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b(theorem)i(extends)f(with)e(almost)i(no)g(c)m(hanges) h(to)g(the)f(case)h(in)e(whic)m(h)f Fk(S)j Fm(=)25 b Fk(G=H)37 b Fm(where)30 b Fk(G)g Fm(is)0 5007 y(a)g(unimo)s(dular)25 b(Lie)k(group,)g Fk(H)36 b Fm(is)28 b(a)h(compact)i(subgroup)c(of)j Fk(H)36 b Fm(and)28 b Fk(G)i Fm(satis\014es)e(the)i(follo)m(wing)d (prop)s(ert)m(y)-8 b(.)40 b(F)-8 b(or)1938 5309 y(2)p eop %%Page: 3 3 3 2 bop 0 191 a Fm(ev)m(ery)35 b(compact)g(subset)f Fk(C)40 b Fm(of)35 b Fk(G)f Fm(there)h(exists)f(a)g(discrete)g(co)s(compact)i (subgroup)c Fk(G)3046 158 y Fg(0)3102 191 y Fk(<)f(G)j Fm(suc)m(h)g(that)h(there)0 304 y(exists)j(a)g(fundamen)m(tal)f(domain) g Fk(F)51 b Fm(for)38 b Fk(G)1533 271 y Fg(0)1594 304 y Fm(and)g Fk(C)44 b Fj(\032)38 b Fk(F)13 b Fm(.)64 b(In)37 b(this)g(setting,)j(`congruen)m(t)f(copies)f(of)g(a)h(b)s(o)s(dy)0 417 y Fk(K)32 b Fj(\032)25 b Fk(S)5 b Fm(')30 b(means)g(subsets)g(of)g (the)g(form)g Fk(g)s(K)37 b Fm(where)30 b Fk(g)e Fj(2)d Fk(G)31 b Fm(and)e Fk(G)i 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b(obtain)e(the)g(follo)m(wing)f(coun)m(terin)m(tuitiv)m(e)h(result:)0 1488 y Fn(Theorem)k(2.2.)46 b Ff(L)-5 b(et)38 b Fk(S)43 b Ff(b)-5 b(e)38 b Fk(n)p Ff(-dimensional)h(hyp)-5 b(erb)g(olic)40 b(sp)-5 b(ac)g(e)40 b(for)e(some)h Fk(n)34 b(>)h Fm(1)p Ff(.)59 b(L)-5 b(et)38 b Fk(\017)d(>)g Fm(0)p Ff(.)58 b(Then)38 b(ther)-5 b(e)0 1601 y(is)38 b(a)h(b)-5 b(o)g(dy)39 b Fk(K)45 b Ff(in)38 b Fk(S)43 b Ff(that)d(admits)f(a)g(tiling)f Fj(T)61 b Ff(and)39 b(a)f(c)-5 b(ompletely)40 b(satur)-5 b(ate)g(d)41 b(p)-5 b(acking)39 b Fj(P)46 b Ff(with)39 b(the)f(fol)5 b(lowing)0 1714 y(pr)-5 b(op)g(erties.)44 b(F)-7 b(or)34 b(every)e(p)-5 b(oint)34 b Fk(p)25 b Fj(2)g Fk(S)5 b Ff(,)33 b(the)g(limit)1540 1937 y Fm(lim)1516 1991 y Fi(r)r Fg(!1)1716 1875 y Fk(\025)1769 1889 y Fi(S)1820 1875 y Fm(\()p Fk(c)p Fm(\()p Fj(P)7 b Fm(\))22 b Fj(\\)e Fk(B)2206 1889 y Fi(r)2244 1875 y Fm(\()p Fk(p)p Fm(\)\))p 1716 1916 680 4 v 1857 1999 a Fk(\025)1910 2013 y Fi(S)1961 1999 y Fm(\()p Fk(B)2065 2013 y Fi(r)2103 1999 y 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y(lo)s(cally)f(compact)j(group)e Fk(G)987 3505 y Fh(0)1059 3491 y Fm(is)g(determined)f(uniquely)f(up)h (to)j(scalar)e(m)m(ultiplication)e(b)m(y)i(the)h(prop)s(ert)m(y)f(that) 0 3604 y(it)j(is)f(a)h(nonzero)h(Borel)f(measure)g(that)h(is)e(in)m(v) -5 b(arian)m(t)34 b(under)g(left)h(m)m(ultiplication)d(b)m(y)j(ev)m (ery)h(elemen)m(t)g Fk(g)h Fj(2)32 b Fk(G)3881 3618 y Fh(0)0 3717 y Fm(\(i.e.)41 b Fk(\025)245 3731 y Fi(G)300 3740 y Fe(0)339 3717 y Fm(\()p Fk(E)5 b Fm(\))26 b(=)f Fk(\025)656 3731 y Fi(G)711 3740 y Fe(0)750 3717 y Fm(\()p Fk(g)s(E)5 b Fm(\))31 b(for)f(ev)m(ery)h Fk(g)e Fj(2)c Fk(G)30 b Fm(and)g(Borel)g(set)h Fk(E)f Fj(\032)25 b Fk(G)31 b Fm(where)e Fk(\025)2777 3731 y Fi(G)2832 3740 y Fe(0)2901 3717 y Fm(is)g(a)i(Haar)f(measure)g(of)h Fk(G)3821 3731 y Fh(0)3861 3717 y Fm(\).)141 3830 y(Let)k Fk(d)355 3844 y Fi(S)440 3830 y Fm(b)s(e)e(the)h(usual)f(distance)h (function)e(on)i Fk(S)5 b Fm(.)52 b(Let)34 b Fk(d)2167 3844 y Fi(G)2261 3830 y Fm(b)s(e)f(an)m(y)i(left-in)m(v)-5 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Fj(g)p Fk(:)495 b Fm(\(2\))1938 5309 y(3)p eop %%Page: 4 4 4 3 bop 0 191 a Fm(Next)31 b(w)m(e)g(de\014ne)f(a)g(metric)h Fk(d)1025 205 y Fi(K)1124 191 y Fm(on)f Fk(C)1315 205 y Fi(K)1413 191 y Fm(b)m(y:)1201 438 y Fk(d)1248 452 y Fi(K)1317 438 y Fm(\()p Fj(P)1415 452 y Fh(1)1455 438 y Fk(;)15 b Fj(P)1558 452 y Fh(2)1598 438 y Fm(\))26 b(=)f(sup)1757 514 y Fi(n>)p Fh(0)1907 438 y Fm(min)n Fj(f)2113 376 y Fk(d)2160 390 y Fi(n)2208 376 y Fm(\()p Fj(P)2306 390 y Fh(1)2346 376 y Fk(;)15 b Fj(P)2449 390 y Fh(2)2489 376 y Fm(\))p 2113 417 411 4 v 2292 500 a Fk(n)2534 438 y(;)2589 376 y Fm(1)p 2585 417 55 4 v 2585 500 a Fk(n)2649 438 y Fj(g)p Fk(:)1086 b Fm(\(3\))0 690 y(It)34 b(can)g(b)s(e)f(c)m(hec)m(k)m(ed)j(that)e Fk(C)1001 704 y Fi(K)1103 690 y Fm(is)f(a)h(compact)h(metric)e(space)h(on)g(whic) m(h)e Fk(G)j Fm(acts)f(join)m(tly)f(con)m(tin)m(uously)g(in)f(the)0 803 y(ob)m(vious)39 b(w)m(a)m(y)-8 b(,)44 b(i.e.)68 b(for)40 b Fj(P)48 b(2)41 b Fk(C)1163 817 y Fi(K)1271 803 y Fm(and)e Fk(h)i Fj(2)f Fk(G)p Fm(,)j Fk(h)p Fj(P)49 b Fm(=)40 b Fj(f)p Fk(hg)s(K)7 b Fj(j)15 b Fk(g)s(K)50 b Fj(2)40 b(P)7 b(g)41 b Fm(\(cf)f([RaW]\).)i(W)-8 b(e)41 b(note)f(that)h(in)0 915 y(Euclidean)33 b(space,)k(if)e Fk(h)g Fm(is)f(translation)g(b)m(y)i Fk(x)p Fm(,)g(then)f Fk(h)p Fj(P)43 b Fm(is)34 b(commonly)h(denoted)g (b)m(y)g Fj(P)d Fm(+)23 b Fk(x)p Fm(.)55 b(Although)34 b(the)0 1028 y(metric)f Fk(d)333 1042 y Fi(K)436 1028 y Fm(will)e(b)s(e)i(con)m(v)m(enien)m(t)h(in)f(what)g(is)g(to)h(follo)m (w,)g(w)m(e)g(are)h(really)d(only)h(concerned)h(with)e(the)i(top)s (ology)0 1141 y(that)28 b(it)f(induces.)38 b(The)26 b(idea)h(is)f(that) i(t)m(w)m(o)h(pac)m(kings)e(are)g(close)h(in)e Fk(C)2360 1155 y Fi(K)2455 1141 y Fm(if)h(in)e(a)j(large)f(ball)f(ab)s(out)h(the) g(origin,)g(the)0 1254 y(b)s(o)s(dies)i(of)i(one)g(pac)m(king)f(can)i (b)m(y)e(small)f(rigid)g(motions)i(b)s(e)f(made)h(to)g(coincide)f(with) f(the)i(b)s(o)s(dies)e(of)i(the)g(other)0 1367 y(pac)m(king.)141 1480 y(Let)j Fk(M)40 b Fm(=)30 b Fk(M)10 b Fm(\()p Fk(K)d Fm(\))33 b(b)s(e)g(the)g(set)h(of)g(Borel)f(probabilit)m(y)e(measures)i (on)g Fk(C)2658 1494 y Fi(K)2726 1480 y Fm(.)49 b(W)-8 b(e)35 b(let)e Fk(M)3182 1494 y Fi(I)3252 1480 y Fm(=)d Fk(M)3441 1494 y Fi(I)3481 1480 y Fm(\()p Fk(K)7 b Fm(\))33 b(b)s(e)g(the)0 1593 y(subset)d(of)h Fk(M)41 b Fm(consisting)29 b(of)i(all)f(those)h(measures)g(that)g(are)g Fk(G)p Fm(-in)m(v)-5 b(arian)m(t.)42 b(In)30 b(other)h(w)m(ords,)f Fk(\026)c Fj(2)f Fk(M)41 b Fm(is)30 b(in)f Fk(M)3881 1607 y Fi(I)0 1706 y Fm(if)g(and)h(only)g(if)f Fk(\026)p Fm(\()p Fk(g)s(E)5 b Fm(\))27 b(=)e Fk(\026)p Fm(\()p Fk(E)5 b Fm(\))31 b(for)f(ev)m(ery)h Fk(g)e Fj(2)c Fk(G)31 b Fm(and)e(Borel)i(set)g Fk(E)f Fj(\032)25 b Fk(C)2594 1720 y Fi(K)2663 1706 y Fm(.)141 1819 y(Although)31 b(w)m(e)h(will)d(not)i(use)h(it,)f(w)m(e)h (sho)m(w)g(here)f(ho)m(w)h(to)g(construct)g(a)g(natural)e(class)i(of)f (measures)h(in)e Fk(M)3856 1833 y Fi(I)3896 1819 y Fm(.)0 1932 y(If)g Fj(P)37 b Fm(is)30 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Fm(b)m(y)g(left)g(m)m(ultiplication.)47 b(There)33 b(is)f(a)i(natural)0 2383 y(measure)k(on)h Fk(G=)p Fm(\006)679 2397 y Fg(P)777 2383 y Fm(induced)d(from)i Fk(\025)1399 2397 y Fi(G)1497 2383 y Fm(since)g(the)h(quotien)m(t)f (map)g(from)g Fk(G)h Fm(to)h Fk(G=)p Fm(\006)3104 2397 y Fg(P)3202 2383 y Fm(is)d(a)i(co)m(v)m(ering)h(map)0 2496 y(in)g(whic)m(h)g(the)h(co)m(v)m(ering)h(transformations)f(are)g (all)f(measure-preserving)g(\(in)h(fact)h(they)f(are)h(isometries)e(of) 0 2609 y Fk(G)p Fm(\).)51 b(This)32 b(measure)h(is)g(in)m(v)-5 b(arian)m(t)33 b(under)f(the)i(action)g(of)f Fk(G)h Fm(since)f Fk(\025)2417 2623 y Fi(G)2510 2609 y Fm(is)g(in)m(v)-5 b(arian)m(t.)49 b(No)m(w)34 b(a)g(measure)41 b(^)-52 b Fk(\026)3733 2623 y Fg(P)3825 2609 y Fm(on)0 2722 y Fk(O)s Fm(\()p Fj(P)7 b Fm(\))35 b(can)g(b)s(e)e(induced)f(\(via)i Fk(q)s Fm(\))g(from)g(the)g(measure)g(on)g Fk(G=)p Fm(\006)2237 2736 y Fg(P)2296 2722 y Fm(.)52 b(This)32 b(measure)i(can)g(then)g(b)s (e)f(extended)h(to)0 2835 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y(for)33 b(any)g(optimal)5 b(ly)35 b(dense)e(me)-5 b(asur)g(e)34 b Fk(\026)e Ff(for)h Fk(K)7 b Ff(,)33 b Fk(\026)p Fm(\()p Fk(U)10 b Fm(\))25 b(=)g(0)p Ff(.)0 2486 y Fm(Theorem)30 b(2.1)i(follo)m(ws)d(immediately) g(from)h(this)f(one)i(and)e(the)i(follo)m(wing)e(existence)i(theorem.)0 2673 y Fn(Theorem)j(3.2.)46 b Ff(F)-7 b(or)34 b(any)f(b)-5 b(o)g(dy)34 b Fk(K)e Fj(\032)25 b Fk(S)5 b Ff(,)32 b(ther)-5 b(e)34 b(exists)f(an)g(optimal)5 b(ly)35 b(dense)d(me)-5 b(asur)g(e)34 b Fk(\026)f Ff(for)g Fk(K)7 b Ff(.)0 2861 y Fm(A)45 b(pro)s(of)f(of)h(this)f(theorem)h(app)s(ears)f(in)g([BoR,)i (Theorem)f(3.2].)85 b(W)-8 b(e)46 b(include)d(the)i(same)h(pro)s(of)e (here)g(for)0 2974 y(completeness.)0 3161 y Ff(Pr)-5 b(o)g(of.)47 b Fm(W)-8 b(e)39 b(let)e Fk(M)675 3175 y Fi(I)753 3161 y Fm(ha)m(v)m(e)i(the)e(w)m(eak*)i(top)s(ology)-8 b(.)64 b(Equiv)-5 b(alen)m(tly)35 b Fk(\026)2425 3175 y Fi(n)2509 3161 y Fj(!)j Fk(\026)f Fj(2)g Fk(M)2916 3175 y Fi(I)2993 3161 y 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Fm(of)g(con)m(tin)m(uous)g(real)g(v)-5 b(alued)20 b(functions)f(on)i Fk(C)3602 3627 y Fi(K)3691 3613 y Fm(whic)m(h)0 3726 y(con)m(v)m(erge)32 b(p)s(oin)m(t)m(wise)c (to)j Fk(\037)940 3740 y Fi(Z)997 3726 y Fm(.)40 b(Cho)s(ose)30 b(a)g(sequence)g Fk(\026)1884 3741 y Fi(k)1952 3726 y Fj(2)25 b(M)2147 3740 y Fi(I)2217 3726 y Fm(suc)m(h)k(that)i Fk(D)s Fm(\()p Fk(\026)2786 3741 y Fi(k)2828 3726 y Fm(\))26 b(=)2985 3653 y Fb(R)3028 3758 y Fi(C)3078 3769 y Fd(K)3156 3726 y Fk(F)3214 3740 y Fg(O)3292 3726 y Fk(d\026)3394 3741 y Fi(k)3462 3726 y Fj(!)f Fk(D)s Fm(\()p Fk(K)7 b Fm(\))30 b(as)0 3839 y Fk(k)e Fj(!)e(1)p Fm(,)k(and,)f(using)f(the)i (compactness)h(of)f Fj(M)1670 3853 y Fi(I)1710 3839 y Fm(,)g(assume)g(without)e(loss)h(of)h(generalit)m(y)g(that)h Fk(\026)3362 3854 y Fi(k)3434 3839 y Fm(con)m(v)m(erges)g(to)0 3952 y(some)h Fk(\026)284 3966 y Fg(1)386 3952 y Fj(2)c(M)584 3966 y Fi(I)624 3952 y Fm(.)45 b(Then)933 3879 y Fb(R)976 3984 y Fi(C)1026 3995 y Fd(K)1104 3952 y Fk(f)1149 3966 y Fi(j)1200 3952 y Fk(d\026)1302 3967 y Fi(k)1372 3952 y Fj(!)1491 3879 y Fb(R)1534 3984 y Fi(C)1584 3995 y Fd(K)1662 3952 y Fk(f)1707 3966 y Fi(j)1758 3952 y Fk(d\026)1860 3966 y Fg(1)1967 3952 y Fm(as)32 b Fk(j)h Fj(!)28 b(1)p Fm(,)k(and)2595 3879 y Fb(R)2638 3984 y Fi(C)2688 3995 y Fd(K)2767 3952 y Fk(f)2812 3966 y Fi(j)2863 3952 y Fk(d\026)2965 3966 y Fg(1)3067 3952 y Fj(&)c Fk(D)s Fm(\()p Fk(\026)3354 3966 y Fg(1)3428 3952 y Fm(\))33 b(as)f Fk(k)f Fj(!)c(1)p Fm(.)0 4076 y(Since)231 4003 y Fb(R)274 4108 y Fi(C)324 4119 y Fd(K)402 4076 y Fk(f)447 4090 y Fi(j)498 4076 y Fk(d\026)600 4091 y Fi(k)668 4076 y Fj(\025)e Fk(D)s Fm(\()p Fk(\026)932 4091 y Fi(k)975 4076 y Fm(\))f(and)f Fk(D)s Fm(\()p Fk(\026)1372 4091 y Fi(k)1415 4076 y Fm(\))i Fj(!)h Fk(D)s Fm(\()p Fk(K)7 b Fm(\))24 b(as)g Fk(k)k Fj(!)d(1)p Fm(,)h Fk(D)s Fm(\()p Fk(\026)2454 4090 y Fg(1)2528 4076 y Fm(\))g Fj(\025)f Fk(D)s Fm(\()p Fk(K)7 b Fm(\).)38 b(F)-8 b(rom)25 b(the)f(Krein-Milman) 0 4201 y(theorem)32 b(there)f(exists)g(an)f(ergo)s(dic)h(measure)38 b(~)-52 b Fk(\026)26 b Fj(2)g(M)1915 4215 y Fi(I)1987 4201 y Fm(for)k(whic)m(h)g Fk(D)s Fm(\()7 b(~)-52 b Fk(\026)p Fm(\))27 b(=)2714 4128 y Fb(R)2757 4233 y Fi(C)2807 4244 y Fd(K)2885 4201 y Fk(\037)2942 4215 y Fi(Z)3014 4201 y Fk(d)7 b Fm(~)-52 b Fk(\026)26 b Fj(\025)h Fk(D)s Fm(\()p Fk(\026)3408 4215 y Fg(1)3482 4201 y Fm(\),)32 b(and)f(th)m(us)0 4314 y Fk(D)s Fm(\()7 b(~)-52 b Fk(\026)p Fm(\))26 b Fj(\025)e Fk(D)s Fm(\()p Fk(K)7 b Fm(\).)41 b(But)31 b(then)f(from)g(the)g(de\014nition)e(of)j Fk(D)s Fm(\()p Fk(K)7 b Fm(\),)31 b Fk(D)s Fm(\()7 b(~)-52 b Fk(\026)p Fm(\))25 b(=)g Fk(D)s Fm(\()p Fk(K)7 b Fm(\).)p 3855 4314 4 62 v 3859 4256 55 4 v 3859 4314 V 3913 4314 4 62 v 0 4600 a Fl(4)135 b(A)44 b(F)-11 b(amily)45 b(of)h(saturating)g (maps)0 4803 y Fm(Before)34 b(pro)m(ving)e(the)h(theorem,)h(w)m(e)f (outline)f(the)h(case)h(in)d(whic)m(h)h Fk(S)38 b Fm(is)32 b(the)h(Euclidean)e(plane.)47 b(Let)33 b Fk(T)3599 4817 y Fi(j)3669 4803 y Fm(b)s(e)f(the)0 4916 y(usual)25 b(square)i(tiling)d (b)m(y)j(squares)f(of)h(side)e(length)h Fk(j)5 b Fm(.)40 b(W)-8 b(e)28 b(assume)f(that)g(the)g(origin)e(is)g(in)h(the)h(cen)m (ter)g(of)g(a)g(tile)f(of)0 5029 y Fk(T)53 5043 y Fi(j)121 5029 y Fm(for)31 b(all)g Fk(j)5 b Fm(.)44 b(Let)32 b Fk(G)735 5043 y Fi(j)804 5029 y Fm(b)s(e)f(the)g(group)g(of)h (translations)e(whic)m(h)g(\014x)h Fk(T)2391 5043 y Fi(j)2428 5029 y Fm(.)44 b(Let)32 b Fk(U)2723 5043 y Fi(j)2791 5029 y Fm(b)s(e)f(the)g(set)i(of)e(pac)m(kings)g(whic)m(h)1938 5309 y(5)p eop %%Page: 6 6 6 5 bop 0 191 a Fm(are)32 b(not)f(saturated)h(in)e(the)h(square)g(of)h Fk(T)1430 205 y Fi(j)1498 191 y Fm(that)g(con)m(tains)f(the)h(origin.) 42 b(T)-8 b(o)31 b(b)s(e)g(precise,)g Fj(P)k(2)26 b Fk(U)3338 205 y Fi(j)3406 191 y Fm(if)k(and)h(only)f(if)0 304 y(there)i(is)f(a)i (\014nite)e(pac)m(king)h Fj(F)1039 318 y Fh(1)1106 304 y Fj(\032)c(P)40 b Fm(and)31 b(a)i(\014nite)d(pac)m(king)i Fj(F)2197 318 y Fh(2)2269 304 y Fm(suc)m(h)g(that)g Fj(F)2739 318 y Fh(2)2811 304 y Fm(has)g(more)g(elemen)m(ts)g(than)g Fj(F)3856 318 y Fh(1)3896 304 y Fm(,)0 417 y(for)27 b(all)f Fk(g)s(K)32 b Fj(2)25 b(F)565 431 y Fh(1)618 417 y Fj([)13 b(F)757 431 y Fh(2)797 417 y Fm(,)28 b Fk(g)s(K)34 b Fm(is)26 b(con)m(tained)h(in)f(the)h(square)f(of)i Fk(T)2195 431 y Fi(j)2258 417 y Fm(that)g(con)m(tains)f(the)g(origin)e(and)i(\()p Fj(P)21 b(\000)13 b(F)3654 431 y Fh(1)3694 417 y Fm(\))g Fj([)g(F)3881 431 y Fh(2)0 530 y Fm(is)29 b(a)i(pac)m(king.)141 643 y(F)-8 b(or)31 b(eac)m(h)h Fk(j)5 b Fm(,)31 b(there)g(exists)f(a)g (Borel)h(map)f(\010)1676 657 y Fi(j)1737 643 y Fm(:)c Fk(C)1853 657 y Fi(K)1946 643 y Fj(!)f Fk(C)2127 657 y Fi(K)2226 643 y Fm(satisfying:)111 810 y(1.)46 b Fk(g)s(K)33 b Fj(2)25 b(P)35 b Fm(and)27 b Fk(g)s(K)35 b Fm(in)m(tersects)28 b(an)f(edge)i(of)e Fk(T)1780 824 y Fi(j)1845 810 y Fm(if)f(and)h(only)g (if)g Fk(g)s(K)32 b Fj(2)25 b Fm(\010)2684 824 y Fi(j)2720 810 y Fm(\()p Fj(P)7 b Fm(\))29 b(and)e Fk(g)s(K)35 b Fm(in)m(tersects)28 b(an)f(edge)227 923 y(of)k Fk(T)384 937 y Fi(j)420 923 y Fm(;)111 1103 y(2.)46 b(F)-8 b(or)32 b(an)m(y)g(giv)m(en)f(square)g(in)f Fk(T)1249 1117 y Fi(j)1285 1103 y Fm(,)i(the)f(n)m(um)m(b)s(er)f(of)h(b)s(o)s(dies)e(of) j(\010)2390 1117 y Fi(j)2426 1103 y Fm(\()p Fj(P)7 b Fm(\))32 b(that)g(are)f(con)m(tained)h(in)e(that)h(square)227 1216 y(is)f(as)g(large)h(as)f(p)s(ossible)e(giv)m(en)j(the)f(ab)s(o)m (v)m(e)i(constrain)m(t;)111 1396 y(3.)46 b(\010)293 1410 y Fi(j)360 1396 y Fm(comm)m(utes)31 b(with)e Fk(G)1065 1410 y Fi(j)1102 1396 y Fm(.)0 1563 y(The)h(idea)g(is)f(that)i(\010)733 1577 y Fi(j)800 1563 y Fm(replaces)f Fj(P)j(2)25 b Fk(C)1391 1577 y Fi(K)1490 1563 y Fm(with)k(a)i(\010)1839 1577 y Fi(j)1875 1563 y Fm(\()p Fj(P)7 b Fm(\))31 b(that)g(is)f(saturated)h (with)e(resp)s(ect)h(to)h(the)g(squares)f(of)0 1676 y Fk(T)53 1690 y Fi(j)90 1676 y Fm(.)39 b(Because)27 b(\010)563 1690 y Fi(j)625 1676 y Fm(comm)m(utes)g(with)e Fk(G)1322 1690 y Fi(j)1359 1676 y Fm(,)i(it)f(induces)e(a)j(map)e(\010)2152 1690 y Fi(j)t Fg(\003)2249 1676 y Fm(:)h Fk(M)35 b Fj(!)25 b Fk(M)36 b Fm(suc)m(h)26 b(that)h(if)e Fk(\026)g Fj(2)f Fk(M)3389 1690 y Fi(I)3456 1676 y Fm(then)h(\010)3724 1690 y Fi(j)t Fg(\003)3796 1676 y Fm(\()p Fk(\026)p Fm(\))0 1789 y(is)e(in)m(v)-5 b(arian)m(t)23 b(under)g Fk(G)783 1803 y Fi(j)820 1789 y Fm(.)38 b(Since)23 b Fk(G)1185 1803 y Fi(j)1246 1789 y Fm(is)g(co)s(compact)j(and)d(discrete)h(w)m(e)g (can)g(a)m(v)m(erage)j(\010)2948 1803 y Fi(j)t Fg(\003)3019 1789 y Fm(\()p Fk(\026)p Fm(\))e(o)m(v)m(er)g(a)f(fundamen)m(tal)0 1902 y(domain)34 b(for)h Fk(G)543 1916 y Fi(j)616 1902 y Fm(in)f Fk(G)h Fm(to)h(obtain)f(a)h(measure)f(that)h(is)e(in)m(v)-5 b(arian)m(t)35 b(under)f(all)g(of)h Fk(G)p Fm(.)56 b(This)34 b(new)h(measure)g(will)0 2015 y(ha)m(v)m(e)h(a)g(densit)m(y)e(greater)i (than)f Fk(\026)g Fm(unless)e Fk(\026)p Fm(\()p Fk(U)1658 2029 y Fi(j)1695 2015 y Fm(\))h(=)e(0.)56 b(Since)34 b Fj([)2296 2029 y Fi(j)2332 2015 y Fk(U)2394 2029 y Fi(j)2464 2015 y Fm(=)e Fk(U)45 b Fm(is)35 b(the)g(set)g(of)h(all)e (non-completely)0 2128 y(saturated)26 b(pac)m(kings,)g(this)f(sho)m(ws) g(that)h(an)f(optimally)f(dense)h(measure)h(m)m(ust)f(satisfy)g Fk(\026)p Fm(\()p Fk(U)10 b Fm(\))26 b(=)e(0)i(whic)m(h)f(pro)m(v)m(es) 0 2241 y(the)31 b(theorem.)141 2354 y(By)d(a)h(fundamen)m(tal)e(domain) g Fk(F)41 b Fm(of)28 b(a)g(subgroup)e Fk(H)33 b(<)25 b(G)p Fm(,)k(w)m(e)f(shall)e(mean)j(a)f(connected)h(set)f(in)f Fk(S)33 b Fm(equal)27 b(to)0 2467 y(the)k(closure)e(of)i(its)f(in)m (terior)f(suc)m(h)h(that)h Fj(f)p Fk(hF)13 b Fj(j)i Fk(h)27 b Fj(2)e Fk(H)7 b Fj(g)30 b Fm(is)g(a)h(tiling)d(b)m(y)i Fk(F)13 b Fm(.)141 2579 y(F)-8 b(or)29 b(the)g(general)f(case,)i(let)f Fj(f)p Fk(G)1230 2593 y Fi(j)1267 2579 y Fj(g)1312 2546 y Fg(1)1312 2605 y Fi(j)t Fh(=0)1468 2579 y Fm(b)s(e)e(a)i(sequence)g (of)f(discrete)h(co)s(compact)h(subgroups)c(of)j Fk(G)g Fm(suc)m(h)f(that)0 2692 y(there)40 b(exist)g(fundamen)m(tal)f(domains) g Fk(F)1429 2706 y Fi(j)1506 2692 y Fm(\(in)g Fk(S)5 b Fm(\))40 b(for)g Fk(G)2013 2706 y Fi(j)2090 2692 y Fm(with)f Fk(B)2376 2706 y Fi(j)2454 2692 y Fj(\032)i Fk(F)2624 2706 y Fi(j)2661 2692 y Fm(.)70 b(W)-8 b(e)41 b(will)d(also)i(assume)g(that)g Fk(F)3884 2706 y Fi(j)0 2805 y Fm(and)34 b Fk(G)252 2819 y Fi(j)323 2805 y Fm(ha)m(v)m(e)h(b)s (een)f(c)m(hosen)h(so)f(that)h(for)f(all)f Fk(g)j Fj(2)31 b Fk(\031)1865 2772 y Fg(\000)p Fh(1)1960 2805 y Fm(\()p Fk(F)2053 2819 y Fi(j)2090 2805 y Fm(\),)36 b Fk(g)2232 2772 y Fg(\000)p Fh(1)2358 2805 y Fj(2)c Fk(\031)2506 2772 y Fg(\000)p Fh(1)2600 2805 y Fm(\()p Fk(F)2693 2819 y Fi(j)2731 2805 y Fm(\).)52 b(F)-8 b(or)35 b(the)g(Euclidean)d(case,) 37 b(w)m(e)0 2918 y(could,)e(for)f(example,)h(let)g Fk(G)1009 2932 y Fi(j)1080 2918 y Fm(b)s(e)f(the)g(cubic)g(lattice)g(generated)i (b)m(y)e(the)h(translations)e Fk(\034)3141 2932 y Fi(i;j)3253 2918 y Fm(:=)f(\()p Fk(x)3468 2932 y Fh(1)3508 2918 y Fk(;)15 b(:::;)g(x)3715 2932 y Fi(n)3763 2918 y Fm(\))32 b Fj(!)0 3031 y Fm(\()p Fk(x)87 3045 y Fh(1)127 3031 y Fk(;)15 b(:::;)g(x)334 3045 y Fi(i)385 3031 y Fm(+)22 b Fk(j;)15 b(x)607 3045 y Fi(i)p Fh(+1)726 3031 y Fk(;)g(:::;)g(x)933 3045 y Fi(n)981 3031 y Fm(\))34 b(for)e(1)e Fj(\024)f Fk(i)g Fj(\024)g Fk(n)p Fm(.)47 b(Then)32 b(w)m(e)h(could)f(c)m(ho)s (ose)i Fk(F)2623 3045 y Fi(j)2692 3031 y Fm(to)g(b)s(e)e(the)h(cub)s(e) f(of)h(side)e(length)i Fk(j)0 3144 y Fm(whose)28 b(cen)m(ter)i(is)e (the)h(origin)e(and)h(whose)g(faces)i(are)f(parallel)e(to)i(the)g(co)s (ordinate)g(planes.)39 b(F)-8 b(or)29 b(the)g(h)m(yp)s(erb)s(olic)0 3257 y(case,)j(w)m(e)f(refer)f(to)h([FKK,)f(Theorem)g(4.1])i(for)e(the) h(existence)g(of)f Fj(f)p Fk(G)2412 3271 y Fi(j)2450 3257 y Fj(g)2495 3224 y Fg(1)2495 3282 y Fi(j)t Fh(=0)2622 3257 y Fm(.)141 3370 y(F)-8 b(or)41 b Fj(P)49 b(2)40 b Fk(C)592 3384 y Fi(K)701 3370 y Fm(and)f Fk(F)54 b Fj(\032)41 b Fk(S)5 b Fm(,)42 b(let)e Fj(P)35 b(\003)27 b Fk(F)53 b Fm(b)s(e)39 b(the)h(pac)m(king)g(consisting)f(of)h(all)e (elemen)m(ts)j Fk(g)s(K)48 b Fj(2)41 b(P)48 b Fm(suc)m(h)0 3483 y(that)36 b(the)f(in)m(terior)f(of)h Fk(g)s(K)42 b Fm(in)m(tersects)36 b Fk(F)48 b Fm(non)m(trivially)-8 b(.)52 b(Also)35 b(let)g Fk(@)5 b(F)49 b Fm(denote)35 b(the)h(b)s(oundary)d(of)i Fk(F)13 b Fm(,)36 b(i.e.)55 b(the)0 3596 y(in)m(tersection)36 b(of)h Fk(F)50 b Fm(with)35 b(the)i(closure)e(of)i(its)f(complemen)m(t.)60 b(Let)37 b Fk(X)2446 3610 y Fi(j)2518 3596 y Fm(=)e Fj(f)p Fk(x)h Fj(2)f Fk(C)2918 3610 y Fi(K)2987 3596 y Fj(j)p Fk(x)24 b Fj(\003)h Fk(@)5 b(F)3269 3610 y Fi(j)3342 3596 y Fm(=)35 b Fk(x)p Fj(g)i Fm(with)e(the)0 3709 y(subspace)j(top)s(ology)-8 b(.)64 b(F)-8 b(or)39 b(an)m(y)g Fk(x)f Fj(2)f Fk(X)1426 3723 y Fi(j)1463 3709 y Fm(,)j(a)f(\014lling)c Fk(f)48 b Fm(for)37 b Fk(x)h Fm(is)g(a)g(pac)m(king)g(b)m(y)g Fk(K)45 b Fm(suc)m(h)38 b(that)g Fk(f)d Fj(\003)26 b Fk(@)5 b(F)3660 3723 y Fi(j)3735 3709 y Fm(=)38 b Fk(x)p Fm(,)0 3821 y Fk(f)29 b Fj(\003)21 b Fk(F)198 3835 y Fi(j)260 3821 y Fm(=)k Fk(f)40 b Fm(and)30 b(the)g(n)m(um)m(b)s(er)f (of)i(elemen)m(ts)g(of)f Fk(f)40 b Fm(is)29 b(as)i(large)f(as)h(p)s (ossible)d(giv)m(en)i(these)h(constrain)m(ts.)0 3989 y Fn(Lemma)i(4.1.)46 b Ff(F)-7 b(or)36 b(e)-5 b(ach)36 b Fk(j)g Fj(\025)29 b Fm(0)p Ff(,)36 b(ther)-5 b(e)37 b(exists)e(a)h(Bor)-5 b(el)36 b(map)g Fk(\036)2303 4003 y Fi(j)2370 3989 y Fm(:)30 b Fk(X)2500 4003 y Fi(j)2567 3989 y Fj(!)g Fk(C)2753 4003 y Fi(K)2856 3989 y Ff(such)36 b(that)g(for)g(e)-5 b(ach)36 b Fk(x)30 b Fj(2)g Fk(X)3857 4003 y Fi(j)3893 3989 y Ff(,)0 4102 y Fk(\036)54 4116 y Fi(j)91 4102 y Fm(\()p Fk(x)p Fm(\))j Ff(is)f(a)h(\014l)5 b(ling)33 b(for)g Fk(x)p Ff(.)0 4270 y(Pr)-5 b(o)g(of.)47 b Fm(Let)d Fk(j)53 b Fj(\025)47 b Fm(0)d(b)s(e)f(\014xed)g(and)g(let)h Fk(X)55 b Fm(=)47 b Fk(X)1786 4284 y Fi(j)1823 4270 y Fm(.)80 b(Giv)m(en)44 b Fk(x)j Fj(2)g Fk(X)7 b Fm(,)48 b Fk(f)k Fm(a)45 b(\014lling)40 b(for)k Fk(x)f Fm(and)g Fk(m)48 b(>)f Fm(0,)g(let)0 4383 y Fk(Y)20 b Fm(\()p Fk(f)5 b(;)15 b(m)p Fm(\))30 b(=)g Fj(f)p Fk(x)541 4350 y Fg(0)595 4383 y Fj(2)f Fk(X)7 b Fj(j)34 b Fm(there)f(exists)g(a)h (\014lling)c Fk(f)1704 4350 y Fg(0)1760 4383 y Fm(for)j Fk(x)1954 4350 y Fg(0)2010 4383 y Fm(with)f Fk(D)2295 4397 y Fi(K)2364 4383 y Fm(\()p Fk(f)2454 4350 y Fg(0)2477 4383 y Fk(;)15 b(f)10 b Fm(\))30 b Fk(<)g Fm(2)2783 4350 y Fg(\000)p Fi(m)2905 4383 y Fj(g)p Fm(.)49 b(F)-8 b(rom)34 b(the)f(compactness)0 4496 y(of)c Fk(F)160 4510 y Fi(j)225 4496 y Fm(it)g(follo)m(ws)e(that)j(giv)m(en)e Fk(m)d(>)g Fm(0,)30 b(there)f(exists)f(a)h(\014nite)f(n)m(um)m(b)s(er)f(of)i(pac)m (kings)f Fk(x)2978 4510 y Fh(1)3018 4496 y Fk(;)15 b(:::;)g(x)3225 4510 y Fi(p)3291 4496 y Fj(2)25 b Fk(X)36 b Fm(and)28 b(\014llings)0 4609 y Fk(f)45 4624 y Fi(k)117 4609 y Fm(for)j Fk(x)309 4624 y Fi(k)382 4609 y Fm(suc)m(h)f(that)h Fk(X)h Fm(=)987 4540 y Fb(S)1063 4564 y Fi(p)1063 4638 y(k)r Fh(=1)1211 4609 y Fk(Y)20 b Fm(\()p Fk(f)1364 4624 y Fi(k)1406 4609 y Fk(;)15 b(m)p Fm(\).)141 4722 y(Th)m(us)21 b(for)h(a)h(giv)m(en)f(in)m(teger)g Fk(m)j(>)g Fm(0,)g(there)d(exists)g (a)g(\014nite)f(partition)g Fj(f)p Fk(A)2621 4737 y Fi(m;k)2746 4722 y Fj(g)2791 4682 y Fi(r)2823 4690 y Fd(m)2791 4751 y Fi(k)r Fh(=1)2947 4722 y Fm(of)h Fk(X)30 b Fm(suc)m(h)21 b(that)i(eac)m(h)g Fk(A)3796 4737 y Fi(m;k)0 4834 y Fm(is)i(Borel)i (and)e(con)m(tained)i(in)e(some)h Fk(Y)20 b Fm(\()p Fk(f)5 b(;)15 b(m)p Fm(\).)40 b(W)-8 b(e)27 b(will)d(assume)i(that)h Fj(f)p Fk(A)2544 4849 y Fi(m)p Fh(+1)p Fi(;k)2759 4834 y Fj(g)2804 4787 y Fi(r)2836 4796 y Fd(m)p Fe(+1)2804 4864 y Fi(k)r Fh(=1)3003 4834 y Fm(re\014nes)e Fj(f)p Fk(A)3395 4850 y Fi(m;k)3516 4831 y Fa(0)3543 4834 y Fj(g)3588 4795 y Fi(r)3620 4803 y Fd(m)3588 4864 y Fi(k)3627 4845 y Fa(0)3649 4864 y Fh(=1)3770 4834 y Fm(\(i.e.)0 4947 y(for)g(ev)m(ery)h Fk(A)437 4962 y Fi(m)p Fh(+1)p Fi(;k)652 4947 y Fm(,)h(there)e(exists)g(an)g Fk(A)1366 4963 y Fi(m;k)1487 4944 y Fa(0)1539 4947 y Fm(suc)m(h)g(that)h Fk(A)1999 4962 y Fi(m)p Fh(+1)p Fi(;k)2239 4947 y Fj(\032)f Fk(A)2403 4963 y Fi(m;k)2524 4944 y Fa(0)2550 4947 y Fm(\).)39 b(F)-8 b(or)26 b(eac)m(h)h Fk(m)e(>)g Fm(0)h(and)e(1)i Fj(\024)f Fk(k)j Fj(\024)d Fk(r)3829 4961 y Fi(m)3896 4947 y Fm(,)0 5060 y(c)m(ho)s(ose)31 b Fk(a)p Fm(\()p Fk(m;)15 b(k)s Fm(\))27 b Fj(2)e Fk(A)757 5075 y Fi(m;k)912 5060 y Fm(and)30 b Fk(f)10 b Fm(\()p Fk(m;)15 b(k)s Fm(\))30 b(a)h(\014lling)d(for)i Fk(a)p Fm(\()p Fk(m;)15 b(k)s Fm(\))31 b(so)g(that)g(the)f(follo)m(wing)f(are)i(satis\014ed:)1938 5309 y(6)p eop %%Page: 7 7 7 6 bop 111 191 a Fm(1.)46 b Fj(f)p Fk(a)p Fm(\()p Fk(m;)15 b(k)s Fm(\))p Fj(g)605 151 y Fi(r)637 159 y Fd(m)605 220 y Fi(k)r Fh(=1)765 191 y Fj(\032)25 b(f)p Fk(a)p Fm(\()p Fk(m)c Fm(+)f(1)p Fk(;)15 b(k)1316 158 y Fg(0)1341 191 y Fm(\))p Fj(g)1421 143 y Fi(r)1453 152 y Fd(m)p Fe(+1)1421 221 y Fi(k)1460 202 y Fa(0)1482 221 y Fh(=1)1623 191 y Fm(for)30 b(all)g Fk(m)25 b(>)g Fm(0.)111 379 y(2.)46 b(F)-8 b(or)34 b(ev)m(ery)g Fk(m)c(>)f Fm(0,)35 b(1)30 b Fj(\024)g Fk(k)j Fj(\024)c Fk(r)1348 393 y Fi(m)1415 379 y Fm(,)34 b Fk(a)c Fj(2)f Fk(A)1710 394 y Fi(m;k)1868 379 y Fm(there)k(is)g(a)g(\014lling)d Fk(f)43 b Fm(for)33 b Fk(a)g Fm(suc)m(h)g(that)g Fk(d)3296 393 y Fi(K)3365 379 y Fm(\()p Fk(f)5 b(;)15 b(f)10 b Fm(\()p Fk(m;)15 b(k)s Fm(\)\))31 b Fk(<)227 491 y Fm(2)272 459 y Fg(\000)p Fi(m)p Fh(+1)484 491 y Fm(.)67 b(This)37 b(is)h(p)s(ossible)f(since)h (for)h(eac)m(h)h Fk(A)1903 506 y Fi(m;k)2067 491 y Fm(there)f(exists)g Fk(x)h Fj(2)f Fk(X)46 b Fm(and)39 b(a)g(\014lling)d Fk(f)49 b Fm(for)39 b Fk(x)f Fm(with)227 604 y Fk(A)295 619 y Fi(m;k)445 604 y Fj(\032)25 b Fk(Y)20 b Fm(\()p Fk(f)5 b(;)15 b(m)p Fm(\).)111 792 y(3.)46 b(If)30 b(for)g(some)h Fk(m)f Fm(and)g Fk(k)s(;)15 b(k)1112 759 y Fg(0)1136 792 y Fm(,)31 b Fk(A)1260 807 y Fi(m)p Fh(+1)p Fi(;k)1471 788 y Fa(0)1522 792 y Fj(\032)25 b Fk(A)1686 807 y Fi(m;k)1811 792 y Fm(,)31 b(then)f Fk(d)2121 806 y Fi(K)2189 792 y Fm(\()p Fk(f)10 b Fm(\()p Fk(m)20 b Fm(+)g(1)p Fk(;)15 b(k)2640 759 y Fg(0)2665 792 y Fm(\))p Fk(;)g(f)10 b Fm(\()p Fk(m;)15 b(k)s Fm(\)\))26 b Fk(<)f Fm(2)3237 759 y Fg(\000)p Fi(m)p Fh(+1)3450 792 y Fm(.)141 980 y(W)-8 b(e)33 b(de\014ne)d(a)h(map)g Fk(\013)899 994 y Fi(m)993 980 y Fm(:)26 b Fk(X)34 b Fj(!)27 b Fk(C)1336 994 y Fi(K)1435 980 y Fm(b)m(y)k Fk(\013)1620 994 y Fi(m)1687 980 y Fm(\()p Fk(x)p Fm(\))c(=)g Fk(f)10 b Fm(\()p Fk(m;)15 b(k)s Fm(\))31 b(if)f Fk(x)d Fj(2)f Fk(A)2578 995 y Fi(m;k)2703 980 y Fm(.)43 b(Eac)m(h)32 b Fk(\013)3056 994 y Fi(m)3154 980 y Fm(is)e(Borel)h(\(since)g(eac)m(h)0 1093 y Fk(A)68 1108 y Fi(m;k)225 1093 y Fm(is)h(Borel\).)48 b(W)-8 b(e)34 b(claim)d(that)j Fj(f)p Fk(\013)1344 1107 y Fi(m)1411 1093 y Fj(g)1456 1060 y Fg(1)1456 1117 y Fi(m)p Fh(=1)1645 1093 y Fm(con)m(v)m(erges)h(p)s(oin)m(t)m(wise)c(as)i(to)g(a)g(map)g Fk(\036)p Fm(.)47 b(Let)33 b Fk(x)c Fj(2)g Fk(X)7 b Fm(.)48 b(Then)31 b(b)m(y)0 1205 y(the)g(third)d(condition)h(ab)s(o)m(v)m(e,)j (if)d Fk(m)c(<)g(m)1430 1172 y Fg(0)1484 1205 y Fm(then)877 1410 y Fk(D)952 1424 y Fi(K)1020 1410 y Fm(\()p Fk(\013)1113 1424 y Fi(m)1180 1410 y Fm(\()p Fk(x)p Fm(\))p Fk(;)15 b(\013)1400 1425 y Fi(m)1462 1406 y Fa(0)1490 1410 y Fm(\()p Fk(x)p Fm(\)\))27 b Fj(\024)e Fm(\006)1836 1371 y Fi(m)1898 1348 y Fa(0)1920 1371 y Fg(\000)p Fh(1)1836 1439 y Fi(k)r Fh(=)p Fi(m)2030 1410 y Fm(2)2075 1372 y Fg(\000)p Fi(k)r Fh(+1)2288 1410 y Fj(\024)g Fm(\006)2450 1372 y Fg(1)2450 1433 y Fi(k)r Fh(=)p Fi(m)2625 1410 y Fm(2)2670 1372 y Fg(\000)p Fi(k)r Fh(+1)2883 1410 y Fj(!)g Fm(0)761 b(\(6\))0 1614 y(as)33 b Fk(m)c Fj(!)h(1)p Fm(.)48 b(So)33 b(the)g(sequence)g Fj(f)p Fk(\013)1278 1628 y Fi(m)1345 1614 y Fm(\()p Fk(x)p Fm(\))p Fj(g)1512 1581 y Fg(1)1512 1638 y Fi(m)p Fh(=1)1703 1614 y Fm(is)f(Cauc)m(h)m(y)h (and)f(therefore)i(con)m(v)m(erges)h(in)c Fk(C)3272 1628 y Fi(K)3373 1614 y Fm(to)j(an)f(elemen)m(t)0 1727 y Fk(\036)p Fm(\()p Fk(x)p Fm(\).)41 b(Since)30 b(all)f(the)i(maps)f Fk(\013)1059 1741 y Fi(m)1156 1727 y Fm(are)g(Borel,)h Fk(\036)g Fm(m)m(ust)f(b)s(e)f(Borel,)i(to)s(o.)141 1840 y(W)-8 b(e)35 b(claim)d(that)i Fk(\036)p Fm(\()p Fk(x)p Fm(\))g(is)f(a)g(\014lling)e(for)i Fk(x)g Fm(for)g(eac)m(h)i Fk(x)30 b Fj(2)g Fk(X)7 b Fm(.)50 b(So)33 b(let)g Fk(x)e Fj(2)e Fk(X)41 b Fm(and)33 b(for)g(eac)m(h)h Fk(m)d(>)e Fm(0)34 b(c)m(ho)s(ose)0 1953 y(a)39 b(\014lling)d Fk(f)390 1967 y Fi(m)495 1953 y Fm(for)j Fk(x)f Fm(suc)m(h)h(that)g Fk(d)1199 1967 y Fi(K)1268 1953 y Fm(\()p Fk(f)1348 1967 y Fi(m)1415 1953 y Fk(;)15 b(\013)1513 1967 y Fi(m)1580 1953 y Fm(\()p Fk(x)p Fm(\)\))40 b Fk(<)f Fm(2)1932 1920 y Fg(\000)p Fi(m)p Fh(+1)2144 1953 y Fm(.)66 b(By)40 b(the)f(de\014nition)d(of)j Fk(A)3144 1968 y Fi(m;k)3308 1953 y Fm(and)f(the)h(second)0 2066 y(condition)h(of)h Fk(f)10 b Fm(\()p Fk(m;)15 b(k)s Fm(\))42 b(listed)e(ab)s(o)m(v)m(e,)45 b(suc)m(h)c(an)g Fk(f)1809 2080 y Fi(m)1917 2066 y Fm(exists)f(for)h (all)g Fk(m)i(>)g Fm(0.)74 b(By)41 b(the)h(triangle)e(inequalit)m(y)-8 b(,)0 2178 y Fk(d)47 2192 y Fi(K)116 2178 y Fm(\()p Fk(f)196 2192 y Fi(m)262 2178 y Fk(;)15 b(\036)p Fm(\()p Fk(x)p Fm(\)\))29 b Fj(\024)d Fm(2)684 2146 y Fg(\000)p Fi(m)p Fh(+1)917 2178 y Fm(+)21 b(\006)1075 2146 y Fg(1)1075 2206 y Fi(k)r Fh(=)p Fi(m)1234 2178 y Fm(2)1279 2146 y Fg(\000)p Fi(k)r Fh(+1)1499 2178 y Fm(whic)m(h)30 b(go)s(es)i(to)g (zero)g(as)f Fk(m)g Fm(go)s(es)h(to)g(in\014nit)m(y)-8 b(.)42 b(Hence)32 b(the)g(sequence)0 2291 y Fj(f)p Fk(f)90 2305 y Fi(m)157 2291 y Fj(g)202 2305 y Fi(m>)p Fh(0)384 2291 y Fm(con)m(v)m(erges)c(to)f Fk(\036)p Fm(\()p Fk(x)p Fm(\).)40 b(Since)24 b Fk(f)1412 2305 y Fi(m)1504 2291 y Fm(is)h(a)h(\014lling)d(for)j Fk(x)f Fm(for)h(eac)m(h)h Fk(m)p Fm(,)f(it)g(follo)m(ws)e(that)j Fk(\036)p Fm(\()p Fk(x)p Fm(\))f(is)f(also)h(a)g(\014lling)0 2404 y(for)k Fk(x)p Fm(.)41 b(No)m(w)31 b(w)m(e)g(are)f(done.)p 3855 2517 4 62 v 3859 2459 55 4 v 3859 2517 V 3913 2517 4 62 v 0 2705 a(W)-8 b(e)36 b(c)m(ho)s(ose)g(maps)f(\010)763 2719 y Fi(j)834 2705 y Fm(satisfying)f(the)i(conclusion)d(of)j(the)f (ab)s(o)m(v)m(e)h(lemma.)55 b(De\014ne)35 b(\010)3077 2719 y Fi(j)3146 2705 y Fm(:)f Fk(C)3270 2719 y Fi(K)3371 2705 y Fj(!)g Fk(C)3561 2719 y Fi(K)3664 2705 y Fm(b)m(y)h(the)0 2818 y(follo)m(wing)29 b(prop)s(erties:)111 3005 y(1.)46 b(for)30 b(an)m(y)h Fj(P)i(2)25 b Fk(C)785 3019 y Fi(K)853 3005 y Fm(,)31 b(\010)975 3019 y Fi(j)1011 3005 y Fm(\()p Fj(P)7 b Fm(\))22 b Fj(\003)e Fk(F)1296 3019 y Fi(j)1359 3005 y Fm(=)k Fk(\036)1508 3019 y Fi(j)1545 3005 y Fm(\()p Fj(P)k(\\)20 b Fk(@)5 b(F)1863 3019 y Fi(j)1901 3005 y Fm(\))20 b Fj(\003)h Fk(F)2080 3019 y Fi(j)2117 3005 y Fm(;)111 3193 y(2.)46 b(for)30 b(an)m(y)h Fk(g)e Fj(2)c Fk(G)767 3207 y Fi(j)834 3193 y Fm(and)30 b Fj(P)j(2)25 b Fk(C)1258 3207 y Fi(K)1326 3193 y Fm(,)31 b(\010)1448 3207 y Fi(j)1484 3193 y Fm(\()p Fk(g)s Fj(P)7 b Fm(\))27 b(=)e Fk(g)s Fm(\010)1905 3207 y Fi(j)1942 3193 y Fm(\()p Fj(P)7 b Fm(\).)0 3381 y(It)40 b(should)e(b)s(e)h(clear)h(from)f(this)g (de\014nition)e(that)k(for)e(eac)m(h)i Fk(j)5 b Fm(,)43 b(\010)2335 3395 y Fi(j)2411 3381 y Fm(is)c(Borel.)69 b(Let)40 b Fk(\026)h Fj(2)g Fk(M)3275 3395 y Fi(I)3354 3381 y Fm(b)s(e)f(giv)m(en.)69 b(Let)0 3494 y Fk(\026)55 3461 y Fg(0)55 3519 y Fi(j)126 3494 y Fm(=)34 b(\010)297 3508 y Fi(j)t Fg(\003)368 3494 y Fm(\()p Fk(\026)p Fm(\),)k(i.e.)57 b(for)36 b(ev)m(ery)g(Borel)g(set)g Fk(E)k Fj(\032)34 b Fk(C)1790 3508 y Fi(K)1859 3494 y Fm(,)j Fk(\026)1976 3461 y Fg(0)1976 3519 y Fi(j)2012 3494 y Fm(\()p Fk(E)5 b Fm(\))36 b(=)e Fk(\026)p Fm(\(\010)2451 3455 y Fg(\000)p Fh(1)2451 3521 y Fi(j)2545 3494 y Fm(\()p Fk(E)5 b Fm(\)\).)58 b(By)36 b(the)g(ab)s(o)m(v)m(e,)j Fk(\026)3460 3461 y Fg(0)3460 3519 y Fi(j)3532 3494 y Fm(is)c(a)h(Borel)0 3606 y(probabilit)m(y)28 b(measure)i(that)h(is)f(in)m(v)-5 b(arian)m(t)29 b(under)g Fk(G)1818 3620 y Fi(j)1855 3606 y Fm(.)41 b(Let)31 b Fk(\026)2139 3620 y Fi(j)2200 3606 y Fj(2)25 b Fk(M)2374 3620 y Fi(I)2444 3606 y Fm(b)s(e)30 b(de\014ned)f(b)m(y)1133 3943 y Fk(\026)1188 3957 y Fi(j)1225 3943 y Fm(\()p Fk(E)5 b Fm(\))26 b(=)1611 3881 y(1)p 1499 3922 270 4 v 1499 4005 a Fk(\025)1552 4019 y Fi(S)1603 4005 y Fm(\()p Fk(F)1696 4019 y Fi(j)1733 4005 y Fm(\))1793 3819 y Fb(Z)1844 4025 y Fi(\031)1887 4006 y Fa(\000)p Fe(1)1969 4025 y Fh(\()p Fi(F)2041 4035 y Fd(j)2074 4025 y Fh(\))2121 3943 y Fk(\026)2176 3905 y Fg(0)2176 3965 y Fi(j)2212 3943 y Fm(\()p Fk(g)2293 3905 y Fg(\000)p Fh(1)2388 3943 y Fk(E)5 b Fm(\))15 b Fk(d\025)2610 3957 y Fi(G)2671 3943 y Fm(\()p Fk(g)s Fm(\))1018 b(\(7\))0 4208 y(for)39 b(an)m(y)g(Borel)g Fk(E)44 b Fj(\032)39 b Fk(C)864 4222 y Fi(K)933 4208 y Fm(.)66 b(It)39 b(is)e(easy)j(to)f (sho)m(w)g(that)h Fk(\026)2053 4222 y Fi(j)2128 4208 y Fm(is)e Fk(G)p Fm(-in)m(v)-5 b(arian)m(t)38 b(and)h(therefore)g (really)f(is)f(in)h Fk(M)3856 4222 y Fi(I)3896 4208 y Fm(.)0 4321 y(The)27 b(next)h(t)m(w)m(o)h(lemmas)e(will)e(pro)m(vide)h (to)s(ols)i(for)f(calculating)g Fk(D)s Fm(\()p Fk(\026)p Fm(\))h(and)f Fk(D)s Fm(\()p Fk(\026)2754 4335 y Fi(j)2790 4321 y Fm(\))h(with)e(resp)s(ect)i(to)g(the)g(relativ)m(e)0 4434 y(densit)m(y)i(within)d Fk(F)654 4448 y Fi(j)691 4434 y Fm(.)0 4621 y Fn(Lemma)33 b(4.2.)1258 4851 y Fk(D)s Fm(\()p Fk(\026)1426 4865 y Fi(j)1462 4851 y Fm(\))26 b(=)1619 4727 y Fb(Z)1669 4933 y Fi(C)1719 4944 y Fd(K)1807 4789 y Fk(\025)1860 4803 y Fi(S)1911 4789 y Fm(\()p Fk(c)p Fm(\()p Fj(P)7 b Fm(\))22 b Fj(\\)e Fk(F)2286 4803 y Fi(j)2323 4789 y Fm(\))p 1807 4830 552 4 v 1948 4913 a Fk(\025)2001 4927 y Fi(S)2052 4913 y Fm(\()p Fk(F)2145 4927 y Fi(j)2182 4913 y Fm(\))2384 4851 y Fk(d\026)2486 4813 y Fg(0)2486 4873 y Fi(j)2522 4851 y Fm(\()p Fj(P)7 b Fm(\))1143 b(\(8\))1938 5309 y(7)p eop %%Page: 8 8 8 7 bop 0 191 a Ff(Pr)-5 b(o)g(of.)47 b Fm(Let)30 b Fk(Z)i Fm(=)25 b Fj(fP)33 b(2)25 b Fk(C)929 205 y Fi(K)998 191 y Fj(jO)j(2)d Fk(c)p Fm(\()p Fj(P)7 b Fm(\))p Fj(g)p Fm(.)42 b(Let)31 b Fk(\037)1720 205 y Fi(Z)1806 191 y Fm(denote)f(the)g(c)m(haracteristic)h(function)d(of)i Fk(Z)7 b Fm(.)40 b(By)31 b(de\014nition)0 304 y(of)g(densit)m(y)e(and)h (of)g Fk(\026)752 318 y Fi(j)789 304 y Fm(,)837 621 y Fk(D)s Fm(\()p Fk(\026)1005 635 y Fi(j)1041 621 y Fm(\))84 b(=)e Fk(\026)1368 635 y Fi(j)1405 621 y Fm(\()p Fk(Z)7 b Fm(\))2261 b(\(9\))1160 805 y(=)1435 743 y(1)p 1323 784 270 4 v 1323 867 a Fk(\025)1376 881 y Fi(S)1427 867 y Fm(\()p Fk(F)1520 881 y Fi(j)1557 867 y Fm(\))1618 681 y Fb(Z)1668 887 y Fi(\031)1711 868 y Fa(\000)p Fe(1)1794 887 y Fh(\()p Fi(F)1866 897 y Fd(j)1898 887 y Fh(\))1945 805 y Fk(\026)2000 767 y Fg(0)2000 827 y Fi(j)2036 805 y Fm(\()p Fk(g)2117 767 y Fg(\000)p Fh(1)2213 805 y Fk(Z)7 b Fm(\))15 b Fk(d\025)2432 819 y Fi(G)2491 805 y Fm(\()p Fk(g)s Fm(\))1153 b(\(10\))1160 1064 y(=)1435 1002 y(1)p 1323 1043 V 1323 1126 a Fk(\025)1376 1140 y Fi(S)1427 1126 y Fm(\()p Fk(F)1520 1140 y Fi(j)1557 1126 y Fm(\))1618 940 y Fb(Z)1668 1146 y Fi(\031)1711 1127 y Fa(\000)p Fe(1)1794 1146 y Fh(\()p Fi(F)1866 1156 y Fd(j)1898 1146 y Fh(\))1945 940 y Fb(Z)1996 1146 y Fi(C)2046 1157 y Fd(K)2124 1064 y Fk(\037)2181 1083 y Fi(g)2217 1064 y Fa(\000)p Fe(1)2299 1083 y Fi(Z)2356 1064 y Fm(\()p Fj(P)7 b Fm(\))15 b Fk(d\026)2613 1026 y Fg(0)2613 1086 y Fi(j)2651 1064 y Fm(\()p Fj(P)7 b Fm(\))15 b Fk(d\025)2906 1078 y Fi(G)2967 1064 y Fm(\()p Fk(g)s Fm(\))677 b(\(11\))1160 1323 y(=)1313 1199 y Fb(Z)1364 1405 y Fi(C)1414 1416 y Fd(K)1614 1261 y Fm(1)p 1502 1302 V 1502 1385 a Fk(\025)1555 1399 y Fi(S)1606 1385 y Fm(\()p Fk(F)1699 1399 y Fi(j)1736 1385 y Fm(\))1797 1199 y Fb(Z)1847 1405 y Fi(\031)1890 1386 y Fa(\000)p Fe(1)1973 1405 y Fh(\()p Fi(F)2045 1415 y Fd(j)2077 1405 y Fh(\))2124 1323 y Fk(\037)2181 1343 y Fi(g)2217 1324 y Fa(\000)p Fe(1)2299 1343 y Fi(Z)2356 1323 y Fm(\()p Fj(P)7 b Fm(\))15 b Fk(d\025)2611 1337 y Fi(G)2672 1323 y Fm(\()p Fk(g)s Fm(\))g Fk(d\026)2905 1285 y Fg(0)2905 1345 y Fi(j)2943 1323 y Fm(\()p Fj(P)7 b Fm(\))677 b(\(12\))0 1588 y(Hence)31 b(w)m(e)g(will)d(b)s(e)h(done)i (once)g(w)m(e)g(sho)m(w)f(that)h(for)f(an)m(y)h Fj(P)i(2)24 b Fk(C)2237 1602 y Fi(K)2306 1588 y Fm(,)1115 1705 y Fb(Z)1166 1911 y Fi(\031)1209 1892 y Fa(\000)p Fe(1)1291 1911 y Fh(\()p Fi(F)1363 1921 y Fd(j)1396 1911 y Fh(\))1443 1829 y Fk(\037)1500 1849 y Fi(g)1536 1830 y Fa(\000)p Fe(1)1618 1849 y Fi(Z)1675 1829 y Fm(\()p Fj(P)7 b Fm(\))15 b Fk(d\025)1930 1843 y Fi(G)1991 1829 y Fm(\()p Fk(g)s Fm(\))26 b(=)f Fk(\025)2282 1843 y Fi(S)2333 1829 y Fm(\()p Fk(c)p Fm(\()p Fj(P)7 b Fm(\))23 b Fj(\\)c Fk(F)2708 1843 y Fi(j)2745 1829 y Fm(\))p Fk(:)955 b Fm(\(13\))0 2093 y(So,)698 2305 y Fb(Z)749 2512 y Fi(\031)792 2493 y Fa(\000)p Fe(1)874 2512 y Fh(\()p Fi(F)946 2522 y Fd(j)979 2512 y Fh(\))1026 2429 y Fk(\037)1083 2449 y Fi(g)1119 2430 y Fa(\000)p Fe(1)1201 2449 y Fi(Z)1258 2429 y Fm(\()p Fj(P)7 b Fm(\))15 b Fk(d\025)1513 2443 y Fi(G)1574 2429 y Fm(\()p Fk(g)s Fm(\))84 b(=)f Fk(\025)1981 2443 y Fi(G)2040 2429 y Fm(\()p Fj(f)p Fk(g)30 b Fj(2)25 b Fk(\031)2334 2392 y Fg(\000)p Fh(1)2428 2429 y Fm(\()p Fk(F)2521 2443 y Fi(j)2558 2429 y Fm(\))p Fj(j)15 b(P)34 b(2)25 b Fk(g)2862 2392 y Fg(\000)p Fh(1)2957 2429 y Fk(Z)7 b Fj(g)p Fm(\))654 b(\(14\))1774 2645 y(=)83 b Fk(\025)1981 2659 y Fi(G)2040 2645 y Fm(\()p Fj(f)p Fk(g)30 b Fj(2)25 b Fk(\031)2334 2607 y Fg(\000)p Fh(1)2428 2645 y Fm(\()p Fk(F)2521 2659 y Fi(j)2558 2645 y Fm(\))p Fj(j)15 b Fk(g)s Fj(P)35 b(2)25 b Fk(Z)7 b Fj(g)p Fm(\))748 b(\(15\))1774 2783 y(=)83 b Fk(\025)1981 2797 y Fi(G)2040 2783 y Fm(\()p Fj(f)p Fk(g)30 b Fj(2)25 b Fk(\031)2334 2745 y Fg(\000)p Fh(1)2428 2783 y Fm(\()p Fk(F)2521 2797 y Fi(j)2558 2783 y Fm(\))p Fj(j)15 b(O)29 b(2)c Fk(c)p Fm(\()p Fk(g)s Fj(P)7 b Fm(\))p Fj(g)p Fm(\))635 b(\(16\))1774 2921 y(=)83 b Fk(\025)1981 2935 y Fi(G)2040 2921 y Fm(\()p Fj(f)p Fk(g)30 b Fj(2)25 b Fk(\031)2334 2883 y Fg(\000)p Fh(1)2428 2921 y Fm(\()p Fk(F)2521 2935 y Fi(j)2558 2921 y Fm(\))p Fj(j)15 b Fk(g)2679 2883 y Fg(\000)p Fh(1)2775 2921 y Fj(O)28 b(2)d Fk(c)p Fm(\()p Fj(P)7 b Fm(\))p Fj(g)p Fm(\))540 b(\(17\))1774 3058 y(=)83 b Fk(\025)1981 3072 y Fi(G)2040 3058 y Fm(\()p Fj(f)p Fk(g)30 b Fj(2)25 b Fk(\031)2334 3021 y Fg(\000)p Fh(1)2428 3058 y Fm(\()p Fk(F)2521 3072 y Fi(j)2558 3058 y Fm(\))p Fj(j)15 b Fk(g)s Fj(O)30 b(2)25 b Fk(c)p Fm(\()p Fj(P)7 b Fm(\))p Fj(g)p Fm(\))634 b(\(18\))1774 3196 y(=)83 b Fk(\025)1981 3210 y Fi(G)2040 3196 y Fm(\()p Fk(\031)2130 3159 y Fg(\000)p Fh(1)2225 3196 y Fm(\()p Fk(c)p Fm(\()p Fj(P)7 b Fm(\))22 b Fj(\\)e Fk(F)2600 3210 y Fi(j)2637 3196 y Fm(\)\))1053 b(\(19\))1774 3334 y(=)83 b Fk(\025)1981 3348 y Fi(S)2032 3334 y Fm(\()p Fk(c)p Fm(\()p Fj(P)7 b Fm(\))22 b Fj(\\)e Fk(F)2407 3348 y Fi(j)2444 3334 y Fm(\))1281 b(\(20\))141 3538 y(Equation)37 b(\(18\))j(holds)c(b)s(ecause)i Fk(\025)1383 3552 y Fi(G)1480 3538 y Fm(is)f(in)m(v)m(ersion-in)m(v)-5 b(arian)m(t)36 b(\(see)j([Rat])g(for)f(the)g(h)m(yp)s(erb)s(olic)d (case\))40 b(and)0 3651 y(b)s(ecause)c Fk(\031)397 3618 y Fg(\000)p Fh(1)492 3651 y Fm(\()p Fk(F)585 3665 y Fi(j)622 3651 y Fm(\))g(=)f Fk(\031)854 3618 y Fg(\000)p Fh(1)948 3651 y Fm(\()p Fk(F)1041 3665 y Fi(j)1078 3651 y Fm(\))1113 3618 y Fg(\000)p Fh(1)1208 3651 y Fm(.)59 b(The)36 b(last)g(equation)g (holds)f(since)h(for)g(an)m(y)h(Borel)f(set)h Fk(E)j Fj(\032)35 b Fk(S)5 b Fm(,)38 b Fk(\025)3621 3665 y Fi(S)3672 3651 y Fm(\()p Fk(E)5 b Fm(\))37 b(=)0 3764 y Fk(\025)53 3778 y Fi(G)112 3764 y Fm(\()p Fk(\031)202 3731 y Fg(\000)p Fh(1)297 3764 y Fm(\()p Fk(E)5 b Fm(\)\).)p 3855 3877 4 62 v 3859 3819 55 4 v 3859 3877 V 3913 3877 4 62 v 0 4065 a Fn(Lemma)33 b(4.3.)1365 4294 y Fk(D)s Fm(\()p Fk(\026)p Fm(\))25 b(=)1689 4171 y Fb(Z)1740 4377 y Fi(C)1790 4388 y Fd(K)1878 4232 y Fk(\025)1931 4246 y Fi(S)1982 4232 y Fm(\()p Fk(c)p Fm(\()p Fj(P)7 b Fm(\))22 b Fj(\\)e Fk(F)2357 4246 y Fi(j)2394 4232 y Fm(\))p 1878 4274 552 4 v 2019 4357 a Fk(\025)2072 4371 y Fi(S)2123 4357 y Fm(\()p Fk(F)2216 4371 y Fi(j)2253 4357 y Fm(\))2454 4294 y Fk(d\026)1204 b Fm(\(21\))1938 5309 y(8)p eop %%Page: 9 9 9 8 bop 0 191 a Ff(Pr)-5 b(o)g(of.)47 b Fm(By)31 b(equation)f(\(13\))i (in)d(the)h(previous)f(lemma,)602 327 y Fb(Z)652 533 y Fi(C)702 544 y Fd(K)790 389 y Fk(\025)843 403 y Fi(S)894 389 y Fm(\()p Fk(c)p Fm(\()p Fj(P)7 b Fm(\))23 b Fj(\\)c Fk(F)1269 403 y Fi(j)1306 389 y Fm(\))p 790 430 552 4 v 931 513 a Fk(\025)984 527 y Fi(S)1035 513 y Fm(\()p Fk(F)1128 527 y Fi(j)1165 513 y Fm(\))1367 451 y Fk(d\026)p Fm(\()p Fj(P)7 b Fm(\))84 b(=)1847 327 y Fb(Z)1898 533 y Fi(C)1948 544 y Fd(K)2026 327 y Fb(Z)2076 533 y Fi(\031)2119 514 y Fa(\000)p Fe(1)2202 533 y Fh(\()p Fi(F)2274 543 y Fd(j)2306 533 y Fh(\))2363 389 y Fk(\037)2420 403 y Fi(Z)2477 389 y Fm(\()p Fk(g)2558 356 y Fg(\000)p Fh(1)2653 389 y Fj(P)7 b Fm(\))p 2363 430 396 4 v 2426 513 a Fk(\025)2479 527 y Fi(S)2530 513 y Fm(\()p Fk(F)2623 527 y Fi(j)2660 513 y Fm(\))2784 451 y Fk(d\025)2884 465 y Fi(G)2944 451 y Fm(\()p Fk(g)s Fm(\))15 b Fk(d\026)p Fm(\()p Fj(P)7 b Fm(\))443 b(\(22\))1693 723 y(=)1847 600 y Fb(Z)1898 806 y Fi(\031)1941 787 y Fa(\000)p Fe(1)2023 806 y Fh(\()p Fi(F)2095 816 y Fd(j)2128 806 y Fh(\))2174 600 y Fb(Z)2225 806 y Fi(C)2275 817 y Fd(K)2363 662 y Fk(\037)2420 676 y Fi(Z)2477 662 y Fm(\()p Fk(g)2558 629 y Fg(\000)p Fh(1)2653 662 y Fj(P)7 b Fm(\))p 2363 702 V 2426 786 a Fk(\025)2479 800 y Fi(S)2530 786 y Fm(\()p Fk(F)2623 800 y Fi(j)2660 786 y Fm(\))2784 723 y Fk(d\026)p Fm(\()p Fj(P)g Fm(\))15 b Fk(d\025)3141 737 y Fi(G)3202 723 y Fm(\()p Fk(g)s Fm(\))442 b(\(23\))1693 988 y(=)1847 865 y Fb(Z)1898 1071 y Fi(\031)1941 1052 y Fa(\000)p Fe(1)2023 1071 y Fh(\()p Fi(F)2095 1081 y Fd(j)2128 1071 y Fh(\))2174 865 y Fb(Z)2225 1071 y Fi(C)2275 1082 y Fd(K)2370 927 y Fk(\037)2427 941 y Fi(Z)2484 927 y Fm(\()p Fj(P)7 b Fm(\))p 2363 968 270 4 v 2363 1051 a Fk(\025)2416 1065 y Fi(S)2467 1051 y Fm(\()p Fk(F)2560 1065 y Fi(j)2597 1051 y Fm(\))2658 988 y Fk(d\026)p Fm(\()p Fj(P)g Fm(\))15 b Fk(d\025)3015 1002 y Fi(G)3076 988 y Fm(\()p Fk(g)s Fm(\))568 b(\(24\))1693 1254 y(=)1847 1130 y Fb(Z)1898 1336 y Fi(\031)1941 1317 y Fa(\000)p Fe(1)2023 1336 y Fh(\()p Fi(F)2095 1346 y Fd(j)2128 1336 y Fh(\))2217 1192 y Fk(D)s Fm(\()p Fk(\026)p Fm(\))p 2184 1233 V 2184 1316 a Fk(\025)2237 1330 y Fi(S)2288 1316 y Fm(\()p Fk(F)2381 1330 y Fi(j)2418 1316 y Fm(\))2464 1254 y Fk(d\025)2564 1268 y Fi(G)2623 1254 y Fm(\()p Fk(g)s Fm(\))1021 b(\(25\))1693 1457 y(=)83 b Fk(D)s Fm(\()p Fk(\026)p Fm(\))1710 b(\(26\))0 1661 y(The)30 b(third)e(inequalit)m(y)h(holds)g(b)s(ecause)h Fk(\026)g Fm(is)g Fk(G)p Fm(-in)m(v)-5 b(arian)m(t.)p 3855 1661 4 62 v 3859 1603 55 4 v 3859 1661 V 3913 1661 4 62 v 0 1849 a Ff(Pr)g(o)g(of.)47 b(\(of)38 b(The)-5 b(or)g(em)39 b(3.1\))d Fm(Let)g Fk(U)1239 1863 y Fi(j)1311 1849 y Fm(b)s(e)f(the)g(set)h(of)g(all)e(pac)m(kings)h(in)g Fk(C)2539 1863 y Fi(K)2642 1849 y Fm(that)h(are)g(unsaturated)f (relativ)m(e)g(to)0 1962 y Fk(F)58 1976 y Fi(j)95 1962 y Fm(,)f(i.e.)51 b(for)33 b(an)m(y)h Fj(P)k(2)30 b Fk(U)892 1976 y Fi(j)929 1962 y Fm(,)k(there)g(is)f(a)g(pac)m(king)h Fj(P)1804 1929 y Fg(0)1861 1962 y Fm(suc)m(h)g(that)g Fj(P)c(\003)23 b Fk(@)5 b(F)2542 1976 y Fi(j)2609 1962 y Fm(=)31 b Fj(P)2781 1929 y Fg(0)2827 1962 y Fj(\003)23 b Fk(@)5 b(F)3006 1976 y Fi(j)3076 1962 y Fm(but)33 b Fj(P)3316 1929 y Fg(0)3363 1962 y Fj(\003)22 b Fk(F)3488 1976 y Fi(j)3559 1962 y Fm(has)33 b(more)0 2075 y(elemen)m(ts)e(than)f Fj(P)e(\003)20 b Fk(F)798 2089 y Fi(j)835 2075 y Fm(.)906 2418 y Fk(D)s Fm(\()p Fk(\026)1074 2432 y Fi(j)1110 2418 y Fm(\))84 b(=)1382 2294 y Fb(Z)1433 2500 y Fi(C)1483 2511 y Fd(K)1571 2356 y Fk(\025)1624 2370 y Fi(S)1675 2356 y Fm(\()p Fk(c)p Fm(\()p Fj(P)7 b Fm(\))22 b Fj(\\)e Fk(F)2050 2370 y Fi(j)2087 2356 y Fm(\))p 1571 2397 552 4 v 1712 2480 a Fk(\025)1765 2494 y Fi(S)1816 2480 y Fm(\()p Fk(F)1909 2494 y Fi(j)1946 2480 y Fm(\))2147 2418 y Fk(d\026)2249 2380 y Fg(0)2249 2440 y Fi(j)2286 2418 y Fm(\()p Fj(P)7 b Fm(\))1334 b(\(27\))1229 2675 y(=)1382 2551 y Fb(Z)1433 2757 y Fi(C)1483 2768 y Fd(K)1571 2613 y Fk(\025)1624 2627 y Fi(S)1675 2613 y Fm(\()p Fk(c)p Fm(\(\010)1850 2627 y Fi(j)1887 2613 y Fm(\()p Fj(P)7 b Fm(\)\))22 b Fj(\\)e Fk(F)2223 2627 y Fi(j)2260 2613 y Fm(\))p 1571 2654 725 4 v 1798 2737 a Fk(\025)1851 2751 y Fi(S)1902 2737 y Fm(\()p Fk(F)1995 2751 y Fi(j)2033 2737 y Fm(\))2320 2675 y Fk(d\026)p Fm(\()p Fj(P)7 b Fm(\))1198 b(\(28\))1229 2932 y(=)1382 2808 y Fb(Z)1433 3014 y Fi(U)1481 3024 y Fd(j)1542 2870 y Fk(\025)1595 2884 y Fi(S)1646 2870 y Fm(\()p Fk(c)p Fm(\(\010)1821 2884 y Fi(j)1858 2870 y Fm(\()p Fj(P)7 b Fm(\)\))22 b Fj(\\)e Fk(F)2194 2884 y Fi(j)2231 2870 y Fm(\))p 1542 2911 V 1770 2994 a Fk(\025)1823 3008 y Fi(S)1873 2994 y Fm(\()p Fk(F)1966 3008 y Fi(j)2004 2994 y Fm(\))2292 2932 y Fk(d\026)p Fm(\()p Fj(P)7 b Fm(\))1226 b(\(29\))1229 3197 y(+)1382 3074 y Fb(Z)1433 3280 y Fi(C)1483 3291 y Fd(K)1542 3280 y Fg(\000)p Fi(U)1645 3290 y Fd(j)1706 3135 y Fk(\025)1759 3149 y Fi(S)1810 3135 y Fm(\()p Fk(c)p Fm(\(\010)1985 3149 y Fi(j)2022 3135 y Fm(\()p Fj(P)7 b Fm(\)\))22 b Fj(\\)e Fk(F)2358 3149 y Fi(j)2395 3135 y Fm(\))p 1706 3176 V 1934 3260 a Fk(\025)1987 3274 y Fi(S)2037 3260 y Fm(\()p Fk(F)2130 3274 y Fi(j)2168 3260 y Fm(\))2455 3197 y Fk(d\026)p Fm(\()p Fj(P)7 b Fm(\))1063 b(\(30\))1229 3463 y Fj(\025)1382 3339 y Fb(Z)1433 3545 y Fi(U)1481 3555 y Fd(j)1542 3401 y Fk(\025)1595 3415 y 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y(uses)30 b(Lemma)g(4.3.)141 4735 y(If)42 b Fk(\026)h Fm(is)e(optimally)g(dense)h (then)h(b)m(y)f(de\014nition)e Fk(D)s Fm(\()p Fk(\026)p Fm(\))46 b Fj(\025)g Fk(D)s Fm(\()p Fk(\026)2432 4749 y Fi(j)2468 4735 y Fm(\).)78 b(So,)45 b Fk(\026)p Fm(\()p Fk(U)2924 4749 y Fi(j)2961 4735 y Fm(\))h(=)g(0)d(for)f(all)f Fk(j)5 b Fm(.)78 b(Hence)0 4848 y Fk(\026)p Fm(\()90 4780 y Fb(S)166 4875 y Fi(j)218 4848 y Fk(U)280 4862 y Fi(j)316 4848 y Fm(\))26 b(=)g(0.)42 b(Since)30 b Fk(B)893 4862 y Fi(j)955 4848 y Fj(\032)25 b Fk(F)1109 4862 y Fi(j)1177 4848 y Fm(for)30 b(all)g Fk(j)5 b Fm(,)32 b Fk(U)j Fm(=)1736 4780 y Fb(S)1812 4875 y Fi(j)1863 4848 y Fk(U)1925 4862 y Fi(j)1992 4848 y Fm(is)30 b(the)h(set)g(of)g(all)f (pac)m(kings)g(that)h(are)h(not)e(completely)0 4961 y(saturated.)p 3855 4961 4 62 v 3859 4903 55 4 v 3859 4961 V 3913 4961 4 62 v 1938 5309 a(9)p eop %%Page: 10 10 10 9 bop 0 191 a Fl(5)135 b(Tiles)45 b(with)h(small)g(optimal)g(densit) l(y)0 394 y Fm(In)29 b(this)g(section)g(w)m(e)h(will)d(pro)m(v)m(e)k (Theorem)e(2.2.)42 b(W)-8 b(e)31 b(assume)e(from)g(no)m(w)h(on)g(that)g Fk(S)35 b Fm(is)28 b Fk(n)p Fm(-dimensional)f(h)m(yp)s(er-)0 507 y(b)s(olic)32 b(space.)51 b(W)-8 b(e)34 b(will)d(need)i(a)h(lemma)f (that)h(follo)m(ws)f(from)g(m)m(uc)m(h)h(more)f(general)h(ergo)s(dic)f (theory)h(results)e(of)0 620 y(Nev)m(o)g(and)e(Stein.)39 b(It)31 b(is)e(an)h(immediate)g(corollary)g(of)g(Theorem)g(3)h(in)e ([NeS].)0 803 y Fn(Lemma)k(5.1.)46 b Ff(If)32 b Fk(G)g Ff(acts)h(c)-5 b(ontinuously)34 b(on)e(a)h(c)-5 b(omp)g(act)34 b(metric)f(sp)-5 b(ac)g(e)33 b Fk(X)40 b Ff(such)32 b(that)h(ther)-5 b(e)33 b(is)g(a)f(Bor)-5 b(el)33 b(pr)-5 b(ob-)0 916 y(ability)42 b(me)-5 b(asur)g(e)44 b Fk(\026)d Ff(on)i Fk(X)49 b Ff(that)43 b(is)f(invariant)h(and)g(er)-5 b(go)g(dic)43 b(under)f(this)h(action,)h(then)f(for)f(every)g(function)0 1029 y Fk(f)35 b Fj(2)24 b Fk(L)227 996 y Fi(p)267 1029 y Fm(\()p Fk(X)r(;)15 b(\026)p Fm(\))34 b Ff(\()p Fm(1)26 b Fk(<)f(p)g(<)g 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Fk(\025)1977 2019 y Fi(S)2028 2005 y Fm(\()p Fk(c)p Fm(\()p Fj(P)7 b Fm(\))22 b Fj(\\)e Fk(B)2414 2019 y Fi(r)2452 2005 y Fm(\))p 1924 2045 564 4 v 2065 2128 a Fk(\025)2118 2142 y Fi(S)2168 2128 y Fm(\()p Fk(B)2272 2142 y Fi(r)2311 2128 y Fm(\))2497 2066 y Fk(:)1238 b Fm(\(36\))0 2314 y Ff(Pr)-5 b(o)g(of.)47 b Fm(By)31 b(the)f(ab)s(o)m(v)m(e)i(lemma,)e (\(since)g Fk(\037)1475 2328 y Fi(Z)1557 2314 y Fj(2)25 b Fk(L)1705 2281 y Fi(p)1744 2314 y Fm(\()p Fk(C)1844 2328 y Fi(K)1913 2314 y Fk(;)15 b(\026)p Fm(\))31 b(for)f(all)f Fk(p)p Fm(\))h(for)h Fk(\026)p Fm(-almost)f(ev)m(ery)h(pac)m(king)g Fj(P)38 b Fm(in)29 b Fk(C)3814 2328 y Fi(K)3882 2314 y Fm(,)1076 2556 y Fk(D)s Fm(\()p Fk(\026)p Fm(\))83 b(=)1516 2432 y Fb(Z)1566 2638 y Fi(C)1616 2649 y Fd(K)1695 2556 y Fk(\037)1752 2570 y Fi(Z)1823 2556 y Fk(d\026)1835 b Fm(\(37\))1362 2806 y(=)107 b(lim)1516 2861 y Fi(r)r Fg(!1)1838 2745 y Fm(1)p 1716 2785 290 4 v 1716 2880 a Fk(\025)1769 2894 y Fi(G)1828 2880 y Fm(\()1885 2857 y(~)1863 2880 y Fk(B)1932 2894 y Fi(r)1970 2880 y Fm(\))2031 2683 y Fb(Z)2098 2872 y Fh(~)2081 2889 y Fi(B)2134 2897 y Fd(r)2188 2806 y Fk(\037)2245 2820 y Fi(Z)2302 2806 y Fm(\()p Fk(g)s Fj(P)7 b Fm(\))15 b Fk(d\025)2603 2820 y Fi(G)2665 2806 y Fm(\()p Fk(g)s Fm(\))979 b(\(38\))1362 3060 y(=)107 b(lim)1516 3114 y Fi(r)r Fg(!1)1834 2998 y Fm(1)p 1716 3039 282 4 v 1716 3122 a Fk(\025)1769 3136 y Fi(S)1820 3122 y Fm(\()p Fk(B)1924 3136 y Fi(r)1962 3122 y Fm(\))2022 2936 y Fb(Z)2089 3125 y Fh(~)2073 3142 y Fi(B)2126 3150 y Fd(r)2180 3060 y Fk(\037)2237 3079 y Fi(g)2273 3060 y Fa(\000)p Fe(1)2355 3079 y Fi(Z)2412 3060 y Fm(\()p Fj(P)7 b Fm(\))15 b Fk(d\025)2667 3074 y Fi(G)2728 3060 y Fm(\()p Fk(g)s Fm(\))916 b(\(39\))0 3329 y(If)30 b(w)m(e)h(replace)f Fk(\031)588 3296 y Fg(\000)p Fh(1)682 3329 y Fm(\()p Fk(F)775 3343 y Fi(j)813 3329 y Fm(\))g(with)g Fk(\031)1141 3296 y Fg(\000)p Fh(1)1235 3329 y Fm(\()p Fk(B)1339 3343 y Fi(r)1377 3329 y Fm(\))c(=)1555 3306 y(~)1534 3329 y Fk(B)1603 3343 y Fi(r)1671 3329 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Fk(K)38 b Fm(ev)m(ery)32 b(protrusion)e(in)g(an)m(y)i(cop)m(y)h(of)f Fk(K)38 b Fm(m)m(ust)31 b(\014t)h(in)m(to)g(an)f(inden)m(tation)g(of)h(another)f (cop)m(y)1915 5309 y(10)p eop %%Page: 11 11 11 10 bop 0 191 a Fm(of)34 b Fk(K)7 b Fm(.)51 b(There)33 b(will)e(b)s(e)j(more)g(inden)m(tations)e(than)i(protrusions)e (implying)f(at)j(least,)h(that)g(it)e(do)s(es)h(not)g(admit)0 304 y(a)d(tiling)e(of)i(a)g(closed)g(manifold.)40 b(T)-8 b(o)31 b(sho)m(w)g(that)g(an)m(y)g(in)m(v)-5 b(arian)m(t)30 b(measure)h(in)f Fk(C)2814 318 y Fi(K)2913 304 y Fm(has)g(small)g (densit)m(y)-8 b(,)31 b(w)m(e)g(will)0 417 y(sho)m(w)j(that)h(the)f (probabilit)m(y)d(that)k(the)f(origin)f(is)g(in)f(a)j(protrusion)d(is)h (ab)s(out)h(equal)f(to)i(the)f(probabilit)m(y)e(that)0 530 y(the)k(origin)e(is)h(in)f(an)m(y)i(giv)m(en)g(inden)m(tation.)55 b(Since)35 b(there)h(are)g(man)m(y)f(more)h(inden)m(tations)f(than)g (protrusions,)0 643 y(the)f(probabilit)m(y)e(that)i(the)h(origin)d(is)h 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(a)h(scalar)f(m)m(ultiple)e(of)j Fk(\025)1196 3250 y Fi(G)1255 3236 y Fm(.)41 b(Since)29 b Fk(D)s Fm(\()p Fk(\026)p Fm(\))c(=)g Fk(\026)p Fm(\()p Fk(C)2037 3250 y Fi(K)2106 3236 y Fm(\()p Fk(K)7 b Fm(\)\))26 b Fj(6)p Fm(=)f(0,)1291 3513 y Fk(\026)p Fm(\()p Fk(C)1446 3527 y Fi(K)1515 3513 y Fm(\()p Fk(L)1612 3527 y Fh(1)1651 3513 y Fm(\)\))p 1291 3554 431 4 v 1291 3637 a Fk(\026)p Fm(\()p Fk(C)1446 3651 y Fi(K)1515 3637 y Fm(\()p Fk(L)1612 3651 y Fh(2)1651 3637 y Fm(\)\))1815 3575 y(=)1990 3513 y Fk(\025)2043 3480 y Fg(0)2043 3539 y Fi(G)2102 3513 y Fm(\(\011)2208 3527 y Fh(1)2247 3513 y Fm(\()p Fk(C)2347 3527 y Fi(K)2416 3513 y Fm(\()p Fk(L)p Fm(\)\)\))p 1979 3554 652 4 v 1979 3637 a Fk(\025)2032 3606 y Fg(0)2032 3666 y Fi(G)2091 3637 y Fm(\(\011)2197 3651 y Fh(2)2236 3637 y Fm(\()p Fk(C)2336 3651 y Fi(K)2405 3637 y Fm(\()p Fk(K)7 b Fm(\)\)\))3760 3575 y(\(42\))1815 3829 y(=)1990 3767 y Fk(\025)2043 3781 y Fi(G)2102 3767 y Fm(\(\011)2208 3781 y Fh(1)2247 3767 y Fm(\()p Fk(C)2347 3781 y Fi(K)2416 3767 y Fm(\()p Fk(L)p Fm(\)\)\))p 1979 3808 V 1979 3891 a Fk(\025)2032 3905 y Fi(G)2091 3891 y Fm(\(\011)2197 3905 y Fh(2)2236 3891 y Fm(\()p Fk(C)2336 3905 y Fi(K)2405 3891 y Fm(\()p Fk(K)g Fm(\)\)\))3760 3829 y(\(43\))1815 4084 y(=)1990 4023 y Fk(\025)2043 4037 y Fi(G)2102 4023 y Fm(\()p Fk(\031)2192 3990 y Fg(\000)p Fh(1)2286 4023 y Fm(\()p Fk(L)p Fm(\))2418 3990 y Fg(\000)p Fh(1)2513 4023 y Fm(\))p 1979 4063 582 4 v 1979 4147 a Fk(\025)2032 4161 y Fi(G)2091 4147 y Fm(\()p Fk(\031)2181 4120 y Fg(\000)p Fh(1)2276 4147 y Fm(\()p Fk(K)g Fm(\))2430 4120 y Fg(\000)p Fh(1)2524 4147 y Fm(\))3760 4084 y(\(44\))1815 4340 y(=)1990 4278 y Fk(\025)2043 4292 y Fi(G)2102 4278 y Fm(\()p Fk(\031)2192 4245 y Fg(\000)p Fh(1)2286 4278 y Fm(\()p Fk(L)p Fm(\)\))p 1979 4319 487 4 v 1979 4402 a Fk(\025)2032 4416 y Fi(G)2091 4402 y Fm(\()p Fk(\031)2181 4376 y Fg(\000)p Fh(1)2276 4402 y Fm(\()p Fk(K)g Fm(\)\))3760 4340 y(\(45\))1815 4588 y(=)1990 4526 y Fk(\025)2043 4540 y Fi(S)2093 4526 y Fm(\()p 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