5 b(en)74 b(simpler,)k(since)c(there)h(is)g(no)g (needs)f(to)g(split)i(the)e(sum)f(o)-5 b(v)g(er)75 b(\\go)5 b(o)g(d")74 b(and)h(\\bad")g Fm(a)6974 3481 y Fl(j)7047 3454 y Fo('s)-201 3719 y(\(sets)61 b Fm(A)i Fo(and)f Fm(B)71 b Fo(ab)5 b(o)-5 b(v)g(e\).)84 b(Indeed)62 b(from)e(\(A.2\))h (one)h(gets)g Fm(K)4036 3623 y Ff(\()p Fl(R)q Ff(\))4023 3740 y Fl(\026)4260 3719 y Fo(\()p Fm(q)7 b(;)30 b(")p Fo(\))52 b Fk(\024)4903 3590 y Fg(P)5078 3765 y Fl(j)8 b Fz(2)p Fe(Z)5334 3719 y Fo(\()5405 3590 y Fg(P)5580 3765 y Fl(k)t Fz(2)p Fe(Z)5878 3719 y Fm(a)5974 3749 y Fl(k)6099 3673 y Fo(~)6059 3719 y Fm(R)r Fo(\()p Fm(j)51 b Fk(\000)41 b Fm(k)6 b Fo(\)\))6822 3653 y Fl(q)6897 3719 y Fo(.)85 b(A)-201 3945 y(Jensen)60 b(inequalit)-5 b(y)61 b(ends)f(the)g(pro)5 b(of)61 b(in)g(this)f(case.)-201 4490 y Fp(References)-118 4854 y Fx([1])91 b(J.-M.)39 b(Barbaroux,)j(J.-M.)d(Com)-5 b(b)5 b(es,)42 b(R.)c(Mon)-5 b(tc)g(ho,)42 b(Remarks)d(on)f(the)f(relation)h(b)5 b(et)-5 b(w)g(een)38 b(quan)-5 b(tum)38 b(dynamics)148 5053 y(and)56 b(fractal)f(sp)5 b(ectra,)55 b Fy(J.)k(Math.)h(A)l(nal)g(and)f(Appl.)d Fv(213)h Fx(\(1997\),)e(698-722.)-118 5344 y([2])91 b(J.-M.)50 b(Barbaroux,)g(F.)f(Germinet,)h(S.)e(Tc)-5 b(heremc)g(han)g(tsev,)51 b(Nonlinear)e(v)-9 b(ariation)48 b(of)g(di\013usion)h(exp)5 b(onen)-5 b(ts)49 b(in)148 5543 y(quan)-5 b(tum)56 b(dynamics,)h Fy(C.)i(R.)g(A)-8 b(c)g(ad.)59 b(Sci.)h(Paris)p Fx(,)55 b(t.)g(330,)h(S)-5 b(\023)-78 b(erie)55 b(I)g(\(2000\),)g(409-414.)-118 5834 y([3])91 b(J.-M.)60 b(Barbaroux,)f(F.)g(Germinet,)h(S.)e(Tc)-5 b(heremc)g(han)g(tsev,)61 b(F)-14 b(ractal)58 b(dimensions)j(and)e(the) f(phenomenon)i(of)148 6033 y(in)-5 b(termittency)55 b(in)h(quan)-5 b(tum)56 b(dynamics,)g(to)f(app)5 b(ear)56 b(in)f(DUKE.)-118 6324 y([4])91 b(L.)50 b(Barreira,)h(Y.B.)e(P)-5 b(esin,)52 b(J.)e(Sc)-5 b(hmeling,)52 b(Dimension)f(and)g(Pro)5 b(duct)49 b(Structure)g(of)h(Hyp)5 b(erb)g(olic)48 b(Measures,)148 6523 y Fy(A)l(nnals)61 b(of)e(Math.)c Fv(149)i Fx(\(1999\),)e(755-783.) -118 6814 y([5])91 b(A.)64 b(Batakis,)h(Y.)f(Heurteaux,)h(On)f (relations)g(b)5 b(et)-5 b(w)g(een)64 b(en)-5 b(trop)g(y)64 b(and)h(Hausdor\013)f(dimension)h(of)f(measures,)148 7013 y Fy(Pr)-8 b(eprint)56 b Fx(Orsa)-5 b(y)56 b(\(1998\).)-118 7304 y([6])91 b(J.)55 b(Bellissard,)h(H.)e(Sc)-5 b(h)g(ulz-Baldes,)56 b(Sub)5 b(di\013usiv)-5 b(e)56 b(quan)-5 b(tum)54 b(transp)5 b(ort)55 b(for)f(3-D)g(Hamiltonians)h(with)f(abso-)148 7503 y(lutely)h(con)-5 b(tin)g(uous)57 b(sp)5 b(ectra,)55 b Fy(J.)60 b(Stat.)g(Phys.)54 b Fv(99)i Fx(\(2000\),)f(587-594.)-118 7794 y([7])91 b(J.-M.)66 b(Com)-5 b(b)5 b(es,)68 b Fy(Conne)-8 b(ction)69 b(b)-8 b(etwe)g(en)70 b(quantum)f(dynamics)e(and)i(sp)-8 b(e)g(ctr)g(al)67 b(pr)-8 b(op)g(erties)68 b(of)g(time)g(evolution)148 7994 y(op)-8 b(er)g(ators)43 b Fx(in)g(\\Di\013eren)-5 b(tial)44 b(Equations)f(and)h(Applications)g(in)g(Mathematical)f(Ph)-5 b(ysics",)47 b(Eds.)d(W.F.)g(Ames,)148 8193 y(E.M.)56 b(Harrel,)f(J.V.)h(Hero)5 b(d)54 b(\(Academic)h(Press)h(1993\),)f (59-69.)-118 8484 y([8])91 b(C.D.)57 b(Cutler,)f(Some)h(results)g(on)g (the)e(b)5 b(eha)-5 b(viour)57 b(and)g(estimation)g(of)f(the)g(fractal) f(dimensions)k(of)d(distribu-)148 8683 y(tions)g(on)g(attractors,)e Fy(J.)60 b(Stat.)g(Phys.)54 b Fv(62)p Fx(,)j(Nos)e(3/4)g(\(1991\),)g (651-708.)-118 8974 y([9])91 b(K.J.)39 b(F)-14 b(alconer,)41 b Fy(The)i(Ge)-8 b(ometry)44 b(of)e(F)-13 b(r)-8 b(actal)44 b(Sets)p Fx(,)e(Cam)-5 b(bridge)39 b(T)-14 b(racts)38 b(in)f(Mathematics,)42 b(No)37 b(85,)k(Cam)-5 b(bridge)148 9173 y(Univ.)55 b(Press,)i(UK,)e(1985.)-201 9464 y([10])91 b(K.J.)84 b(F)-14 b(alconer,)89 b Fy(F)-13 b(r)-8 b(actal)86 b(Ge)-8 b(ometry-Mathematic)g(al)84 b(F)-13 b(oundations)86 b(and)e(Applic)-8 b(ation)p Fx(,)90 b(John)83 b(Wiley)f(and)148 9663 y(Sons,)57 b(New)d(Y)-14 b(ork,)55 b(1990.)-201 9954 y([11])91 b(K.J.)56 b(F)-14 b(alconer,)57 b Fy(T)-13 b(e)-8 b(chniques)60 b(in)g(F)-13 b(r)-8 b(actal)60 b(Ge)-8 b(ometry)p Fx(,)56 b(John)g(Wiley)f(and)h(Sons,)g(New)f(Y)-14 b(ork,)55 b(1997.)-201 10245 y([12])91 b(P)-14 b(.)59 b(Grassb)5 b(erger,)59 b(I.)f(Pro)5 b(caccia,)58 b(Measuring)h(the)f (strangeness)h(of)f(strange)g(attractors,)g Fy(Phys.)i(D)f Fv(9)f Fx(\(1983\),)148 10444 y(189-208.)-201 10735 y([13])91 b(I.)80 b(Guarneri,)87 b(Sp)5 b(ectral)79 b(prop)5 b(erties)80 b(of)f(quan)-5 b(tum)80 b(di\013usion)h(on)f(discrete)g(lattices,)85 b Fy(Eur)-8 b(ophys.)80 b(L)-8 b(ett.)80 b Fv(10)148 10934 y Fx(\(1989\),)43 b(95-100;)j(On)41 b(an)g(estimate)g(concerning) g(quan)-5 b(tum)41 b(di\013usion)h(in)f(the)f(presence)h(of)f(a)h (fractal)f(sp)5 b(ectrum,)148 11133 y Fy(Eur)-8 b(ophys.)58 b(L)-8 b(ett.)56 b Fv(21)g Fx(\(1993\),)f(729-733.)3393 11631 y Fo(34)p eop %%Page: 35 35 35 34 bop -201 1158 a Fx([14])91 b(I.)66 b(Guarneri,)i(Singular)e(con) -5 b(tin)g(uous)67 b(sp)5 b(ectra)65 b(and)h(discrete)f(w)-5 b(a)g(v)g(e)66 b(pac)-5 b(k)g(et)65 b(dynamics,)j Fy(J.)h(Math.)f(Phys) c Fv(37)148 1358 y Fx(\(1996\),)55 b(5195-5206.)-201 1657 y([15])91 b(I.)56 b(Guarneri,)h(H.)e(Sc)-5 b(h)g(ulz-Baldes,)58 b(Lo)-5 b(w)g(er)56 b(b)5 b(ounds)56 b(on)g(w)-5 b(a)g(v)g(e)56 b(pac)-5 b(k)g(et)56 b(propagation)g(b)-5 b(y)56 b(pac)-5 b(king)56 b(dimensions)148 1856 y(of)g(sp)5 b(ectral)55 b(measures,)h Fy(Math.)j(Phys.)g(Ele)-8 b(c.)59 b(J.)d Fv(5)g Fx(\(1999\),)e(pap)5 b(er)56 b(1.)-201 2155 y([16])91 b(I.)i(Guarneri,)102 b(H.)92 b(Sc)-5 b(h)g(ulz-Baldes,)104 b(In)-5 b(termitten)g(t)92 b(lo)-5 b(w)g(er)92 b(b)5 b(ound)93 b(on)g(quan)-5 b(tum)93 b(di\013usion,)103 b Fy(L)-8 b(ett.)94 b(Math.)148 2354 y(Phys.)55 b Fv(49)h Fx(\(1999\),)f(317-324.)-201 2653 y([17])91 b(C.A.)80 b(Guerin,)88 b(M.)81 b(Holsc)-5 b(hneider,)87 b(Three)81 b(equiv)-9 b(alen)k(t)79 b(de\014nitions)j(of)e(the)g(correlation)g (dimension)i(of)e(a)148 2852 y(probabilit)-5 b(y)57 b(measure,)f Fy(J.)j(Stat.)h(Phys.)54 b Fv(29)j Fx(\(1996\),)d(7651-7667.)-201 3151 y([18])91 b(T.C.)73 b(Halsey)-14 b(,)76 b(M.)d(Jensen,)k(L.)c (Kaddano)-5 b(v,)77 b(I.)c(Pro)5 b(caccia,)76 b(B)c(Sc)-5 b(hraiman,)78 b(F)-14 b(ractal)72 b(measures)i(and)f(their)148 3350 y(singularities:)j(the)55 b(c)-5 b(haracterization)55 b(of)g(strange)g(sets,)h Fy(Phys.)j(R)-8 b(ev.)59 b(A)d Fv(33)g Fx(\(1986\),)f(1141-1151.)-201 3649 y([19])91 b(H.G.E.)76 b(Hen)-5 b(tsc)g(hel,)80 b(I.)c(Pro)5 b(caccia,)79 b(The)c(in\014nite)h(n)-5 b(um)g(b)5 b(er)76 b(of)f(generalized)h (dimensions)h(of)e(fractals)g(and)148 3848 y(strange)56 b(attractors,)f Fy(Phys.)j(D)e Fv(8)f Fx(\(1983\),)g(435-444.)-201 4147 y([20])91 b(Y.)56 b(Heurteaux,)f(Estimations)i(de)f(la)g (dimension)h(inf)-5 b(\023)-78 b(erieure)56 b(et)f(de)h(la)g(dimension) i(sup)-5 b(\023)-78 b(erieure)57 b(des)f(mesures,)148 4346 y Fy(A)l(nn.)61 b(Inst.)e(Henri)i(Poinc)-8 b(ar)l(\023)-81 b(e)55 b Fx(V)-14 b(ol.)55 b(34,)h(No)f(3)g(\(1998\),)g(309-338.)-201 4645 y([21])91 b(X.)59 b(Hu,)i(J.)f(T)-14 b(a)-5 b(ylor,)61 b(F)-14 b(ractal)59 b(prop)5 b(erties)60 b(of)f(pro)5 b(ducts)59 b(and)h(pro)9 b(jections)60 b(of)f(measures)i(in)f Fq(R)6111 4585 y Fs(d)6197 4645 y Fx(,)h Fy(Math.)h(Pr)-8 b(o)g(c.)148 4845 y(Camb.)60 b(Phil.)f(So)-8 b(c.)55 b Fv(115)i Fx(\(1994\),)e(527-544.)-201 5143 y([22])91 b(Y.)g(Last,)99 b(Quan)-5 b(tum)92 b(dynamics)g(and)f(decomp)5 b(osition)92 b(of)e(singular)i(con)-5 b(tin)g(uous)93 b(sp)5 b(ectrum,)100 b Fy(J.)92 b(F)-13 b(unct.)148 5343 y(A)l(nal.)57 b Fv(142)f Fx(\(1996\),)f(406-445.)-201 5642 y([23])91 b(K.S.)56 b(Lau,)g(F)-14 b(ractal)55 b(measures)i(and)f (mean)g Fu(p)p Fx(-v)-9 b(ariations,)55 b Fy(J.)60 b(F)-13 b(unct.)61 b(A)l(nal.)56 b Fv(108)g Fx(\(1992\),)f(427-457.)-201 5940 y([24])91 b(J.)48 b(Lu)-5 b(\023)-78 b(ev)-9 b(ano,)48 b(V.)f(P)-5 b(enn)g(\023)-78 b(e,)50 b(S.)d(V)-14 b(aien)-5 b(ti,)49 b(Multifractal)e(sp)5 b(ectrum)47 b(via)g(return)g(times,)i Fy(Pr)-8 b(eprint)47 b Fx(\(2000\),)h(mp-arc)148 6140 y(00-232.)-201 6439 y([25])91 b(L.)56 b(Olsen,)g(A)f(m)-5 b(ultifractal)55 b(formalism,)i Fy(A)-8 b(dv.)60 b(Math.)55 b Fv(116)h Fx(\(1995\),)f(82-195.)-201 6738 y([26])91 b(Y.B.)65 b(P)-5 b(esin,)70 b Fy(Dimension)f(The)-8 b(ory)68 b(in)h(Dynamic)-8 b(als)69 b(Systems:)95 b(Contemp)-8 b(or)g(ary)67 b(Views)j(and)f(Applic)-8 b(ations)p Fx(,)148 6937 y(Univ.)55 b(Chicago)h(Press,)g(\(1996\).)-201 7236 y([27])91 b(Y.B.)65 b(P)-5 b(esin,)68 b(H.)c(W)-14 b(eiss,)68 b(The)d(Multifractal)g(Analysis)g(of)g(Gibbs)h(Measures:)94 b(Motiv)-9 b(ation,)66 b(Mathematical)148 7435 y(F)-14 b(oundation,)57 b(and)f(Examples,)g Fy(Chaos)f Fv(7)h Fx(\(1997\),)e(89-106.)-201 7734 y([28])91 b(H.)73 b(Sc)-5 b(h)g(ulz-Baldes,)79 b(J.)73 b(Bellissard,)78 b(Anomalous)73 b(transp)5 b(ort:)109 b(a)72 b(mathematical)h(framew)-5 b(ork,)77 b Fy(R)-8 b(ev.)76 b(Math.)148 7933 y(Phys.)55 b Fv(10)h Fx(\(1998\),)f(1-46.)-201 8232 y([29])91 b(F.)49 b(T)-14 b(ak)-5 b(ens,)50 b(E.A.)e(V)-14 b(erbitskii,)49 b(Multifractal)f(analysis)h(of)f(lo)5 b(cal)48 b(en)-5 b(tropies)48 b(for)g(expansiv)-5 b(e)48 b(homeomorphisms)148 8431 y(with)56 b(sp)5 b(eci\014cation,)55 b Fy(Commun.)k(Math.)g(Phys.) c Fv(203)h Fx(\(1999\),)f(593-612.)-201 8730 y([30])91 b(E.A.)56 b(V)-14 b(erbitskii,)55 b(F.)h(T)-14 b(ak)-5 b(ens,)56 b(Generalized)f(en)-5 b(tropies,)57 b Fy(Math.)i(Notes)d Fv(63)g Fx(\(1998\),)f(112-115.)-201 9029 y([31])91 b(L.)77 b(Y)-14 b(oung,)82 b Fy(Dimension,)i(entr)-8 b(opy)79 b(and)g(Lyap)-8 b(ounov)78 b(exp)-8 b(onents,)83 b Fx(Ergo)5 b(d.)76 b(Th.)h(Dynam.)g(Sys.,)82 b Fv(2)77 b Fx(\(1982\),)148 9228 y(109-124.)3393 11631 y Fo(35)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0009061145911--