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Fj(\017)1300 2509 y Fh(\003)1324 2515 y Ff(N)1355 2497 y Fl(:)p Fm(\(\()p Fk(\000)p Fm(1\))1489 2476 y Fj(\017)p Fh(+1)1550 2497 y Fl(A)p Fm(\))1606 2504 y Fj(@)1625 2510 y Ff(R)1649 2504 y Fh(\003)1673 2510 y Ff(N)1704 2497 y Fl(:x)1746 2476 y Fj(\017)1746 2509 y Fh(\003)1770 2498 y Ff(c)1770 2521 y(N)1801 2497 y Fm(\))p Fl(;)962 2699 y Fm(8)p eop %%Page: 9 9 9 8 bop 0 26 a Fm(or,)13 b(more)f(explicitely)d(w)o(e)j(c)o(ho)q(ose)646 14 y(\026)637 26 y Fl(U)675 8 y Fj(N)722 26 y Fm(to)h(b)q(e)g(the)f(energy)g (of)h(our)g(system)f(forced)g(to)h(b)q(e)g(at)g(the)f(extremal)0 86 y(states)19 b(on)f(the)g(b)q(oundary)i(of)e(the)g(cub)q(e)g(\003)812 93 y Fj(N)846 86 y Fm(.)27 b(It)18 b(is)g(not)g(hard)h(to)g(see)f(that)g(the) g(Mark)o(o)o(v)g(semigroups)0 147 y Fk(fP)60 154 y Fj(t)74 147 y Fl(;)8 b(t)14 b Fk(\025)f Fm(0)p Fk(g)j Fm(\()g(resp.)21 b Fk(f)438 134 y Fm(\026)427 147 y Fk(P)466 129 y Fj(N)462 159 y(t)499 147 y Fl(;)8 b(t)13 b Fk(\025)h Fm(0)p Fk(g)i Fm(\))g(can)g(b)q 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1951 y(to)h(the)f(deca)o(y)g(of)h(correlations.)28 b(The)19 b(ab)q(o)o(v)o(e)g(relations)f(for)h(a)g(family)d(of)j(\014nite)f(sets)h(a)g (giv)o(en)f(in)o(teger)0 2011 y(n)o(um)o(b)q(er)i Fl(r)j Fm(and)g(an)f Fl(\013)i Fk(\034)g Fm(1,)f(are)f(the)g(main)e(step)i(to)o(w)o(ards)h (logarithmic)d(Sob)q(olev)i(inequalities)e(as)0 2072 y(pro)o(v)o(ed)c(in)f ([25)q(]-[27].)21 b(Indeed,)15 b(let)g Fl(r)j Fm(b)q(e)e(an)h(in)o(teger)e(n) o(um)o(b)q(er)g(\()h(to)g(b)q(e)h(sp)q(eci\014ed)f(later)f(\))i(and)f (de\014ne)g(:)415 2182 y Fl(Y)443 2189 y Fj(r)462 2182 y Fm(\(0\))e(=)g Fk(f)p Fl(k)i Fk(2)e Fl(Z)-21 b(Z)752 2161 y Fj(d)772 2182 y Fl(;)8 b Fk(\000)p Fl(r)k Fk(\000)f Fl(R)j Fk(\024)g Fl(k)1046 2189 y Fj(i)1074 2182 y Fk(\024)g Fl(r)e Fm(+)f Fl(R)49 b Fm(1)15 b Fk(\024)e Fl(i)h Fk(\024)f Fl(d)p Fk(g)p Fl(;)0 2292 y Fm(and,)j(for)h Fl(k)f Fk(2)e Fl(Z)-21 b(Z)320 2271 y Fj(d)340 2292 y Fm(,)16 b Fl(Y)398 2299 y Fj(r)417 2292 y Fm(\()p Fl(k)r Fm(\))e(=)g Fl(k)f Fm(+)e Fl(Y)663 2299 y Fj(r)682 2292 y Fm(\(0\).)0 2352 y(Moreo)o(v)o(er,)k(let)481 2433 y Fl(\036)510 2440 y Fj(K;r)569 2433 y Fm(\()p Fl(\015)s Fm(\))e(=)h Fl(\015)g Fm(+)837 2400 y Fl(K)p 793 2422 133 2 v 793 2467 a Fm(1)e Fk(\000)f Fl(\015)907 2453 y Fh(2)940 2392 y Fi(X)942 2484 y Fj(l)p Fg(\025)p Fj(r)1000 2433 y Fm(\(2)p Fl(l)h Fm(+)f(1\))1162 2413 y Fj(d)p Fg(\000)p Fh(1)1227 2433 y Fl(\015)1255 2413 y Fh(2)p Fj(lr)q(=)p Fh(\(9)p Fj(R)p Fh(+6)p Fj(r)q Fh(\))1454 2433 y Fl(;)950 2699 y Fm(13)p eop %%Page: 14 14 14 13 bop 0 26 a Fm(and,)18 b(for)g Fl(m)f Fk(\025)f Fm(2,)i Fl(\036)386 8 y Fj(m)386 39 y(K;r)445 26 y Fm(\()p Fl(\015)s Fm(\))e(=)h Fl(\036)611 33 y Fj(K;r)678 -22 y Fi(\020)703 26 y Fl(\036)732 6 y Fj(m)p Fg(\000)p Fh(1)732 38 y Fj(K;r)810 26 y Fm(\()p Fl(\015)s Fm(\))876 -22 y Fi(\021)901 26 y Fm(.)26 b(Finally)l(,)16 b(if)h Fl(C)1199 33 y Fj(d)1237 26 y Fm(denotes)h(the)g(v)o (olume)d(of)j(the)g(sphere)0 99 y(with)e(radius)h(one)f(in)g Fl(Z)-21 b(Z)453 78 y Fj(d)473 99 y Fm(,)16 b(w)o(e)g(de\014ne)g(for)g Fl(m)e Fk(2)g Fm([1)p Fl(;)8 b(d)j Fk(\000)g Fm(1],)16 b(an)o(y)g Fl(r)f(>)f Fm(0,)i Fl(\013)e(>)g Fm(0)i(:)522 247 y Fl(\030)543 254 y Fj(m)576 247 y Fm(\()p Fl(\013;)8 b(A;)g(r)q Fm(\))14 b(=)g Fl(\036)844 226 y Fj(m)844 259 y(A;r)907 161 y Fi(0)907 236 y(@)944 247 y Fl(C)979 254 y Fj(d)1010 205 y Fi(X)1007 296 y Fj(n)p Fg(\025)p Fj(r)1073 247 y Fm(\(2)p Fl(n)d Fm(+)g(1\))1248 226 y Fj(d)p Fg(\000)p Fh(1)1314 247 y Fl(\013)1345 226 y Fj(n)1369 161 y Fi(1)1369 236 y(A)1413 247 y Fl(:)0 394 y Fm(Then,)16 b(it)g(is)g(pro)o(v)o(en)f(in)h([25])g(that)h(:)0 512 y Fb(Theorem)g(4.1)24 b Fe(L)n(et)15 b Fm(\003)h Fe(b)n(e)h(a)f(subset)h(of)f Fl(Z)-21 b(Z)814 491 y Fj(d)834 512 y Fe(.)22 b(If)16 b(ther)n(e)g(exist)h(\014nite)g (c)n(onstants)g Fm(\()p Fl(A)1546 519 y Fh(\003)1572 512 y Fl(;)8 b(\013)1625 519 y Fh(\003)1652 512 y Fl(;)g(r)1696 519 y Fh(\003)1722 512 y Fm(\))16 b Fe(such)h(that)73 573 y(1\))g Fl(\030)156 580 y Fj(d)p Fg(\000)p Fh(1)222 573 y Fm(\()p Fl(\013)272 580 y Fh(\003)299 573 y Fl(;)8 b(A)358 580 y Fh(\003)384 573 y Fl(;)g(r)428 580 y Fh(\003)454 573 y Fm(\))14 b Fl(<)f Fm(1)p Fe(,)73 633 y(2\))j(F)l(or)g(any)g Fl(k)g Fk(2)e Fm([\()p Fl(r)459 640 y Fh(\003)494 633 y Fm(+)8 b(2)p Fl(R)p Fm(\))p Fl(Z)-21 b(Z)5 b Fm(])684 615 y Fj(d)712 633 y Fk(\\)758 620 y Fm(\026)754 633 y(\003)788 615 y Fj(R)816 633 y Fe(,)17 b(the)f(c)n(onditional)i(exp)n(e) n(ctations)f(asso)n(ciate)n(d)f(to)g Fl(Y)1759 640 y Fj(r)1775 646 y Fc(\003)1800 633 y Fm(\()p Fl(k)r Fm(\))8 b Fk(\\)h Fm(\003)73 693 y Fe(satisfy)17 b Fl(S)s(O)q(R)p Fm(\()p Fl(\013)383 700 y Fh(\003)411 693 y Fl(;)8 b(A)470 700 y Fh(\003)496 693 y Fl(;)g(r)540 700 y Fh(\003)566 693 y Fm(\))p Fe(,)73 753 y(3\))17 b(If)h(we)g(denote)324 857 y Fl(Y)364 836 y Fj(l)352 869 y(r)377 857 y Fm(\()p Fl(k)r Fm(\))13 b(=)h Fk([)540 869 y Fj(n)561 874 y Fc(1)579 869 y Fj(;)p Fg(\001\001\001)o Fj(;n)649 875 y Ff(l)662 869 y Fg(2)p Fj(Z)-23 b(Z)713 857 y Ff(l)t Fk(f)p Fl(Y)780 864 y Fj(r)799 857 y Fm(\()p Fl(k)13 b Fm(+)e Fl(n)934 864 y Fh(1)954 857 y Fm(\()p Fl(r)h Fm(+)f(2)p Fl(R)p Fm(\))p Fl(e)1159 864 y Fh(1)1191 857 y Fm(+)g Fk(\001)d(\001)g(\001)g Fl(n)1335 864 y Fj(l)1348 857 y Fm(\()p Fl(r)k Fm(+)f(2)p Fl(R)p Fm(\))p Fl(e)1553 864 y Fj(l)1567 857 y Fm(\))p Fk(g)p Fl(;)73 961 y Fe(for)16 b(any)g Fl(l)e Fk(2)g Fm([1)p Fl(;)8 b(d)g Fk(\000)g Fm(1])p Fe(,)17 b(for)e(any)h Fl(k)g Fk(2)e(f)p Fm(0)p Fk(g)858 942 y Fj(l)872 961 y Fk(\002)o Fm([\()p Fl(r)9 b Fm(+)f(2)p Fl(R)p Fm(\))p Fl(Z)-21 b(Z)5 b Fm(])1164 942 y Fj(d)p Fg(\000)p Fj(l)1231 961 y Fk(\\)1277 948 y Fm(\026)1272 961 y(\003)1306 942 y Fj(R)1335 961 y Fe(,)16 b(the)h(c)n(onditional)g(exp)n(e)n(ctations)73 1021 y(asso)n(ciate)n(d)g(to)h Fl(Y)399 1003 y Fj(l)388 1033 y(r)404 1039 y Fc(\003)429 1021 y Fm(\()p Fl(k)r Fm(\))11 b Fk(\\)g Fm(\003)17 b Fe(satisfy)g Fl(S)s(O)q(R)p Fm(\()p Fl(A)916 1028 y Fh(\003)943 1021 y Fm(\))p Fe(,)73 1081 y(then)i(the)e(lo)n(garithmic) h(Sob)n(olev)h(c)n(o)n(e\016cient)f(of)g Fl(E)1010 1088 y Fh(\003)1054 1081 y Fe(c)n(an)f(b)n(e)h(b)n(ounde)n(d,)g(for)e(any)i Fl(r)d Fk(\025)f Fl(r)1687 1088 y Fh(\003)1713 1081 y Fe(,)j(by)h(:)345 1226 y Fl(c)p Fm(\(\003\))13 b Fk(\024)504 1172 y Fj(d)p Fg(\000)p Fh(1)509 1185 y Fi(Y)508 1277 y Fj(l)p Fh(=0)575 1153 y Fi( )710 1193 y Fm(2)p Fl(A)771 1200 y Fh(\003)808 1193 y Fm(+)e(3)p 613 1215 365 2 v 613 1261 a(1)h Fk(\000)f Fl(\030)720 1268 y Fj(l)733 1261 y Fm(\()p Fl(\013)783 1268 y Fh(\003)809 1261 y Fl(;)d(A)868 1268 y Fh(\003)894 1261 y Fl(;)g(r)q Fm(\))958 1246 y Fh(2)983 1153 y Fi(!)1128 1226 y Fm(max)1024 1261 y Fj(k)q Fg(2)p Fh([\()p Fj(r)q Fh(+2)p Fj(R)p Fh(\))p Fj(Z)-23 b(Z)r Fh(])1231 1251 y Ff(d)1249 1261 y Fg(\\)1276 1252 y Fh(\026)1273 1261 y(\003)1297 1251 y Ff(R)1330 1226 y Fl(c)p Fm(\()p Fl(Y)1398 1233 y Fj(r)1417 1226 y Fm(\()p Fl(k)13 b Fk(\\)f Fm(\003\)\))p Fl(:)257 b Fm(\(29\))73 1367 y(Th)o(us,)18 b(our)f(task)h(is)f(reduced)f(to)i (sho)o(w)g(that,)f(for)h Fl(I)-13 b(E)20 b Fm(almost)c(all)h(in)o (teractions,)g(the)g(sw)o(eeping)f(out)0 1427 y(relations)g(of)h(Theorem)f (4.1)h(are)f(v)o(eri\014ed)f(when)i(\003)g(is)f(a)h(large)g(cub)q(e)f(cen)o (tered)f(at)i(the)g(origin.)22 b(Indeed,)0 1487 y(w)o(e)16 b(will)f(pro)o(v)o(e)g(that)i(:)0 1594 y Fb(Theorem)g(4.2)73 b Fe(L)n(et)22 b Fm(\003)506 1601 y Fj(n)551 1594 y Fm(=)f([)p Fk(\000)p Fl(n;)8 b(n)p Fm(])757 1576 y Fj(d)776 1594 y Fe(.)35 b(If)43 b(I)-12 b(H1)21 b(and)44 b(I)-12 b(H2)21 b(ar)n(e)g(satis\014e)n(d)h (with)g(p)n(ositive)f Fl(J)1795 1601 y Fh(0)1837 1594 y Fe(smal)r(l)0 1654 y(enough,)h(then)f(ther)n(e)f(exist)h(a)e(p)n(ositive)i(r)n(e)n(al)e (numb)n(er)h Fl(\013)f(<)g Fm(1)h Fe(\()g Fl(\013)f Fk(!)f Fm(0)j Fe(when)g Fl(J)1537 1661 y Fh(0)1575 1654 y Fk(!)d Fm(0)j Fe(\),)f(and)g(\014nite)0 1715 y(c)n(onstants)h Fl(\034)26 b Fe(and)20 b Fl(\034)388 1696 y Fg(0)400 1715 y Fe(,)h(\()p Fl(\034)j Fk(!)19 b Fm(0)i Fe(when)g Fl(J)769 1722 y Fh(0)808 1715 y Fk(!)e Fm(0)p Fe(\),)i(such)g(that)f(for)g Fl(I)-13 b(E)23 b Fe(almost)d(every)h Fl(J)k Fe(ther)n(e)20 b(exists)h(a)0 1775 y(\014nite)e(inte)n(ger)f(numb)n(er)g Fl(N)5 b Fm(\()p Fl(J)g Fm(\))17 b Fe(for)g(which)h(for)f(any)g(inte)n(ger)h(numb)n(er)g Fl(n)c Fk(\025)g Fl(N)5 b Fm(\()p Fl(J)g Fm(\))p Fe(,)73 1870 y(a\))25 b(for)f(any)h Fl(k)k Fk(2)f Fm([\(2)p Fl(R)17 b Fm(+)g Fl(\034)6 b Fm(\(log)j Fl(n)p Fm(\))775 1852 y Fj(d)795 1870 y Fm(\))p Fl(Z)-21 b(Z)t Fm(])877 1852 y Fj(d)913 1870 y Fk(\\)968 1858 y Fm(\026)963 1870 y(\003)997 1852 y Fj(R)997 1883 y(n)1026 1870 y Fe(,)26 b(the)g(c)n(onditional)f(exp)n(e)n(ctation)h(asso)n(ciate)n(d) f(to)0 1937 y Fl(Y)28 1946 y Fj(\034)t Fh(\(log)5 b Fj(n)p Fh(\))148 1937 y Ff(d)t Fm(\()p Fl(k)r Fm(\))11 b Fk(\\)h Fm(\003)323 1944 y Fj(n)363 1937 y Fe(satis\014es)18 b(the)g(c)n(ondition)g Fl(S)s(O)q(R)958 1889 y Fi(\020)982 1937 y Fl(\013;)8 b Fm(exp)p Fk(f)p Fl(\034)1162 1919 y Fg(0)1173 1937 y Fm(\(log)i Fl(n)p Fm(\))1312 1919 y Fh(\()p Fj(d)p Fg(\000)p Fh(1\))1405 1937 y Fk(g)p Fl(;)e(\034)e Fm(\(log)i Fl(n)p Fm(\))1616 1919 y Fh(\()p Fj(d)p Fg(\000)p Fh(1\))1709 1889 y Fi(\021)1734 1937 y Fe(,)73 2042 y(b\))23 b(for)f(any)h Fl(l)h Fk(2)f Fm([1)p Fl(;)8 b(d)15 b Fk(\000)g Fm(1])p Fe(,)24 b(any)e Fl(k)k Fk(2)e(f)p Fm(0)p Fk(g)926 2024 y Fj(l)939 2042 y Fk(\002)p Fm([\(2)p Fl(R)15 b Fm(+)g Fl(\034)6 b Fm(\(log)j Fl(n)p Fm(\))1305 2024 y Fj(d)1325 2042 y Fm(\))p Fl(Z)-21 b(Z)t Fm(])1407 2024 y Fj(d)p Fg(\000)p Fj(l)1480 2042 y Fk(\\)1533 2030 y Fm(\026)1528 2042 y(\003)1562 2024 y Fj(R)1562 2055 y(n)1591 2042 y Fe(,)24 b(the)f(c)n(onditional)0 2103 y(exp)n(e)n(ctation)c(asso)n(ciate)n(d)d(to)i Fl(Y)577 2085 y Fj(l)566 2118 y(\034)t Fh(\(log)5 b Fj(n)p Fh(\))686 2108 y Ff(d)p Fd(\000)p Fc(1)745 2103 y Fm(\()p Fl(k)r Fm(\))11 b Fk(\\)h Fm(\003)900 2110 y Fj(n)941 2103 y Fe(satis\014es)18 b(the)g(c)n(orr)n(esp)n(onding)e(c)n(ondition)0 2180 y Fl(S)s(O)q(R)117 2132 y Fi(\020)142 2180 y Fm(exp)p Fk(f)p Fl(\034)269 2162 y Fg(0)280 2180 y Fm(\(log)9 b Fl(n)p Fm(\))418 2162 y Fh(\()p Fj(d)p Fg(\000)p Fh(1\))511 2180 y Fk(g)536 2132 y Fi(\021)561 2180 y Fe(.)0 2287 y Fm(As)16 b(a)h(consequence,)e(if)g Fl(\013)i Fm(is)f(small)f(enough)i(so)f(that)369 2391 y Fl(\017)p Fm(\()p Fl(d;)8 b(\013)p Fm(\))14 b Fk(\021)f Fm(sup)597 2425 y Fj(n)653 2391 y Fl(\030)674 2398 y Fj(d)p Fg(\000)p Fh(1)739 2391 y Fm(\()p Fl(\013;)8 b Fm(exp)p Fk(f)p Fl(\034)938 2370 y Fg(0)949 2391 y Fm(\(log)i Fl(n)p Fm(\))1088 2370 y Fh(\()p Fj(d)p Fg(\000)p Fh(1\))1181 2391 y Fk(g)p Fl(;)e(\034)e Fm(\(log)i Fl(n)p Fm(\))1392 2370 y Fj(d)p Fg(\000)p Fh(1)1458 2391 y Fm(\))14 b Fl(<)f Fm(1)p Fl(;)0 2515 y Fm(\(29\))j(implies)c(that)j(there)f(exists)g(a)h (\014nite)g(in)o(teger)e(n)o(um)o(b)q(er)g Fl(N)5 b Fm(\()p Fl(J)g Fm(\))15 b(for)g(whic)o(h)f(for)h(an)o(y)g(in)o(teger)f(n)o(um)o(b)q (er)0 2575 y Fl(n)g Fk(\025)g Fl(N)5 b Fm(\()p Fl(J)g Fm(\),)950 2699 y(14)p eop %%Page: 15 15 15 14 bop 117 159 a Fl(c)p Fm(\(\003\))13 b Fk(\024)276 86 y Fi( )314 125 y Fm(2)8 b(exp)p Fk(f)p Fl(\034)473 107 y Fg(0)484 125 y Fm(\(log)h Fl(n)p Fm(\))622 107 y Fj(d)p Fg(\000)p Fh(1)688 125 y Fk(g)i Fm(+)g(3)p 314 147 484 2 v 435 193 a(1)g Fk(\000)g Fl(\017)p Fm(\()p Fl(d;)d(\013)p Fm(\))656 178 y Fh(2)802 86 y Fi(!)835 97 y Fj(d)p Fg(\000)p Fh(1)1092 159 y Fm(max)909 193 y Fj(k)q Fg(2)p Fh([\()p Fj(\034)t Fh(\(log)t Fj(n)p Fh(\))1095 183 y Ff(d)p Fd(\000)p Fc(1)1152 193 y Fh(+2)p Fj(R)p Fh(\))p Fj(Z)-23 b(Z)r Fh(])1275 183 y Ff(d)1293 193 y Fg(\\)1320 185 y Fh(\026)1317 193 y(\003)1341 183 y Ff(R)1341 201 y(n)1375 159 y Fl(c)p Fm(\()p Fl(Y)1443 168 y Fj(\034)t Fh(\(log)5 b Fj(n)p Fh(\))1563 159 y Ff(d)p Fd(\000)p Fc(1)t Fm(\()p Fl(k)r Fm(\))11 b Fk(\\)g Fm(\003)1776 166 y Fj(n)1800 159 y Fm(\))p Fl(:)73 293 y Fm(Th)o(us,)23 b(it)e(is)h(enough)g(to)g(b)q(ound)h(the)f (logarithmic)e(co)q(e\016cien)o(ts)g(of)i(the)f Fl(Y)1503 303 y Fj(\034)t Fh(\(log)5 b Fj(n)p Fh(\))1623 293 y Ff(d)p Fd(\000)p Fc(1)g Fm(\()p Fl(k)r Fm(\)'s)21 b(to)i(\014nd)0 354 y(Theorem)16 b(3.2.)26 b(This)18 b(last)g(step)f(can)h(b)q(e)g(easily)f(p)q(erformed)f(as) i(e.g.)25 b(in)18 b([12])f(\()h(see)f(Lemma)f(5.5)i(\))f(and)0 414 y(one)f(can)h(\014nd)f(:)0 521 y Fb(Lemma)g(4.3)24 b Fe(F)l(or)e(any)h Fl(\034)30 b(>)24 b Fm(0)p Fe(,)h(ther)n(e)e(exists)h(a)f(\014nite)h(c)n (onstants)g Fl(\034)6 b Fm(")p Fe(,)25 b(satisfying)e Fl(\034)6 b Fm(")24 b Fk(!)g Fm(0)g Fe(when)0 581 y Fl(\034)c Fk(!)13 b Fm(0)p Fe(,)18 b(and)f(such)g(that)g(for)g Fl(I)-13 b(E)19 b Fe(almost)f(every)f Fl(J)5 b Fe(,)17 b(ther)n(e)g(exists)g(an)h(inte)n(ger) f(numb)n(er)h Fl(N)5 b Fm(\()p Fl(J)g Fm(\))16 b Fe(such)i(that,)0 641 y(for)f(any)g Fl(n)d Fk(\025)g Fl(N)5 b Fm(\()p Fl(J)g Fm(\))p Fe(,)17 b(for)g(any)g Fl(k)f Fk(2)e Fm([\()p Fl(\034)6 b Fm(\(log)j Fl(n)p Fm(\))870 623 y Fj(d)p Fg(\000)p Fh(1)946 641 y Fm(+)i(2)p Fl(R)p Fm(\))p Fl(Z)-21 b(Z)5 b Fm(])1139 623 y Fj(d)1159 641 y Fe(,)17 b(we)h(have)517 761 y Fl(c)p Fm(\()p Fl(Y)585 771 y Fj(\034)t Fh(\(log)5 b Fj(n)p Fh(\))705 762 y Ff(d)726 761 y Fm(\()p Fl(k)r Fm(\))11 b Fk(\\)851 749 y Fm(\026)846 761 y(\003)880 741 y Fj(R)880 774 y(n)909 761 y Fm(\))i Fk(\024)h Fm(exp)p Fk(f)p Fl(\034)6 b Fm("\(log)j Fl(n)p Fm(\))1283 741 y Fh(\()p Fj(d)p Fg(\000)p Fh(1\))1374 729 y Fc(2)1393 761 y Fk(g)p Fl(:)0 868 y Fm(It)15 b(is)g(then)g(easy)g(to)h (c)o(hec)o(k)d(that)j(Lemma)d(4.3)i(and)h(Theorem)e(4.2)h(imply)e(Theorem)h (3.2.)21 b(As)15 b(a)h(remark,)0 928 y(w)o(e)j(w)o(ould)h(lik)o(e)e(to)j(men) o(tion)c(that,)k(in)f(the)f(case)h(where)g(the)f(in)o(teraction)g(is)h(b)q (ounded)h(but)f(p)q(ossibly)0 989 y(large,)c(the)g(pro)q(ofs)h(of)g(Theorem)d (4.2.b\))j(and)f(of)h(Lemma)d(4.3)i(are)g(m)o(uc)o(h)e(easier.)21 b(Nev)o(ertheless,)14 b(w)o(e)h(do)0 1049 y(not)i(see)f(ho)o(w)g(this)g (additional)h(assumption)f(w)o(ould)g(impro)o(v)o(e)d(Theorem)i(3.2.)73 1109 y(In)e(the)g(follo)o(wing)f(w)o(e)h(will)f(pro)o(v)o(e)g(Theorem)g(4.2)h (b)o(y)g(\014rst)g(b)q(ounding)h(the)f(sw)o(eeping)g(out)g(co)q(e\016cien)o (ts)0 1169 y(for)k(a)f(giv)o(en)g(in)o(teraction)f(and)i(then)f(deducing)g (an)g(almost)g(sure)g(b)q(ound.)0 1313 y Fa(4.1)89 b(Sw)n(eeping)22 b(out)h(relations)0 1405 y Fm(F)l(or)16 b(ev)o(ery)f(realization)h(of)g(the)g (in)o(teraction,)f(the)h(follo)o(wing)g(b)q(ound)h(holds)g(:)0 1535 y Fb(Prop)r(ert)n(y)h(4.4)24 b Fe(Ther)n(e)e(exist)h(\014nite)h(nonne)n (gative)h(numb)n(ers)1201 1487 y Fi(\020)1226 1535 y Fl(\013)1257 1517 y Fh(\003)1257 1547 y Fj(i;j)1297 1535 y Fl(;)8 b(i)13 b Fk(2)h Fm(\003)1430 1517 y Fj(c)1447 1535 y Fl(;)26 b(j)16 b Fk(2)e Fm(\003)1604 1487 y Fi(\021)1652 1535 y Fe(such)23 b(that,)g(for)0 1610 y(any)17 b(smo)n(oth)g(function)i Fl(f)k Fe(in)17 b Fl(L)590 1592 y Fh(2)610 1610 y Fm(\()p Fl(\026)658 1587 y Fj(J)o(;!)658 1623 y Fh(\003)712 1610 y Fm(\))p Fe(,)g(for)g(any)h Fl(i)f Fe(in)h Fm(\003)1063 1592 y Fj(c)1080 1610 y Fe(,)290 1674 y Fi(\014)290 1699 y(\014)290 1724 y(\014)290 1749 y(\014)303 1737 y Fk(r)345 1744 y Fj(i)367 1689 y Fi(\020)392 1737 y Fl(E)428 1744 y Fh(\003)455 1737 y Fm([)p Fl(f)498 1716 y Fh(2)517 1737 y Fm(])531 1689 y Fi(\021)561 1686 y Fc(1)p 561 1692 16 2 v 561 1713 a(2)583 1674 y Fi(\014)583 1699 y(\014)583 1724 y(\014)583 1749 y(\014)611 1737 y Fk(\024)663 1689 y Fi(\020)688 1737 y Fl(E)724 1744 y Fh(\003)759 1689 y Fi(h)779 1737 y Fk(jr)835 1744 y Fj(i)848 1737 y Fl(f)5 b Fk(j)891 1716 y Fh(2)911 1689 y Fi(i\021)960 1686 y Fc(1)p 960 1692 V 960 1713 a(2)994 1737 y Fm(+)11 b Fe(1)-8 b(I)1078 1744 y Fj(i)p Fg(2)p Fj(@)1133 1750 y Ff(R)1157 1744 y Fh(\003)1194 1695 y Fi(X)1192 1787 y Fj(j)r Fg(2)p Fh(\003)1265 1737 y Fl(\013)1296 1716 y Fh(\003)1296 1749 y Fj(i;j)1344 1689 y Fi(\020)1369 1737 y Fl(E)1405 1744 y Fh(\003)1440 1689 y Fi(h)1459 1737 y Fk(jr)1515 1744 y Fj(j)1533 1737 y Fl(f)5 b Fk(j)1576 1716 y Fh(2)1596 1689 y Fi(i)o(\021)1645 1686 y Fc(1)p 1645 1692 V 1645 1713 a(2)1862 1737 y Fm(\(30\))0 1880 y Fe(Mor)n(e)n(over,)16 b(for)h(any)g(inte)n(ger)i(numb)n(er)e Fl(D)q Fe(,)i Fl(D)c(>)f(R)p Fe(,)k(the)g(fol)r(lowing)i(b)n(ound)d(holds)h (:)410 2032 y Fl(\013)441 2011 y Fh(\003)441 2044 y Fj(i;j)495 2032 y Fk(\024)13 b Fl(\027)574 2011 y Fh(\003)571 2044 y Fj(i;j)622 2032 y Fm(+)671 1973 y Fg(j)p Fh(\003)p Fg(j\000)p Fh(1)686 1990 y Fi(X)688 2083 y Fj(l)p Fh(=1)883 1990 y Fi(X)769 2088 y Fg(f)p Fj(k)805 2093 y Fc(1)822 2088 y Fj(;)p Fg(\001\001\001)o Fj(;k)889 2094 y Ff(l)902 2088 y Fg(g2)p Fj(G)971 2077 y Fc(\003)971 2100 y Ff(l;D)1017 2088 y Fh(\()p Fj(i)p Fh(\))1065 2032 y Fl(\027)1092 2011 y Fh(\003)1089 2044 y Fj(i;k)1129 2049 y Fc(1)1157 1946 y Fi(0)1157 2021 y(@)1197 1978 y Fj(l)p Fg(\000)p Fh(1)1198 1990 y Fi(Y)1193 2081 y Fj(p)p Fh(=1)1265 2032 y Fl(\027)1292 2011 y Fh(\003)1289 2044 y Fj(k)1307 2048 y Ff(p)1325 2044 y Fj(;k)1353 2049 y Ff(p)p Fc(+1)1412 1946 y Fi(1)1412 2021 y(A)1457 2032 y Fl(\027)1484 2011 y Fh(\003)1481 2044 y Fj(k)1499 2050 y Ff(l)1511 2044 y Fj(;j)1862 2032 y Fm(\(31\))0 2172 y Fe(wher)n(e)167 2282 y Fl(G)205 2262 y Fh(\003)205 2295 y Fj(l;D)258 2282 y Fm(\()p Fl(i)p Fm(\))41 b(=)434 2222 y Fi(\032)465 2282 y Fk(f)p Fl(k)515 2289 y Fh(1)535 2282 y Fl(;)8 b Fk(\001)g(\001)g(\001)g Fl(;)g(k)670 2289 y Fj(l)683 2282 y Fk(g)14 b(2)g Fm(\003)p Fk(j)p Fl(k)842 2289 y Fj(i)870 2282 y Fk(6)p Fm(=)g Fl(k)947 2289 y Fj(i)p Fh(+1)1024 2282 y Fe(for)j Fl(i)c Fk(2)h Fm([1)p Fl(;)8 b(l)k Fk(\000)f Fm(1])17 b Fe(and)434 2399 y Fl(d)p Fm(\()p Fl(k)503 2406 y Fj(p)523 2399 y Fl(;)8 b(k)570 2406 y Fj(p)p Fh(+1)635 2399 y Fm(\))14 b Fl(>)g(D)37 b Fe(for)17 b Fl(p)d Fk(2)g Fm([1)p Fl(;)8 b(l)j Fk(\000)g Fm(1])50 b Fl(d)p Fm(\()p Fl(k)1253 2406 y Fh(1)1273 2399 y Fl(;)8 b(i)p Fm(\))14 b Fl(>)f Fm(max)o Fk(f)p Fl(D)q(;)8 b Fk(j)p Fl(i)j Fk(\000)g Fl(j)s Fk(jg)1729 2338 y Fi(\033)1760 2399 y Fl(;)0 2575 y Fe(F)l(urthermor)n(e,)16 b(if)i(one)g(denotes,)g(for)f (any)h(subset)g Fl(Y)29 b Fe(of)17 b Fl(Z)-21 b(Z)1105 2554 y Fj(d)1125 2575 y Fe(,)18 b Fl(m)p Fm(\()p Fl(Y)10 b Fm(\))18 b Fe(the)g(sp)n(e)n(ctr)n(al)e(gap)i(c)n(o)n(e\016cient)h(:)950 2699 y Fm(15)p eop %%Page: 16 16 16 15 bop 617 107 a Fl(m)p Fm(\()p Fl(Y)11 b Fm(\))j(=)20 b(inf)802 137 y Fj(!)q Fg(2)p Fh(\012)944 107 y Fm(inf)888 133 y Ff(f)s Fd(2)p Ff(I)-15 b(L)946 140 y Fc(2)964 133 y(\()p Ff(\026)995 125 y(!)995 148 y(J)o(;Y)1046 133 y Fc(\))945 175 y Ff(f)s Fd(?)p Fc(1)e(1)1077 70 y Fl(\026)1106 52 y Fj(!)1106 82 y(J)o(;Y)1165 70 y Fk(jr)1221 77 y Fj(Y)1250 70 y Fl(f)5 b Fk(j)1293 52 y Fh(2)p 1077 95 237 2 v 1093 141 a Fl(\026)1122 124 y Fj(!)1122 153 y(J)o(;Y)1181 141 y Fm(\()p Fl(f)s(;)j(f)d Fm(\))1318 107 y Fl(;)0 248 y Fe(and)18 b(:)607 368 y Fl(m)650 375 y Fh(\003)676 368 y Fm(\()p Fl(D)q Fm(\))d(=)f(inf)822 398 y Fj(i)p Fg(2)p Fh(\003)902 368 y Fm(inf)890 398 y Fj(X)s Fg(\032)p Fh(\003)982 368 y Fl(m)8 b Fm(\()p Fl(X)15 b Fk(\\)c Fl(B)s Fm(\()p Fl(i;)d(D)q Fm(\)\))p Fl(;)0 468 y Fe(then)18 b(one)g(c)n(an)f(\014nd)h(a)f(\014nite)h(c) n(onstant)g Fl(C)t Fe(,)f(which)h(only)f(dep)n(ends)h(on)f Fl(M)23 b Fe(and)17 b Fl(\027)1499 475 y Fj(M)1538 468 y Fe(,)h(such)f(that,) g(for)g(any)0 528 y Fm(\003)g Fe(:)73 664 y Fl(\027)100 644 y Fh(\003)97 677 y Fj(k)q(;k)145 667 y Fd(0)172 664 y Fm(=)224 577 y Fi(8)224 614 y(<)224 689 y(:)269 630 y Fl(C)t Fk(jjr)378 637 y Fj(k)404 617 y Fm(~)398 630 y(\010)433 612 y Fh(\()p Fj(k)q Fh(\))482 630 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)610 637 y Fh(1)629 630 y Fl(m)672 637 y Fh(\003)698 630 y Fm(\()p Fl(D)q Fm(\))777 611 y Fg(\000)809 598 y Fc(1)p 811 604 16 2 v 811 624 a(2)1548 630 y Fe(if)17 b Fk(j)p Fl(k)c Fk(\000)e Fl(k)1724 612 y Fg(0)1735 630 y Fk(j)j(\024)g Fl(D)q(;)269 690 y(C)t Fk(jjr)378 697 y Fj(k)404 677 y Fm(~)398 690 y(\010)433 672 y Fh(\()p Fj(k)q Fh(\))482 690 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)610 697 y Fh(1)637 690 y Fm(exp)p Fk(f)p Fm(3)p Fk(jj)794 677 y Fm(~)789 690 y(\010)824 672 y Fh(\()p Fj(k)857 660 y Fd(0)868 672 y Fh(\))884 690 y Fm(\()p Fl(J)o(;)g(:)p Fm(\))p Fk(jj)1012 697 y Fh(1)1031 690 y Fk(g)1064 657 y Fi(P)1108 701 y Fj(\015)r Fg(2W)1190 690 y Fc(\003)1187 717 y Ff(k)q(;k)1230 710 y Fd(0)1254 657 y Fi(Q)1293 700 y Fj(Y)g Fg(2)p Fj(\015)1375 690 y Fl(\032)1400 697 y Fj(Y)1548 690 y Fe(otherwise,)0 810 y(wher)n(e)466 941 y Fl(\032)491 948 y Fj(Y)536 941 y Fm(=)587 880 y Fi(\032)627 912 y Fm(exp)o Fk(f)p Fm(4)p Fl(J)777 919 y Fh(0)797 912 y Fk(g)j(\000)g Fm(1)117 b Fe(if)68 b Fk(jj)p Fm(\010)1185 919 y Fj(Y)1215 912 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1343 919 y Fj(u)1378 912 y Fk(\024)14 b Fl(J)1458 919 y Fh(0)627 972 y Fm(1)399 b Fe(otherwise,)0 1052 y(and)21 b Fk(W)151 1034 y Fh(\003)147 1064 y Fj(k)q(;k)195 1055 y Fd(0)229 1052 y Fe(denotes)h(the)g(set)f(of)g(c)n(onne)n(cte)n(d)g(p)n(aths)f Fl(\015)j Fm(=)d(\()p Fl(X)1150 1059 y Fh(1)1170 1052 y Fl(;)8 b Fk(\001)g(\001)g(\001)h Fl(;)f(X)1321 1059 y Fj(p)1341 1052 y Fm(\))21 b Fe(of)f(sets)h(interse)n(cting)i Fm(\003)d Fe(with)0 1112 y(diameter)e(less)g(than)g Fl(R)g Fe(and)f(such)h(that)g Fl(k)e Fk(2)e Fl(X)895 1119 y Fh(1)915 1112 y Fe(,)j Fl(X)987 1119 y Fj(i)1013 1112 y Fk(\\)11 b Fl(X)1097 1119 y Fj(i)p Fh(+1)1171 1112 y Fk(6)p Fm(=)i Fk(;)p Fe(,)18 b Fl(k)1307 1094 y Fg(0)1332 1112 y Fk(2)c Fl(X)1419 1119 y Fj(p)1440 1112 y Fe(.)0 1220 y(Pr)n(o)n(of)i(of)h(Pr)n(op)n(erty)f(4.4.)p 0 1237 467 2 v 73 1281 a Fm(W)l(e)g(shall)g(pro)o(v)o(e)g(Prop)q(ert)o(y)g (4.4)g(b)o(y)g(induction)g(on)h(the)f(v)o(olume)e(of)i(\003.)73 1364 y Fe(We)i(\014rst)f(pr)n(ove)g(the)h(pr)n(op)n(erty)e(when)i Fk(j)p Fm(\003)p Fk(j)c Fm(=)f(1)p Fe(.)0 1436 y Fm(In)21 b(that)h(case,)g (since)e(w)o(e)h(consider)g(in)o(teger)f(n)o(um)o(b)q(ers)g Fl(D)k(>)e(R)p Fm(,)h(w)o(e)e(only)g(need)f(to)i(c)o(hec)o(k)d(that)j(the)0 1496 y(sw)o(eeping)d(out)g(co)q(e\016cien)o(ts)f(can)h(b)q(e)h(b)q(ounded)g (roughly)f(sp)q(eaking)h(b)o(y)f(the)f(in)o(v)o(erse)g(of)h Fl(m)1718 1503 y Fh(\003)1744 1496 y Fm(\()p Fl(D)q Fm(\).)31 b(But)0 1557 y(w)o(e)16 b(ha)o(v)o(e)f(:)404 1732 y Fk(r)446 1739 y Fj(i)468 1684 y Fi(\020)493 1732 y Fl(E)529 1739 y Fh(\003)555 1732 y Fm([)p Fl(f)598 1712 y Fh(2)618 1732 y Fm(])632 1684 y Fi(\021)661 1682 y Fc(1)p 661 1688 16 2 v 661 1709 a(2)725 1732 y Fm(=)916 1699 y(1)p 810 1721 237 2 v 810 1779 a(2)8 b(\()p Fl(E)897 1786 y Fh(\003)924 1779 y Fm([)p Fl(f)967 1765 y Fh(2)986 1779 y Fm(]\))1024 1742 y Fc(1)p 1024 1748 16 2 v 1024 1768 a(2)1051 1732 y Fk(r)1093 1739 y Fj(i)1107 1732 y Fl(E)1143 1739 y Fh(\003)1169 1732 y Fm([)p Fl(f)1212 1712 y Fh(2)1232 1732 y Fm(])725 1874 y(=)916 1841 y(1)p 810 1863 237 2 v 810 1921 a(2)g(\()p Fl(E)897 1928 y Fh(\003)924 1921 y Fm([)p Fl(f)967 1907 y Fh(2)986 1921 y Fm(]\))1024 1884 y Fc(1)p 1024 1890 16 2 v 1024 1910 a(2)1060 1826 y Fi(\020)1084 1874 y Fm(2)p Fl(E)1144 1881 y Fh(\003)1171 1874 y Fm([)p Fl(f)d Fk(r)1256 1881 y Fj(i)1270 1874 y Fl(f)g Fm(])11 b(+)g Fl(E)1412 1854 y Fj(i)1409 1887 y Fh(\003)1436 1874 y Fm([)p Fl(f)1479 1854 y Fh(2)1498 1874 y Fm(])1512 1826 y Fi(\021)1862 1874 y Fm(\(32\))0 2052 y(where)16 b(w)o(e)g(ha)o(v)o(e)f(denoted)h(:)572 2172 y Fl(E)611 2152 y Fj(i)608 2185 y Fh(\003)634 2172 y Fm([)p Fl(f)677 2152 y Fh(2)697 2172 y Fm(])d(=)h Fl(E)812 2179 y Fh(\003)847 2124 y Fi(h)867 2172 y Fl(f)896 2152 y Fh(2)924 2172 y Fm(\()p Fl(E)979 2179 y Fh(\003)1006 2172 y Fm([)p Fk(r)1062 2179 y Fj(i)1075 2172 y Fl(H)1115 2179 y Fh(\003)1142 2172 y Fm(])c Fk(\000)h(r)1258 2179 y Fj(i)1272 2172 y Fl(H)1312 2179 y Fh(\003)1339 2172 y Fm(\))1358 2124 y Fi(i)1862 2172 y Fm(\(33\))0 2266 y(It)i(is)h(clear)f(that)i Fl(E)354 2248 y Fj(i)351 2279 y Fh(\003)377 2266 y Fm([)p Fl(f)420 2248 y Fh(2)440 2266 y Fm(])f(is)f(n)o(ull)g(unless)h Fl(d)p Fm(\()p Fl(i;)8 b Fm(\003\))13 b Fk(\024)h Fl(R)g Fm(since)g Fl(H)1161 2273 y Fh(\003)1201 2266 y Fm(is)g(lo)q(calized)f(in)h(\003)6 b Fk([)g Fl(@)1605 2273 y Fj(R)1634 2266 y Fm(\003.)20 b(Th)o(us,)14 b(using)0 2327 y(Cauc)o(h)o(y)i(Sc)o(h)o(w)o(arz)f(inequalit)o(y)l(,)f(the)32 b(righ)o(t)16 b(hand)h(side)f(of)g(\(32\))h(is)f(b)q(ounded)h(b)o(y)f(:)293 2440 y Fi(\014)293 2465 y(\014)293 2490 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16 bop 0 26 a Fm(T)l(o)17 b(b)q(ound)g(the)f(last)h(term)d(in)i(the)32 b(righ)o(t)16 b(hand)h(side)f(of)g(\(34\),)h(w)o(e)f(\014rst)g(notice)g(that) h(:)681 147 y Fk(jjr)751 154 y Fj(i)764 147 y Fl(H)804 154 y Fh(\003)831 147 y Fk(jj)859 154 y Fj(u)895 147 y Fk(\024)d(jjr)1018 154 y Fj(i)1036 134 y Fm(~)1031 147 y(\010)1066 126 y Fh(\()p Fj(i)p Fh(\))1108 147 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1236 154 y Fh(1)1255 147 y Fl(;)593 b Fm(\(35\))0 232 y(so)17 b(that)g(:)285 399 y Fk(j)p Fl(E)338 379 y Fj(i)335 412 y Fh(\003)361 399 y Fm([)p Fl(f)404 379 y Fh(2)424 399 y Fm(])p Fk(j)41 b Fm(=)h Fk(j)p Fl(E)623 406 y Fh(\003)658 351 y Fi(h)677 399 y Fk(r)719 406 y Fj(i)733 399 y Fl(H)773 406 y Fh(\003)808 351 y Fi(\020)833 399 y Fl(E)869 406 y Fh(\003)896 399 y Fm([)p Fl(f)939 379 y Fh(2)958 399 y Fm(])11 b Fk(\000)g Fl(f)1062 379 y Fh(2)1082 351 y Fi(\021i)1135 399 y Fk(j)493 491 y Fm(=)42 b Fk(j)p Fl(E)626 470 y Fg(\012)p Fh(2)623 503 y(\003)681 442 y Fi(h)701 491 y Fk(r)743 498 y Fj(i)757 491 y Fl(H)797 498 y Fh(\003)824 491 y Fm(\()p Fl(x)p Fm(\))898 442 y Fi(\020)922 491 y Fl(f)951 470 y Fh(2)971 491 y Fm(\()p Fl(x)p Fm(\))11 b Fk(\000)g Fl(f)1127 470 y Fh(2)1147 491 y Fm(\()s(~)-27 b Fl(x)p Fm(\))1213 442 y Fi(\021)o(i)1265 491 y Fk(j)493 574 y(\024)41 b(jjr)643 581 y Fj(i)656 574 y Fl(H)696 581 y Fh(\003)723 574 y Fk(jj)751 581 y Fj(u)773 574 y Fk(j)p Fl(E)826 554 y Fg(\012)p Fh(2)823 586 y(\003)882 574 y Fm([)o(\()p Fl(f)5 b Fm(\()p Fl(x)p Fm(\))11 b(+)g Fl(f)5 b Fm(\()s(~)-27 b Fl(x)p Fm(\)\))9 b(\()p Fl(f)c Fm(\()p Fl(x)p Fm(\))11 b Fk(\000)g Fl(f)5 b Fm(\()s(~)-27 b Fl(x)p Fm(\)\)])7 b Fk(j)493 669 y(\024)41 b Fm(2)p Fk(jjr)667 676 y Fj(i)686 657 y Fm(~)681 669 y(\010)716 649 y Fh(\()p Fj(i)p Fh(\))757 669 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)885 676 y Fh(1)913 621 y Fi(\020)937 669 y Fl(E)973 676 y Fh(\003)1000 669 y Fm([)p Fl(f)1043 649 y Fh(2)1063 669 y Fm(])1077 621 y Fi(\021)1106 619 y Fc(1)p 1106 625 16 2 v 1106 646 a(2)1137 621 y Fi(\020)1161 669 y Fl(E)1200 649 y Fg(\012)p Fh(2)1197 682 y(\003)1256 621 y Fi(h)1276 669 y Fm(\()o Fl(f)d Fm(\()p Fl(x)p Fm(\))12 b Fk(\000)e Fl(f)5 b Fm(\()s(~)-27 b Fl(x)p Fm(\)\))1564 645 y Fh(2)1584 621 y Fi(i\021)1633 619 y Fc(1)p 1633 625 V 1633 646 a(2)1862 669 y Fm(\(36\))0 849 y(where)18 b(w)o(e)h(ha)o(v)o(e)f(\014nally)g(used)h(Cauc)o(h)o(y)f(Sc)o(h)o(w)o(arz)g (inequalit)o(y)f(to)i(get)g(the)37 b(righ)o(t)19 b(hand)g(side)f(of)i (\(36\).)0 909 y(Moreo)o(v)o(er,)15 b(b)o(y)g(de\014nition)h(of)h Fl(m)611 916 y Fh(\003)637 909 y Fm(\()p Fl(D)q Fm(\))g(w)o(e)f(ha)o(v)o(e)f (that)i(:)508 1039 y Fl(E)547 1019 y Fg(\012)p Fh(2)544 1051 y(\003)602 991 y Fi(h)622 1039 y Fm(\()p Fl(f)5 b Fm(\()p Fl(x)p Fm(\))11 b Fk(\000)g Fl(f)5 b Fm(\()s(~)-27 b Fl(x)p Fm(\)\))911 1015 y Fh(2)931 991 y Fi(i)964 1039 y Fk(\024)1084 1005 y Fm(2)p 1022 1027 149 2 v 1022 1073 a Fl(m)1065 1080 y Fh(\003)1091 1073 y Fm(\()p Fl(D)q Fm(\))1176 1039 y Fl(E)1212 1046 y Fh(\003)1247 991 y Fi(h)1266 1039 y Fm(\()p Fk(r)1327 1046 y Fh(\003)1353 1039 y Fl(f)5 b Fm(\))1402 1015 y Fh(2)1421 991 y Fi(i)0 1153 y Fm(so)17 b(that)g(\(36\))g(giv)o(es)e(:)273 1342 y Fk(j)p Fl(E)326 1321 y Fj(i)323 1354 y Fh(\003)350 1342 y Fm([)p Fl(f)393 1321 y Fh(2)412 1342 y Fm(])p Fk(j)41 b(\024)g Fm(2)585 1267 y Fi(s)p 627 1267 159 2 v 695 1308 a Fm(2)p 632 1330 149 2 v 632 1376 a Fl(m)675 1383 y Fh(\003)701 1376 y Fm(\()p Fl(D)q Fm(\))786 1342 y Fk(jjr)856 1349 y Fj(i)875 1329 y Fm(~)869 1342 y(\010)904 1321 y Fh(\()p Fj(i)p Fh(\))946 1342 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1074 1349 y Fh(1)1101 1293 y Fi(\020)1126 1342 y Fl(E)1162 1349 y Fh(\003)1189 1342 y Fm([)p Fl(f)1232 1321 y Fh(2)1251 1342 y Fm(])1265 1293 y Fi(\021)1295 1291 y Fc(1)p 1295 1297 16 2 v 1295 1318 a(2)1325 1293 y Fi(\020)1350 1342 y Fl(E)1386 1349 y Fh(\003)1421 1293 y Fi(h)1441 1342 y Fm(\()p Fk(r)1502 1349 y Fh(\003)1528 1342 y Fl(f)d Fm(\))1576 1317 y Fh(2)1596 1293 y Fi(i)o(\021)1645 1291 y Fc(1)p 1645 1297 V 1645 1318 a(2)1862 1342 y Fm(\(37\))0 1521 y(As)16 b(a)h(consequence)e (\(34\))i(and)g(\(37\))g(sho)o(w)g(:)226 1636 y Fi(\014)226 1661 y(\014)226 1686 y(\014)226 1710 y(\014)240 1698 y Fk(r)282 1705 y Fj(i)304 1650 y Fi(\020)329 1698 y Fl(E)365 1705 y Fh(\003)392 1698 y Fm([)p Fl(f)435 1677 y Fh(2)454 1698 y Fm(])468 1650 y Fi(\021)498 1648 y Fc(1)p 498 1654 V 498 1674 a(2)520 1636 y Fi(\014)520 1661 y(\014)520 1686 y(\014)520 1710 y(\014)575 1698 y Fk(\024)655 1650 y Fi(\020)680 1698 y Fl(E)716 1705 y Fh(\003)751 1650 y Fi(h)771 1698 y Fk(jr)827 1705 y Fj(i)840 1698 y Fl(f)5 b Fk(j)883 1677 y Fh(2)903 1650 y Fi(i\021)952 1648 y Fc(1)p 952 1654 V 952 1674 a(2)655 1830 y Fm(+)p Fe(1)-8 b(I)729 1837 y Fj(i)p Fg(2)p Fj(@)784 1843 y Ff(R)808 1837 y Fh(\003)835 1755 y Fi(s)p 876 1755 159 2 v 943 1796 a Fm(2)p 881 1818 149 2 v 881 1864 a Fl(m)924 1871 y Fh(\003)950 1864 y Fm(\()p Fl(D)q Fm(\))1035 1830 y Fk(jjr)1105 1837 y Fj(i)1124 1817 y Fm(~)1118 1830 y(\010)1153 1809 y Fh(\()p Fj(i)p Fh(\))1195 1830 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1323 1837 y Fh(1)1350 1782 y Fi(\020)1375 1830 y Fl(E)1411 1837 y Fh(\003)1446 1782 y Fi(h)1466 1830 y Fm(\()p Fk(r)1527 1837 y Fh(\003)1553 1830 y Fl(f)d Fm(\))1601 1806 y Fh(2)1621 1782 y Fi(i)o(\021)1670 1780 y Fc(1)p 1670 1786 16 2 v 1670 1806 a(2)1701 1830 y Fl(;)147 b Fm(\(38\))0 2009 y(whic)o(h)15 b(pro)o(v)o(es)h(that)g(Prop)q(ert)o(y)g (4.4)g(holds)h(when)f Fk(j)p Fm(\003)p Fk(j)d Fm(=)h(1,)i(as)g(w)o(ell)f(as)i (the)e(b)q(ound)i(\(31\))g(with)f Fl(C)h Fm(=)1883 1968 y Fk(p)p 1924 1968 25 2 v 1924 2009 a Fm(2)0 2069 y(since)e Fl(d)p Fm(\(\003)p Fl(;)8 b(i)p Fm(\))14 b Fk(\024)g Fl(R)g Fk(\024)f Fl(D)19 b Fm(when)d Fl(i)d Fk(2)h Fl(@)714 2076 y Fj(R)743 2069 y Fm(\003.)73 2154 y Fe(The)20 b(induction)i(hyp)n(otheses)e(:)27 b Fm(W)l(e)19 b(no)o(w)h(let)e Fl(m)h Fm(b)q(e)g(an)h(in)o(teger)e(n)o(um)o(b)q(er)g(and)h (assume)g(Prop)q(ert)o(y)0 2214 y(4.4)e(is)f(true)g(with)g(an)o(y)g(\003)g (with)g(v)o(olume)e(smaller)g(or)j(equal)f(to)g Fl(m)11 b Fk(\000)g Fm(1,)16 b(as)h(w)o(ell)e(as)i(the)f(con)o(trol)g(\(31\).)73 2274 y(Let)f(no)o(w)f(\003)h(b)q(e)f(a)h(subset)f(of)h Fl(Z)-21 b(Z)660 2253 y Fj(d)695 2274 y Fm(suc)o(h)14 b(that)h Fk(j)p Fm(\003)p Fk(j)e Fm(=)h Fl(m)g Fm(and)h Fl(i)f Fm(b)q(e)g(a)h(site)f(in)g Fl(@)1487 2281 y Fj(R)1515 2274 y Fm(\003.)21 b(W)l(e)14 b(then)g(consider)0 2334 y(an)k(en)o(umeration)d Fk(f)p Fl(j)395 2341 y Fh(1)415 2334 y Fl(;)8 b(j)457 2341 y Fh(2)477 2334 y Fl(;)g Fk(\001)g(\001)g(\001)g Fl(;)g(j)607 2341 y Fj(m)641 2334 y Fk(g)p Fm(,)17 b(of)g(\003)h(suc)o(h)f (that)h Fk(j)p Fl(j)1057 2341 y Fj(n)1092 2334 y Fk(\000)11 b Fl(i)p Fk(j)16 b(\025)f(j)p Fl(j)1277 2341 y Fj(n)p Fh(+1)1358 2334 y Fk(\000)c Fl(i)p Fk(j)p Fm(,)17 b(for)g(an)o(y)h Fl(n)e Fk(2)f Fm([1)p Fl(;)8 b(m)k Fk(\000)f Fm(1].)0 2394 y(W)l(e)16 b(write)g(\003)243 2401 y Fh(0)276 2394 y Fm(=)e(\003)i(and,)g(for)h Fl(n)d Fk(\025)f Fm(1,)k(\003)745 2401 y Fj(n)782 2394 y Fm(=)d(\003)868 2401 y Fj(n)p Fg(\000)p Fh(1)947 2394 y Fk(\000)d(f)p Fl(j)1042 2401 y Fj(n)1065 2394 y Fk(g)p Fm(.)22 b(W)l(e)16 b(also)g(denote)g Fl(f)1493 2376 y Fh(2)1488 2407 y Fj(n)1527 2394 y Fm(=)e Fl(E)1615 2401 y Fh(\003)1639 2405 y Ff(n)1663 2394 y Fm([)p Fl(f)1706 2376 y Fh(2)1726 2394 y Fm(].)73 2454 y(According)g(to)g(\(32\),)h(w)o(e)f (essen)o(tially)e(need)i(to)g(b)q(ound)h Fk(r)1135 2461 y Fj(i)1149 2454 y Fl(E)1185 2461 y Fh(\003)1211 2454 y Fm([)p Fl(f)1254 2436 y Fh(2)1274 2454 y Fm(].)20 b(Let)14 b Fl(N)20 b Fm(b)q(e)14 b(an)g(in)o(teger)g(n)o(um)o(b)q(er)e(\()0 2515 y(to)h(b)q(e)f(sp)q (eci\014ed)g(latter)g(\))g(smaller)e(than)j Fl(m)p Fm(.)20 b(One)12 b(notices)g(that,)h(b)o(y)f(induction)g(and)g(with)h(the)f (notations)0 2575 y(of)17 b(\(33\))g(:)950 2699 y(17)p eop %%Page: 18 18 18 17 bop 321 161 a Fk(r)363 168 y Fj(i)377 161 y Fl(E)413 168 y Fh(\003)439 161 y Fm([)p Fl(f)482 140 y Fh(2)502 161 y Fm(])41 b(=)637 107 y Fj(N)t Fg(\000)p Fh(1)645 119 y Fi(X)642 210 y Fj(n)p Fh(=0)721 161 y Fl(E)757 168 y Fh(\003)792 113 y Fi(h)812 161 y Fl(E)851 140 y Fj(i)848 173 y Fh(\003)872 177 y Ff(n)896 161 y Fm([)p Fl(f)939 140 y Fh(2)934 173 y Fj(n)p Fh(+1)1002 161 y Fm(])1016 113 y Fi(i)1046 161 y Fm(+)11 b Fl(E)1131 168 y Fh(\003)1166 113 y Fi(h)1186 161 y Fl(E)1225 140 y Fj(i)1222 173 y Fh(\003)1246 179 y Ff(N)1277 161 y Fm([)p Fl(f)1320 140 y Fh(2)1340 161 y Fm(])1354 113 y Fi(i)1384 161 y Fm(+)g Fl(E)1469 168 y Fh(\003)1496 161 y Fm([)p Fk(r)1552 168 y Fj(i)1565 161 y Fl(f)1594 140 y Fh(2)1614 161 y Fm(])234 b(\(39\))0 344 y(One)22 b(can)g(no)o(w)g(b)q(ound)h(the)f(\014rst)g(sum)f(of) i(terms)d(in)i(the)43 b(righ)o(t)22 b(hand)h(side)e(of)i(\(39\))f(using)h (cluster)0 404 y(prop)q(erties)16 b(in)g(the)g(small)f(couplings)h(regions)h (as)g(dev)o(elop)q(ed)e(in)h([7])f(and)i(the)f(induction)g(h)o(yp)q(otheses.) 73 464 y(Indeed,)f(w)o(e)h(\014rst)h(ha)o(v)o(e)e(:)213 644 y Fl(E)252 623 y Fj(i)249 656 y Fh(\003)273 660 y Ff(n)306 596 y Fi(h)325 644 y Fl(f)354 623 y Fh(2)349 656 y Fj(n)p Fh(+1)418 596 y Fi(i)479 644 y Fm(=)563 610 y(1)p 563 632 25 2 v 563 678 a(2)593 644 y Fl(E)632 623 y Fg(\012)p Fh(2)629 656 y(\003)653 660 y Ff(n)687 596 y Fi(h\020)732 644 y Fl(f)761 623 y Fh(2)756 656 y Fj(n)p Fh(+1)824 644 y Fm(\()p Fl(x)p Fm(\))c Fk(\000)g Fl(f)980 623 y Fh(2)975 656 y Fj(n)p Fh(+1)1043 644 y Fm(\()s(~)-27 b Fl(x)p Fm(\))1109 596 y Fi(\021)1142 644 y Fm(\()p Fk(r)1203 651 y Fj(i)1217 644 y Fl(H)1257 651 y Fh(\003)1281 655 y Ff(n)1305 644 y Fm(\()p Fl(x)p Fm(\))11 b Fk(\000)f(r)1473 651 y Fj(i)1487 644 y Fl(H)1527 651 y Fh(\003)1551 655 y Ff(n)1575 644 y Fm(\()s(~)-27 b Fl(x)p Fm(\)\))1660 596 y Fi(i)1862 644 y Fm(\(40\))479 761 y(=)563 727 y(1)p 563 749 V 563 795 a(2)593 761 y Fl(E)632 740 y Fg(\012)p Fh(2)629 773 y(\003)653 777 y Ff(n)679 761 y Fm([)701 712 y Fi(\020)726 761 y Fl(f)755 740 y Fh(2)750 773 y Fj(n)p Fh(+1)818 761 y Fm(\()p Fl(x)p Fm(\))11 b Fk(\000)g Fl(f)974 740 y Fh(2)969 773 y Fj(n)p Fh(+1)1037 761 y Fm(\()s(~)-27 b Fl(x)p Fm(\))1103 712 y Fi(\021)575 813 y(\020)599 861 y Fl(E)635 868 y Fh(\003)659 873 y Ff(n)p Fc(+1)722 861 y Fm([)p Fk(r)778 868 y Fj(i)791 861 y Fl(H)831 868 y Fh(\003)855 872 y Ff(n)879 861 y Fm(]\()p Fl(!)13 b Fk(\017)980 868 y Fj(j)994 873 y Ff(n)p Fc(+1)1067 861 y Fl(x)p Fm(\))e Fk(\000)g Fl(E)1211 868 y Fh(\003)1235 873 y Ff(n)p Fc(+1)1297 861 y Fm([)p Fk(r)1353 868 y Fj(i)1366 861 y Fl(H)1406 868 y Fh(\003)1430 872 y Ff(n)1454 861 y Fm(]\()p Fl(!)i Fk(\017)1555 868 y Fj(j)1569 873 y Ff(n)p Fc(+1)1645 861 y Fm(~)-27 b Fl(x)p Fm(\))1689 813 y Fi(\021)1722 861 y Fm(])0 1044 y(where)17 b(in)f(the)h(last)h(step)f(step)g(w)o(e)f(ha)o (v)o(e)h(used)g(that)g Fl(f)1014 1051 y Fj(n)p Fh(+1)1100 1044 y Fm(is)g Fk(F)1186 1051 y Fh(\003)1210 1040 y Ff(c)1210 1063 y(n)p Fc(+1)1289 1044 y Fm(measurable.)23 b(W)l(e)16 b(no)o(w)i(mak)o(e)d (the)0 1104 y(follo)o(wing)h(crude)g(b)q(ond)h(in)f(the)32 b(righ)o(t)16 b(hand)h(side)f(of)g(\(40\))h(:)153 1224 y Fi(\014)153 1249 y(\014)153 1274 y(\014)o Fl(E)205 1254 y Fj(i)202 1287 y Fh(\003)226 1291 y Ff(n)259 1226 y Fi(h)278 1274 y Fl(f)307 1254 y Fh(2)302 1287 y Fj(n)p Fh(+1)371 1226 y Fi(i)390 1224 y(\014)390 1249 y(\014)390 1274 y(\014)72 1382 y Fk(\024)158 1348 y Fm(1)p 158 1370 V 158 1416 a(2)195 1332 y Fi(\014)195 1357 y(\014)195 1382 y(\014)209 1332 y(\014)209 1357 y(\014)209 1382 y(\014)p Fl(E)259 1389 y Fh(\003)283 1394 y Ff(n)p Fc(+1)345 1382 y Fm([)p Fk(r)401 1389 y Fj(i)415 1382 y Fl(H)455 1389 y Fh(\003)479 1393 y Ff(n)503 1382 y Fm(]\()p Fl(!)12 b Fk(\017)603 1389 y Fj(j)617 1394 y Ff(n)p Fc(+1)691 1382 y Fl(x)p Fm(\))e Fk(\000)h Fl(E)834 1389 y Fh(\003)858 1394 y Ff(n)p Fc(+1)921 1382 y Fm([)p Fk(r)977 1389 y Fj(i)990 1382 y Fl(H)1030 1389 y Fh(\003)1054 1393 y Ff(n)1078 1382 y Fm(]\()p Fl(!)i Fk(\017)1179 1389 y Fj(j)1193 1394 y Ff(n)p Fc(+1)1269 1382 y Fm(~)-27 b Fl(x)o Fm(\))1312 1332 y Fi(\014)1312 1357 y(\014)1312 1382 y(\014)1326 1332 y(\014)1326 1357 y(\014)1326 1382 y(\014)1340 1408 y Fj(u)1371 1382 y Fl(E)1410 1361 y Fg(\012)p Fh(2)1407 1394 y(\003)1431 1398 y Ff(n)1465 1333 y Fi(h)1485 1332 y(\014)1485 1357 y(\014)1485 1382 y(\014)p Fl(f)1528 1361 y Fh(2)1523 1394 y Fj(n)p Fh(+1)1591 1382 y Fm(\()p Fl(x)p Fm(\))11 b Fk(\000)g Fl(f)1747 1361 y Fh(2)1742 1394 y Fj(n)p Fh(+1)1811 1382 y Fm(\()s(~)-27 b Fl(x)o Fm(\))1876 1332 y Fi(\014)1876 1357 y(\014)1876 1382 y(\014)1890 1333 y(i)72 1482 y Fk(\024)153 1432 y Fi(\014)153 1457 y(\014)153 1482 y(\014)166 1432 y(\014)166 1457 y(\014)166 1482 y(\014)p Fl(E)216 1489 y Fh(\003)240 1494 y Ff(n)p Fc(+1)303 1482 y Fm([)p Fk(r)359 1489 y Fj(i)372 1482 y Fl(H)412 1489 y Fh(\003)436 1493 y Ff(n)460 1482 y Fm(]\()p Fl(!)13 b Fk(\017)561 1489 y Fj(j)575 1494 y Ff(n)p Fc(+1)648 1482 y Fl(x)p Fm(\))e Fk(\000)f Fl(E)791 1489 y Fh(\003)815 1494 y Ff(n)p Fc(+1)878 1482 y Fm([)p Fk(r)934 1489 y Fj(i)947 1482 y Fl(H)987 1489 y Fh(\003)1011 1493 y Ff(n)1035 1482 y Fm(]\()p Fl(!)j Fk(\017)1136 1489 y Fj(j)1150 1494 y Ff(n)p Fc(+1)1226 1482 y Fm(~)-27 b Fl(x)p Fm(\))1270 1432 y Fi(\014)1270 1457 y(\014)1270 1482 y(\014)1284 1432 y(\014)1284 1457 y(\014)1284 1482 y(\014)1297 1509 y Fj(u)153 1592 y Fk(\002)o Fl(E)230 1571 y Fg(\012)p Fh(2)227 1604 y(\003)251 1608 y Ff(n)286 1544 y Fi(h)305 1592 y Fm(\()p Fl(f)348 1599 y Fj(n)p Fh(+1)417 1592 y Fm(\()p Fl(x)p Fm(\))11 b Fk(\000)g Fl(f)568 1599 y Fj(n)p Fh(+1)636 1592 y Fm(\()s(~)-27 b Fl(x)p Fm(\)\))721 1568 y Fh(2)741 1544 y Fi(i)765 1542 y Fc(1)p 765 1548 16 2 v 765 1568 a(2)796 1592 y Fl(E)832 1599 y Fh(\003)856 1603 y Ff(n)888 1544 y Fi(h)908 1592 y Fl(f)937 1571 y Fh(2)932 1604 y Fj(n)p Fh(+1)1000 1544 y Fi(i)1025 1542 y Fc(1)p 1025 1548 V 1025 1568 a(2)1862 1592 y Fm(\(41\))0 1775 y(Moreo)o(v)o(er,)15 b(since)g Fl(f)371 1782 y Fj(n)p Fh(+1)440 1775 y Fm(\()p Fl(x)p Fm(\))h(only)g(dep)q(ends)g(on)h(the)f(site)g Fl(j)1079 1782 y Fj(n)p Fh(+1)1164 1775 y Fm(in)g(\003)1255 1782 y Fj(n)1278 1775 y Fm(,)g(w)o(e)g(ha)o(v)o(e)f(that)i(:)323 1955 y Fl(E)362 1935 y Fg(\012)p Fh(2)359 1967 y(\003)383 1971 y Ff(n)417 1907 y Fi(h)437 1955 y Fm(\()p Fl(f)480 1962 y Fj(n)p Fh(+1)549 1955 y Fm(\()p Fl(x)p Fm(\))10 b Fk(\000)h Fl(f)699 1962 y Fj(n)p Fh(+1)768 1955 y Fm(\()s(~)-27 b Fl(x)o Fm(\)\))852 1931 y Fh(2)872 1907 y Fi(i)897 1905 y Fc(1)p 897 1911 V 897 1931 a(2)933 1955 y Fk(\024)243 2055 y(\024)41 b Fm(exp)o Fk(f)p Fm(4)p Fk(jj)479 2043 y Fm(~)474 2055 y(\010)509 2035 y Fh(\()p Fj(j)537 2040 y Ff(n)p Fc(+1)597 2035 y Fh(\))613 2055 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)741 2062 y Fh(1)760 2055 y Fk(g)793 1997 y Fi(Z)843 2055 y Fm(\()p Fl(f)886 2062 y Fj(n)p Fh(+1)955 2055 y Fm(\()p Fl(x)p Fm(\))i Fk(\000)h Fl(f)1105 2062 y Fj(n)p Fh(+1)1174 2055 y Fm(\()s(~)-27 b Fl(x)o Fm(\)\))1258 2031 y Fh(2)1286 2055 y Fl(d\027)1335 2062 y Fj(M)1375 2055 y Fm(\()p Fl(x)1422 2062 y Fj(j)1436 2067 y Ff(n)p Fc(+1)1498 2055 y Fm(\))p Fl(d\027)1566 2062 y Fj(M)1606 2055 y Fm(\()s(~)g Fl(x)1653 2062 y Fj(j)1667 2067 y Ff(n)p Fc(+1)1729 2055 y Fm(\))0 2238 y(And)16 b(w)o(e)g(kno)o(w)g(that)h(there)f(exists)f Fl(m)714 2245 y Fh(0)750 2238 y Fm(suc)o(h)h(that)h(:)100 2359 y Fi(Z)150 2418 y Fm(\()p Fl(f)193 2425 y Fj(n)p Fh(+1)262 2418 y Fm(\()p Fl(x)p Fm(\))10 b Fk(\000)h Fl(f)412 2425 y Fj(n)p Fh(+1)481 2418 y Fm(\()s(~)-27 b Fl(x)o Fm(\)\))565 2394 y Fh(2)593 2418 y Fl(d\027)642 2425 y Fj(M)682 2418 y Fm(\()p Fl(x)729 2425 y Fj(j)743 2430 y Ff(n)p Fc(+1)805 2418 y Fm(\))p Fl(d\027)873 2425 y Fj(M)913 2418 y Fm(\()s(~)g Fl(x)960 2425 y Fj(j)974 2430 y Ff(n)p Fc(+1)1036 2418 y Fm(\))14 b Fk(\024)1145 2384 y Fm(2)p 1126 2406 63 2 v 1126 2452 a Fl(m)1169 2459 y Fh(0)1202 2359 y Fi(Z)1252 2370 y(\020)1276 2418 y Fk(r)1318 2425 y Fj(j)1332 2430 y Ff(n)p Fc(+1)1394 2418 y Fl(f)1418 2425 y Fj(n)p Fh(+1)1487 2370 y Fi(\021)1511 2381 y Fh(2)1539 2418 y Fl(d\027)1588 2425 y Fj(M)1628 2418 y Fm(\()p Fl(x)1675 2425 y Fj(j)1689 2430 y Ff(n)p Fc(+1)1751 2418 y Fm(\))92 b(\(42\))0 2548 y(And)16 b(that)h(w)o(e)f(ha)o(v)o(e)f(again)i(:)950 2699 y(18)p eop %%Page: 19 19 19 18 bop 218 34 a Fi(Z)268 45 y(\020)293 93 y Fk(r)335 100 y Fj(j)349 105 y Ff(n)p Fc(+1)410 93 y Fl(f)434 100 y Fj(n)p Fh(+1)503 45 y Fi(\021)528 56 y Fh(2)556 93 y Fl(d\027)605 100 y Fj(M)645 93 y Fm(\()p Fl(x)692 100 y Fj(j)706 105 y Ff(n)p Fc(+1)768 93 y Fm(\))13 b Fk(\024)h Fm(exp)p Fk(f)p Fm(2)p Fk(jj)1010 80 y Fm(~)1005 93 y(\010)1040 72 y Fh(\()p Fj(j)1068 77 y Ff(n)p Fc(+1)1127 72 y Fh(\))1143 93 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1271 100 y Fh(1)1290 93 y Fk(g)p Fl(E)1351 100 y Fh(\003)1375 104 y Ff(n)1407 32 y Fi(\024)1429 45 y(\020)1454 93 y Fk(r)1496 100 y Fj(j)1510 105 y Ff(n)p Fc(+1)1572 93 y Fl(f)1596 100 y Fj(n)p Fh(+1)1664 45 y Fi(\021)1689 56 y Fh(2)1709 32 y Fi(\025)0 192 y Fm(As)16 b(a)h(consequence,)e(w)o(e)g(get)i(that)f(:)110 378 y Fl(E)149 358 y Fg(\012)p Fh(2)146 390 y(\003)170 394 y Ff(n)204 330 y Fi(h)224 378 y Fm(\()p Fl(f)267 385 y Fj(n)p Fh(+1)335 378 y Fm(\()p Fl(x)p Fm(\))11 b Fk(\000)g Fl(f)486 385 y Fj(n)p Fh(+1)555 378 y Fm(\()s(~)-27 b Fl(x)o Fm(\)\))639 354 y Fh(2)659 330 y Fi(i)684 328 y Fc(1)p 684 334 16 2 v 684 354 a(2)720 378 y Fk(\024)772 301 y Fi(s)p 814 301 73 2 v 838 344 a Fm(2)p 819 366 63 2 v 819 412 a Fl(m)862 419 y Fh(0)894 378 y Fm(exp)p Fk(f)p Fm(3)p Fk(jj)1051 366 y Fm(~)1046 378 y(\010)1081 358 y Fh(\()p Fj(j)1109 363 y Ff(n)p Fc(+1)1169 358 y Fh(\))1185 378 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1313 385 y Fh(1)1332 378 y Fk(g)p Fl(E)1393 385 y Fh(\003)1417 389 y Ff(n)1440 378 y Fm([)p Fk(jr)1510 385 y Fj(j)1524 390 y Ff(n)p Fc(+1)1585 378 y Fl(f)1609 385 y Fj(n)p Fh(+1)1678 378 y Fk(j)1692 358 y Fh(2)1712 378 y Fm(])1731 344 y Fc(1)p 1730 350 16 2 v 1730 371 a(2)1862 378 y Fm(\(43\))0 509 y(But)16 b Fl(f)126 491 y Fh(2)121 521 y Fj(n)p Fh(+1)204 509 y Fm(=)e Fl(E)292 516 y Fh(\003)316 521 y Ff(n)p Fc(+1)378 509 y Fm([)p Fl(f)421 491 y Fh(2)441 509 y Fm(])i(and)h Fk(j)p Fm(\003)614 516 y Fj(n)p Fh(+1)682 509 y Fk(j)d(\024)g Fl(m)d Fk(\000)g Fm(1)17 b(so)g(that)g(w)o(e)f(can)h(use)f(the)h(induction)f(h)o(yp)q(otheses)g(and)0 569 y(\014nd)h(:)104 731 y Fl(E)140 738 y Fh(\003)164 742 y Ff(n)187 731 y Fm([)p Fk(jr)257 738 y Fj(j)271 743 y Ff(n)p Fc(+1)332 731 y Fl(f)356 738 y Fj(n)p Fh(+1)425 731 y Fk(j)439 710 y Fh(2)459 731 y Fm(])478 697 y Fc(1)p 477 703 V 477 723 a(2)541 731 y Fk(\024)41 b Fl(E)657 738 y Fh(\003)681 742 y Ff(n)705 731 y Fm([)p Fk(jr)775 738 y Fj(j)789 743 y Ff(n)p Fc(+1)850 731 y Fl(f)5 b Fk(j)893 710 y Fh(2)913 731 y Fm(])932 697 y Fc(1)p 931 703 V 931 723 a(2)965 731 y Fm(+)11 b Fe(1)-8 b(I)1049 738 y Fj(k)q Fg(2)p Fj(@)1111 744 y Ff(R)1136 738 y Fh(\003)1160 743 y Ff(n)p Fc(+1)1264 689 y Fi(X)1230 782 y Fj(k)q Fg(2)p Fh(\003)1297 787 y Ff(n)p Fc(+1)1366 731 y Fl(\013)1397 706 y Fh(\003)1421 711 y Ff(n)p Fc(+1)1397 743 y Fj(j)1411 748 y Ff(n)p Fc(+1)1471 743 y Fj(;k)1502 731 y Fl(E)1538 738 y Fh(\003)1562 742 y Ff(n)1586 731 y Fm([)p Fk(jr)1656 738 y Fj(k)1676 731 y Fl(f)5 b Fm(\()p Fl(x)p Fm(\))p Fk(j)1785 710 y Fh(2)1805 731 y Fm(])1824 697 y Fc(1)p 1823 703 V 1823 723 a(2)1862 731 y Fm(\(44\))0 912 y(Moreo)o(v)o(er,)17 b(using)h(cluster)f (prop)q(erties)h(in)f(the)h(small)e(couplings)i(regions)g(as)h(dev)o(elopp)q (ed)e(in)g([7])g(\()h(see)0 972 y(Theorem)d(2.1)i(\),)e(one)i(\014nds)g (that,)f(for)g(an)o(y)g Fl(n)e Fk(2)g Fm([0)p Fl(;)8 b(N)16 b Fk(\000)11 b Fm(1[)17 b(;)42 1084 y Fi(\014)42 1109 y(\014)42 1134 y(\014)56 1084 y(\014)56 1109 y(\014)56 1134 y(\014)o Fl(E)105 1141 y Fh(\003)129 1146 y Ff(n)p Fc(+1)192 1134 y Fm([)p Fk(r)248 1141 y Fj(i)261 1134 y Fl(H)t Fm(]\()p Fl(!)c Fk(\017)406 1141 y Fj(j)420 1146 y Ff(n)p Fc(+1)493 1134 y Fl(x)p Fm(\))e Fk(\000)g Fl(E)637 1141 y Fh(\003)661 1146 y Ff(n)p Fc(+1)723 1134 y Fm([)p Fk(r)779 1141 y Fj(i)792 1134 y Fl(H)t Fm(]\()p Fl(!)i Fk(\017)937 1141 y Fj(j)951 1146 y Ff(n)p Fc(+1)1027 1134 y Fm(~)-27 b Fl(x)p Fm(\))1071 1084 y Fi(\014)1071 1109 y(\014)1071 1134 y(\014)1085 1084 y(\014)1085 1109 y(\014)1085 1134 y(\014)1099 1161 y Fj(u)1163 1134 y Fk(\024)41 b Fm(2)p Fk(jjr)1337 1141 y Fj(i)1356 1121 y Fm(~)1351 1134 y(\010)1386 1113 y Fh(\()p Fj(i)p Fh(\))1427 1134 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1555 1141 y Fh(1)1639 1092 y Fi(X)1583 1201 y Fj(\015)r Fg(2W)1665 1178 y Fc(\003)1686 1185 y Ff(n)p Fc(+1)1662 1212 y Ff(i;j)1695 1219 y(n)p Fc(+1)1776 1092 y Fi(Y)1765 1184 y Fj(X)s Fg(2)p Fj(\015)1848 1134 y Fl(\032)1873 1141 y Fj(X)0 1346 y Fm(with)20 b(the)g(notations)h(giv)o(en)e(in)h(prop)q (ert)o(y)g(4.4.)33 b(Th)o(us)21 b(w)o(e)f(get)g(that,)h(for)f(an)o(y)g(in)o (teger)g(n)o(um)o(b)q(er)e Fl(n)i Fm(in)0 1406 y([0)p Fl(;)8 b(N)16 b Fk(\000)11 b Fm(1])16 b(so)h(that)g Fk(j)p Fl(i)10 b Fk(\000)h Fl(j)496 1413 y Fj(n)p Fh(+1)565 1406 y Fk(j)i(\025)h Fl(D)k Fm(and)f(if)f Fl(C)h Fk(\025)d Fm(4,)i Fl(\027)1029 1381 y Fh(\003)1053 1386 y Ff(n)p Fc(+1)1026 1417 y Fj(i;j)1062 1422 y Ff(n)p Fc(+1)1140 1406 y Fm(has)h(b)q(een)f(de\014ned)g(so)h(that)g(:) 42 1528 y Fi(\014)42 1552 y(\014)42 1577 y(\014)56 1528 y(\014)56 1552 y(\014)56 1577 y(\014)p Fl(E)106 1584 y Fh(\003)130 1589 y Ff(n)p Fc(+1)192 1577 y Fm([)p Fk(r)248 1584 y Fj(i)261 1577 y Fl(H)t Fm(]\()p Fl(x)366 1584 y Fj(j)380 1589 y Ff(n)p Fc(+1)442 1577 y Fm(\))11 b Fk(\000)g Fl(E)558 1584 y Fh(\003)582 1589 y Ff(n)p Fc(+1)644 1577 y Fm([)p Fk(r)700 1584 y Fj(i)713 1577 y Fl(H)t Fm(]\()s(~)-27 b Fl(x)818 1584 y Fj(j)832 1589 y Ff(n)p Fc(+1)894 1577 y Fm(\))913 1528 y Fi(\014)913 1552 y(\014)913 1577 y(\014)927 1528 y(\014)927 1552 y(\014)927 1577 y(\014)941 1604 y Fj(u)1005 1577 y Fk(\024)1109 1544 y Fm(1)p 1090 1566 63 2 v 1090 1612 a(2)p Fl(C)1166 1577 y Fm(exp)p Fk(f\000)p Fm(3)p Fk(jjr)1399 1584 y Fj(j)1413 1589 y Ff(n)p Fc(+1)1479 1565 y Fm(~)1474 1577 y(\010)1509 1557 y Fh(\()p Fj(j)1537 1562 y Ff(n)p Fc(+1)1597 1557 y Fh(\))1613 1577 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1741 1584 y Fh(1)1760 1577 y Fk(g)p Fl(\027)1812 1552 y Fh(\003)1836 1557 y Ff(n)p Fc(+1)1809 1589 y Fj(i;j)1845 1594 y Ff(n)p Fc(+1)1862 1665 y Fm(\(45\))0 1767 y(Then)16 b(\(41\),)h(\(43\),)f(\(44\))h(and)g(\(45\))g(sho)o(w)g(that)g(:)523 1825 y Fi(\014)523 1850 y(\014)523 1875 y(\014)p Fl(E)576 1854 y Fj(i)573 1887 y Fh(\003)597 1891 y Ff(n)629 1827 y Fi(h)648 1875 y Fl(f)677 1854 y Fh(2)672 1887 y Fj(n)p Fh(+1)741 1827 y Fi(i)761 1825 y(\014)761 1850 y(\014)761 1875 y(\014)c Fk(\024)250 2032 y(\024)g Fl(\027)329 2007 y Fh(\003)353 2012 y Ff(n)p Fc(+1)326 2044 y Fj(i;j)362 2049 y Ff(n)p Fc(+1)433 1984 y Fi(\020)457 2032 y Fl(E)493 2039 y Fh(\003)517 2043 y Ff(n)550 1984 y Fi(h)569 2032 y Fl(f)598 2012 y Fh(2)618 1984 y Fi(i\021)668 1982 y Fc(1)p 668 1988 16 2 v 668 2009 a(2)698 1947 y Fi(0)698 2021 y(@)735 2032 y Fl(E)771 2039 y Fh(\003)795 2043 y Ff(n)818 2032 y Fm([)p Fk(jr)888 2039 y Fj(j)902 2044 y Ff(n)p Fc(+1)963 2032 y Fl(f)5 b Fk(j)1006 2012 y Fh(2)1026 2032 y Fm(])1045 1998 y Fc(1)p 1045 2004 V 1045 2025 a(2)1078 2032 y Fm(+)1161 1991 y Fi(X)1127 2083 y Fj(k)q Fg(2)p Fh(\003)1194 2088 y Ff(n)p Fc(+1)1263 2032 y Fl(\013)1294 2007 y Fh(\003)1318 2012 y Ff(n)p Fc(+1)1294 2045 y Fj(j)1308 2050 y Ff(n)p Fc(+1)1368 2045 y Fj(;k)1399 2032 y Fl(E)1435 2039 y Fh(\003)1459 2043 y Ff(n)1483 2032 y Fm([)p Fk(jr)1553 2039 y Fj(k)1573 2032 y Fl(f)g Fk(j)1616 2012 y Fh(2)1636 2032 y Fm(])1655 1998 y Fc(1)p 1654 2004 V 1654 2025 a(2)1677 1947 y Fi(1)1677 2021 y(A)1862 2032 y Fm(\(46\))0 2173 y(Quite)15 b(similarly)l(,)e(w)o(e)j(can)g(b)q(ound)h(the)f(second)g (term)e Fl(E)1048 2180 y Fh(\003)1083 2124 y Fi(h)1103 2173 y Fl(E)1142 2154 y Fj(i)1139 2185 y Fh(\003)1163 2191 y Ff(N)1194 2173 y Fm([)p Fl(f)1237 2154 y Fh(2)1257 2173 y Fm(])1271 2124 y Fi(i)1306 2173 y Fm(in)i(\(39\).)22 b(Indeed,)14 b(follo)o(wing)i(\(33\))0 2233 y(w)o(e)g(ha)o(v)o(e)f(:)312 2394 y Fl(E)351 2374 y Fj(i)348 2407 y Fh(\003)372 2413 y Ff(N)404 2394 y Fm([)p Fl(f)447 2374 y Fh(2)467 2394 y Fm(])41 b(=)h Fl(E)638 2401 y Fh(\003)662 2407 y Ff(N)702 2346 y Fi(h)721 2394 y Fk(r)763 2401 y Fj(i)777 2394 y Fl(H)817 2401 y Fh(\003)841 2407 y Ff(N)881 2346 y Fi(\020)906 2394 y Fl(E)942 2401 y Fh(\003)966 2407 y Ff(N)997 2394 y Fm([)p Fl(f)1040 2374 y Fh(2)1060 2394 y Fm(])10 b Fk(\000)h Fl(f)1163 2374 y Fh(2)1183 2346 y Fi(\021i)522 2502 y Fk(\024)41 b Fm(2)p Fk(jjr)696 2509 y Fj(i)715 2490 y Fm(~)709 2502 y(\010)744 2482 y Fh(\()p Fj(i)p Fh(\))786 2502 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)914 2509 y Fh(1)933 2502 y Fm(\()p Fl(E)988 2509 y Fh(\003)1012 2515 y Ff(N)1044 2502 y Fm([)p Fl(f)1087 2482 y Fh(2)1106 2502 y Fm(]\))1144 2468 y Fc(1)p 1144 2474 V 1144 2495 a(2)1166 2502 y Fl(E)1202 2509 y Fh(\003)1226 2515 y Ff(N)1266 2454 y Fi(h)1285 2502 y Fm(\()p Fl(E)1340 2509 y Fh(\003)1364 2515 y Ff(N)1396 2502 y Fm([)p Fl(f)d Fm(])10 b Fk(\000)h Fl(f)5 b Fm(\))1562 2478 y Fh(2)1581 2454 y Fi(i)1606 2452 y Fc(1)p 1606 2458 V 1606 2479 a(2)1862 2502 y Fm(\(47\))950 2699 y(19)p eop %%Page: 20 20 20 19 bop 0 26 a Fm(But,)15 b(b)o(y)h(de\014nition)g(of)g(the)g(sp)q(ectral)h (gap,)409 156 y Fl(E)445 163 y Fh(\003)469 169 y Ff(N)508 108 y Fi(h)528 156 y Fm(\()p Fl(E)583 163 y Fh(\003)607 169 y Ff(N)638 156 y Fm([)p Fl(f)5 b Fm(])11 b Fk(\000)g Fl(f)5 b Fm(\))804 132 y Fh(2)824 108 y Fi(i)848 106 y Fc(1)p 848 112 16 2 v 848 133 a(2)912 156 y Fk(\024)1080 122 y Fm(1)p 997 145 190 2 v 997 153 a Fi(q)p 1039 153 149 2 v 50 x Fl(m)p Fm(\(\003)1135 210 y Fj(N)1168 203 y Fm(\))1192 156 y Fl(E)1228 163 y Fh(\003)1252 169 y Ff(N)1292 108 y Fi(h)1311 156 y Fk(jr)1367 163 y Fh(\003)1391 169 y Ff(N)1422 156 y Fl(f)g Fk(j)1465 136 y Fh(2)1485 108 y Fi(i)1510 106 y Fc(1)p 1510 112 16 2 v 1510 133 a(2)1862 156 y Fm(\(48\))0 353 y(Plugging)17 b(\(46\))g(and)g(this)f(last)g(inequalit) o(y)e(in)i(\(39\),)h(yields)e(:)56 504 y Fi(\014)56 529 y(\014)56 554 y(\014)p Fk(r)112 561 y Fj(i)126 554 y Fl(E)162 561 y Fh(\003)188 554 y Fm([)p Fl(f)231 533 y Fh(2)251 554 y Fm(])265 504 y Fi(\014)265 529 y(\014)265 554 y(\014)41 b Fk(\024)400 506 y Fi(\020)425 554 y Fl(E)461 561 y Fh(\003)496 506 y Fi(h)515 554 y Fl(f)544 533 y Fh(2)564 506 y Fi(i\021)614 504 y Fc(1)p 614 510 V 614 530 a(2)644 500 y Fj(N)t Fg(\000)p Fh(1)653 512 y Fi(X)649 603 y Fj(n)p Fh(=0)729 554 y Fl(\027)756 529 y Fh(\003)780 534 y Ff(n)p Fc(+1)753 565 y Fj(i;j)789 570 y Ff(n)p Fc(+1)862 554 y Fk(\002)912 468 y Fi(0)912 543 y(@)949 554 y Fl(E)985 561 y Fh(\003)1011 554 y Fm([)p Fk(jr)1081 561 y Fj(j)1095 566 y Ff(n)p Fc(+1)1156 554 y Fl(f)5 b Fk(j)1199 533 y Fh(2)1219 554 y Fm(])1238 520 y Fc(1)p 1238 526 V 1238 546 a(2)1271 554 y Fm(+)1353 512 y Fi(X)1320 605 y Fj(k)q Fg(2)p Fh(\003)1387 610 y Ff(n)p Fc(+1)1455 554 y Fl(\013)1486 529 y Fh(\003)1510 534 y Ff(n)p Fc(+1)1486 566 y Fj(j)1500 571 y Ff(n)p Fc(+1)1560 566 y Fj(;k)1591 554 y Fl(E)1627 561 y Fh(\003)1654 554 y Fm([)p Fk(jr)1724 561 y Fj(k)1744 554 y Fl(f)g Fk(j)1787 533 y Fh(2)1807 554 y Fm(])1826 520 y Fc(1)p 1826 526 V 1826 546 a(2)1848 468 y Fi(1)1848 543 y(A)400 700 y Fm(+2)p Fl(m)p Fm(\(\003)558 707 y Fj(N)591 700 y Fm(\))610 679 y Fg(\000)642 666 y Fc(1)p 643 672 V 643 692 a(2)665 700 y Fk(jjr)735 707 y Fj(i)754 687 y Fm(~)748 700 y(\010)783 679 y Fh(\()p Fj(i)p Fh(\))825 700 y Fm(\()p Fl(J)o(;)j(:)p Fm(\))p Fk(jj)953 707 y Fh(1)980 652 y Fi(\020)1005 700 y Fl(E)1041 707 y Fh(\003)1076 652 y Fi(h)1096 700 y Fl(f)1125 679 y Fh(2)1145 652 y Fi(i)o(\021)1194 649 y Fc(1)p 1194 655 V 1194 676 a(2)1225 700 y Fl(E)1261 707 y Fh(\003)1296 652 y Fi(h)1315 700 y Fk(jr)1371 707 y Fh(\003)1395 713 y Ff(N)1426 700 y Fl(f)d Fk(j)1469 679 y Fh(2)1489 652 y Fi(i)1513 649 y Fc(1)p 1513 655 V 1513 676 a(2)1547 700 y Fm(+)11 b Fl(E)1632 707 y Fh(\003)1658 700 y Fm([)p Fk(r)1714 707 y Fj(i)1728 700 y Fl(f)1757 679 y Fh(2)1777 700 y Fm(])71 b(\(49\))0 882 y(Let)16 b(us)h(no)o(w)g(c)o(ho)q(ose)f Fl(N)22 b Fm(suc)o(h)16 b(that)h(:)549 992 y Fl(N)i Fm(=)14 b(inf)s Fk(f)p Fl(n)f Fk(2)h Fm([0)p Fl(;)8 b(m)j Fk(\000)g Fm(1[)16 b Fk(j)g Fm(\003)1114 999 y Fj(n)1151 992 y Fk(\032)e Fl(B)s Fm(\()p Fl(i;)8 b(D)q Fm(\))p Fk(g)p Fl(:)0 1102 y Fm(Then,)16 b(b)o(y)g(de\014nition)f(of)i Fl(m)524 1109 y Fh(\003)550 1102 y Fm(\()p Fl(D)q Fm(\),)g(w)o(e)f(ha)o(v)o(e)f(:)843 1201 y(1)p 781 1223 149 2 v 781 1269 a Fl(m)p Fm(\(\003)877 1276 y Fj(N)910 1269 y Fm(\))948 1235 y Fk(\024)1068 1201 y Fm(1)p 1005 1223 V 1005 1269 a Fl(m)1048 1276 y Fh(\003)1075 1269 y Fm(\()p Fl(D)q Fm(\))1159 1235 y Fl(;)0 1379 y Fm(so)h(that)f(reordering)g(the)31 b(righ)o(t)14 b(hand)i(side)f(of)g(\(50\))h(after)f(the)g(ob)o(vious)h (remark)d(that)j Fl(\027)1654 1361 y Fh(\003)1651 1391 y Fj(i;j)1705 1379 y Fk(\024)e Fl(\027)1785 1361 y Fh(\003)1809 1349 y Fd(0)1782 1391 y Fj(i;j)1838 1379 y Fm(when)0 1439 y(\003)g Fk(\032)f Fm(\003)134 1421 y Fg(0)146 1439 y Fm(,)j(giv)o(es)f(:)152 1591 y Fi(\014)152 1615 y(\014)152 1640 y(\014)p Fk(r)208 1647 y Fj(i)222 1640 y Fl(E)258 1647 y Fh(\003)284 1640 y Fm([)p Fl(f)327 1620 y Fh(2)347 1640 y Fm(])366 1606 y Fc(1)p 365 1612 16 2 v 365 1633 a(2)388 1591 y Fi(\014)388 1615 y(\014)388 1640 y(\014)41 b Fk(\024)523 1586 y Fj(N)t Fg(\000)p Fh(1)532 1599 y Fi(X)528 1689 y Fj(n)p Fh(=0)608 1640 y Fl(\027)635 1615 y Fh(\003)659 1620 y Ff(n)p Fc(+1)632 1652 y Fj(i;j)668 1657 y Ff(n)p Fc(+1)738 1555 y Fi(0)738 1630 y(@)775 1640 y Fl(E)811 1647 y Fh(\003)837 1640 y Fm([)p Fk(jr)907 1647 y Fj(j)921 1652 y Ff(n)p Fc(+1)982 1640 y Fl(f)5 b Fk(j)1025 1620 y Fh(2)1045 1640 y Fm(])1064 1606 y Fc(1)p 1064 1612 V 1064 1633 a(2)1097 1640 y Fm(+)1180 1599 y Fi(X)1146 1691 y Fj(k)q Fg(2)p Fh(\003)1213 1696 y Ff(n)p Fc(+1)1282 1640 y Fl(\013)1313 1615 y Fh(\003)1337 1620 y Ff(n)p Fc(+1)1313 1653 y Fj(j)1327 1658 y Ff(n)p Fc(+1)1387 1653 y Fj(;k)1418 1640 y Fl(E)1454 1647 y Fh(\003)1480 1640 y Fm([)p Fk(jr)1550 1647 y Fj(k)1571 1640 y Fl(f)g Fk(j)1614 1620 y Fh(2)1634 1640 y Fm(])1653 1606 y Fc(1)p 1652 1612 V 1652 1633 a(2)1674 1555 y Fi(1)1674 1630 y(A)523 1770 y Fm(+2)p Fl(m)628 1777 y Fh(\003)655 1770 y Fm(\()p Fl(D)q Fm(\))734 1750 y Fg(\000)766 1736 y Fc(1)p 767 1742 V 767 1763 a(2)789 1770 y Fk(jjr)859 1777 y Fj(i)878 1758 y Fm(~)872 1770 y(\010)907 1750 y Fh(\()p Fj(i)p Fh(\))949 1770 y Fm(\()p Fl(J)o(;)j(:)p Fm(\))p Fk(jj)1077 1777 y Fh(1)1123 1729 y Fi(X)1104 1821 y Fj(k)q Fg(2)p Fh(\003)1171 1827 y Ff(N)1209 1770 y Fl(E)1245 1777 y Fh(\003)1271 1770 y Fm([)p Fk(jr)1341 1777 y Fj(k)1362 1770 y Fl(f)d Fk(j)1405 1750 y Fh(2)1425 1770 y Fm(])1444 1736 y Fc(1)p 1443 1742 V 1443 1763 a(2)1477 1770 y Fm(+)11 b Fl(E)1562 1777 y Fh(\003)1588 1770 y Fm([)p Fk(jr)1658 1777 y Fj(i)1671 1770 y Fl(f)5 b Fk(j)1714 1750 y Fh(2)1734 1770 y Fm(])1753 1736 y Fc(1)p 1753 1742 V 1753 1763 a(2)443 1934 y Fk(\024)527 1893 y Fi(X)523 1985 y Fj(k)q Fg(2)p Fh(\003)599 1934 y Fl(\027)626 1914 y Fh(\003)623 1947 y Fj(i;k)666 1934 y Fl(E)702 1941 y Fh(\003)729 1934 y Fm([)p Fk(jr)799 1941 y Fj(k)819 1934 y Fl(f)g Fk(j)862 1914 y Fh(2)882 1934 y Fm(])901 1900 y Fc(1)p 900 1906 V 900 1927 a(2)934 1934 y Fm(+)985 1893 y Fi(X)983 1985 y Fj(j)r Fg(2)p Fh(\003)1055 1849 y Fi(0)1055 1924 y(@)1172 1893 y(X)1091 1985 y Fj(n)p Fg(\024)p Fj(N)t Fg(\000)p Fh(1:)p Fj(j)r Fg(2)p Fh(\003)1290 1989 y Ff(n)1320 1934 y Fl(\027)1347 1914 y Fh(\003)1344 1947 y Fj(i;j)1380 1951 y Ff(n)1404 1934 y Fl(\013)1435 1914 y Fh(\003)1459 1918 y Ff(n)1435 1947 y Fj(j)1449 1951 y Ff(n)1471 1947 y Fj(;j)1499 1849 y Fi(1)1499 1924 y(A)1543 1934 y Fl(E)1579 1941 y Fh(\003)1606 1934 y Fm([)p Fk(jr)1676 1941 y Fj(j)1693 1934 y Fl(f)g Fk(j)1736 1914 y Fh(2)1756 1934 y Fm(])1775 1900 y Fc(1)p 1774 1906 V 1774 1927 a(2)621 2063 y Fm(+)p Fl(E)695 2070 y Fh(\003)721 2063 y Fm([)p Fk(jr)791 2070 y Fj(i)804 2063 y Fl(f)g Fk(j)847 2043 y Fh(2)867 2063 y Fm(])886 2029 y Fc(1)p 886 2035 V 886 2056 a(2)1862 2063 y Fm(\(50\))0 2246 y(Th)o(us,)14 b(it)e(remains)g(to)i (pro)o(v)o(e)e(that)i(the)f(b)q(ound)h(\(31\))g(holds.)20 b(T)l(o)14 b(this)f(end,)g(\014rst)h(notice)e(that)i(\(50\))g(sho)o(ws)0 2306 y(that)j(w)o(e)e(can)i(tak)o(e)f(:)612 2476 y Fl(\013)643 2456 y Fh(\003)643 2489 y Fj(i;j)725 2476 y Fk(\024)41 b Fl(\027)832 2456 y Fh(\003)829 2489 y Fj(i;j)880 2476 y Fm(+)1010 2435 y Fi(X)929 2527 y Fj(n)p Fg(\024)p Fj(N)t Fg(\000)p Fh(1:)p Fj(j)r Fg(2)p Fh(\003)1128 2531 y Ff(n)1158 2476 y Fl(\027)1185 2456 y Fh(\003)1182 2489 y Fj(i;j)1218 2493 y Ff(n)1242 2476 y Fl(\013)1273 2456 y Fh(\003)1297 2460 y Ff(n)1273 2489 y Fj(j)1287 2493 y Ff(n)1308 2489 y Fj(;j)1862 2476 y Fm(\(51\))950 2699 y(20)p eop %%Page: 21 21 21 20 bop 0 26 a Fm(But,)15 b(for)i Fl(n)d Fk(\025)f Fm(1,)k Fk(j)p Fm(\003)383 33 y Fj(n)406 26 y Fk(j)d(\024)f Fl(m)e Fk(\000)g Fm(1)16 b(so)h(that)g(w)o(e)f(ha)o(v)o(e,)f(according)h(to)h(our)f (induction)g(h)o(yp)q(otheses)h(:)345 178 y Fl(\013)376 158 y Fh(\003)400 162 y Ff(n)376 191 y Fj(j)390 195 y Ff(n)412 191 y Fj(;j)454 178 y Fk(\024)c Fl(\027)533 158 y Fh(\003)530 191 y Fj(j)544 195 y Ff(n)566 191 y Fj(;j)605 178 y Fm(+)654 120 y Fg(j)p Fh(\003)688 124 y Ff(n)709 120 y Fg(j\000)p Fh(1)679 137 y Fi(X)681 229 y Fj(l)p Fh(=1)898 137 y Fi(X)773 238 y Fg(f)p Fj(k)809 243 y Fc(1)826 238 y Fj(;)p Fg(\001\001\001)o Fj(;k)893 244 y Ff(l)905 238 y Fg(g2)p Fj(G)974 223 y Fc(\003)995 227 y Ff(n)974 251 y(l;D)1021 238 y Fh(\()p Fj(j)1049 242 y Ff(n)1071 238 y Fh(\))1093 178 y Fl(\027)1120 158 y Fh(\003)1117 191 y Fj(j)1131 195 y Ff(n)1152 191 y Fj(;k)1180 196 y Fc(1)1208 93 y Fi(0)1208 167 y(@)1248 124 y Fj(l)p Fg(\000)p Fh(1)1249 137 y Fi(Y)1244 227 y Fj(p)p Fh(=1)1315 178 y Fl(\027)1342 158 y Fh(\003)1339 191 y Fj(k)1357 195 y Ff(p)1376 191 y Fj(;k)1404 196 y Ff(p)p Fc(+1)1463 93 y Fi(1)1463 167 y(A)1507 178 y Fl(\027)1534 158 y Fh(\003)1531 191 y Fj(k)1549 197 y Ff(l)1562 191 y Fj(;j)1590 178 y Fl(:)0 328 y Fm(Moreo)o(v)o(er,)h(b)o(y)g(construction,)h(w)o(e)g(kno)o (w)g(that)g Fk(j)p Fl(j)920 335 y Fj(n)952 328 y Fk(\000)9 b Fl(i)p Fk(j)k Fl(>)h(D)j Fm(for)e Fl(n)f Fk(\024)g Fl(N)f Fk(\000)c Fm(1)15 b(and)h(that)f(\003)1693 335 y Fj(n)1730 328 y Fk(\032)f Fm(\003)p Fk(nf)p Fl(j)1887 335 y Fj(n)1910 328 y Fk(g)p Fm(.)0 389 y(F)l(urthermore,)e Fk(j)p Fl(i)7 b Fk(\000)g Fl(j)398 396 y Fj(n)420 389 y Fk(j)14 b(\025)g(j)p Fl(i)7 b Fk(\000)g Fl(j)s Fk(j)12 b Fm(for)j(an)o(y)e Fl(n)i Fm(suc)o(h)f(that)g Fl(j)j Fk(2)d Fm(\003)1169 396 y Fj(n)1192 389 y Fm(.)21 b(Th)o(us,)14 b(the)g(relation)g(\(51\))g(sho)o(ws)h(that)422 590 y Fl(\013)453 570 y Fh(\003)453 603 y Fj(i;j)535 590 y Fk(\024)41 b Fl(\027)642 570 y Fh(\003)639 603 y Fj(i;j)690 590 y Fm(+)739 532 y Fg(j)p Fh(\003)p Fg(j\000)p Fh(1)754 549 y Fi(X)756 641 y Fj(l)p Fh(=1)951 549 y Fi(X)837 647 y Fg(f)p Fj(k)873 652 y Fc(1)890 647 y Fj(;)p Fg(\001\001\001)o Fj(;k)957 653 y Ff(l)970 647 y Fg(g2)p Fj(G)1039 635 y Fc(\003)1039 659 y Ff(l;D)1086 647 y Fh(\()p Fj(i)p Fh(\))1133 590 y Fl(\027)1160 570 y Fh(\003)1157 603 y Fj(i;k)1197 608 y Fc(1)1229 537 y Fj(l)p Fg(\000)p Fh(1)1230 549 y Fi(Y)1225 640 y Fj(p)p Fh(=1)1296 590 y Fl(\027)1323 570 y Fh(\003)1320 603 y Fj(k)1338 607 y Ff(p)1357 603 y Fj(;k)1385 608 y Ff(p)p Fc(+1)1444 590 y Fl(\027)1471 570 y Fh(\003)1468 603 y Fj(k)1486 609 y Ff(l)1498 603 y Fj(;j)0 795 y Fm(whic)o(h)16 b(ends)g(the)g(pro)q(of)h(of)g(Prop)q(ert)o(y)f(4.4.) 1910 855 y Fk(})0 1024 y Fa(4.2)66 b(Almost)22 b(sure)h(b)r(ound)g(on)g(the)f (sw)n(eeping)g(out)h(co)r(e\016cien)n(ts)0 1117 y Fm(W)l(e)16 b(are)g(no)o(w)h(ready)f(to)h(\014nish)f(the)g(pro)q(of)h(of)g(Theorem)e (4.2,)h(and)h(th)o(us)f(the)g(one)h(of)f(Theorem)f(3.2.)73 1177 y(Let)k(us)g(\014rst)g(notice)f(that,)i(in)e(the)h(high)g(temp)q (erature)e(regime)g(\()h Fl(J)1361 1184 y Fh(0)1400 1177 y Fm(small)f(\),)i(the)f(in)o(teraction)g(is)0 1237 y(exp)q(ected)h(to)h(b)q(e) f(w)o(eak)h(in)f(general.)31 b(This)20 b(feature)f(can)h(b)q(e)g(expressed)f (b)o(y)g(means)g(of)h(the)f(follo)o(wing)0 1297 y(p)q(ercolation)d(result)g (:)0 1399 y Fb(Lemma)g(4.5)24 b Fe(L)n(et)484 1474 y Fl(\025)p Fm(\()p Fl(i;)8 b(j)s Fm(\))14 b(=)45 b(inf)678 1517 y Fj(\015)r Fg(2W)760 1505 y Ff(Z)-23 b(Z)780 1498 y(d)757 1529 y(j;i)813 1440 y Fm(#)p Fk(f)p Fl(X)18 b Fk(2)c Fl(\015)j Fm(:)k Fk(jj)p Fm(\010)1124 1447 y Fj(X)1158 1440 y Fm(\()p Fl(J)o(;)8 b Fk(\001)p Fm(\))p Fk(jj)1286 1447 y Fj(u)1322 1440 y Fk(\024)13 b Fl(J)1401 1447 y Fh(0)1421 1440 y Fk(g)p 813 1462 633 2 v 1018 1508 a Fm(#)p Fk(f)p Fl(X)k Fk(2)d Fl(\015)s Fk(g)1451 1474 y Fl(;)0 1607 y Fe(Then,)21 b(for)e(any)g Fl(\021)h Fk(2)e Fm(\(0)p Fl(;)8 b Fm(1\))p Fe(,)21 b(ther)n(e)f(exists)g(a)g(p)n(ositive)f(function)k Fm(\026)-26 b Fl(r)1257 1583 y Fj(\021)1256 1617 y Fh(0)1279 1607 y Fm(\()p Fl(p)1322 1614 y Fh(1)1342 1607 y Fl(;)8 b(\021)r(;)g(d)p Fm(\))p Fe(,)22 b Fm(\026)-26 b Fl(r)1514 1583 y Fj(\021)1513 1617 y Fh(0)1535 1607 y Fm(\()p Fl(p)1578 1614 y Fh(1)1598 1607 y Fl(;)8 b(\021)r(;)g(d)p Fm(\))18 b Fk(!)g Fm(0)i Fe(when)0 1667 y Fl(p)24 1674 y Fh(1)58 1667 y Fk(!)14 b Fm(0)p Fe(,)j(such)h(that,)g (for)f(any)g Fl(r)597 1674 y Fh(0)631 1667 y Fk(\025)e Fm(\026)-26 b Fl(r)706 1643 y Fj(\021)705 1678 y Fh(0)728 1667 y Fm(\()p Fl(p)771 1674 y Fh(1)791 1667 y Fl(;)8 b(\021)r(;)g(d)p Fm(\))17 b Fe(:)280 1742 y Fi(X)299 1829 y Fj(n)348 1783 y Fl(I)-13 b(E)404 1735 y Fi(n)432 1783 y Fk(9)p Fm(\()p Fl(i;)8 b(j)s Fm(\))13 b Fk(2)h Fm([)p Fk(\000)p Fl(n;)8 b(n)p Fm(])767 1762 y Fj(d)804 1783 y Fk(j)17 b Fl(\025)p Fm(\()p Fl(i;)8 b(j)s Fm(\))14 b Fk(\024)f Fl(\021)19 b Fe(and)f Fk(j)p Fl(i)11 b Fk(\000)f Fl(j)s Fk(j)k(\025)g Fl(r)1384 1790 y Fh(0)1412 1783 y Fm(log)9 b Fl(n)1512 1735 y Fi(o)1554 1783 y Fl(<)k Fk(1)p Fl(;)0 1924 y Fe(so)j(that,)h(for)f(almost)h(al)r(l)h(inter)n(action,)f(ther) n(e)g(exists)g(a)g(\014nite)h(inte)n(ger)f(numb)n(er)g Fl(N)5 b Fm(\()p Fl(J)g Fm(\))17 b Fe(so)f(that,)h(for)f(any)0 1984 y Fl(n)e Fk(\025)g Fl(N)5 b Fm(\()p Fl(J)g Fm(\))p Fe(,)17 b(for)g(any)g Fm(\()p Fl(i;)8 b(j)s Fm(\))17 b Fe(in)h Fm([)p Fk(\000)p Fl(n;)8 b(n)p Fm(])737 1966 y Fj(d)773 1984 y Fe(such)18 b(that)g Fk(j)p Fl(j)13 b Fk(\000)e Fl(i)p Fk(j)j(\025)f Fl(r)1200 1991 y Fh(0)1228 1984 y Fm(log)c Fl(n)p Fe(,)858 2094 y Fl(\025)p Fm(\()p Fl(i;)f(j)s Fm(\))13 b Fk(\025)h Fl(\021)r(:)0 2204 y Fm(This)i(result)f(is)g(pro)o(v)o(en)g(in)g([15)q(])g(in)g(the)g(nearest)h (neigh)o(b)q(ors)g(in)o(teraction)f(setting)g(and)h(in)g([12]\()f(Lemma)0 2265 y(5.4)i(\))f(in)g(the)g(general)g(setting.)21 b(On)16 b(the)g(other)g(hand,)h(w)o(e)f(are)g(going)h(to)g(see)f(that)g(:)0 2379 y Fb(Prop)r(ert)n(y)i(4.6)24 b Fe(L)n(et)c Fm(\004)g Fe(b)n(e)h(a)f (\014nite)i(subset)g(of)e Fl(Z)-21 b(Z)969 2358 y Fj(d)990 2379 y Fe(,)21 b(such)g(that)f(ther)n(e)h(exists)g(a)g(r)n(e)n(al)f(numb)n (er)g Fl(\021)i(>)d Fm(0)0 2439 y Fe(and)f(an)f(inte)n(ger)i(numb)n(er)e Fl(N)23 b Fe(such)18 b(that,)f(for)g(any)h Fm(\()p Fl(i;)8 b(j)s Fm(\))17 b Fe(in)g Fm(\004)g Fe(satisfying)h Fk(j)p Fl(i)11 b Fk(\000)f Fl(j)s Fk(j)k(\025)g Fl(N)5 b Fe(,)17 b(we)h(have)h(:)864 2549 y Fl(\025)p Fm(\()p Fl(i;)8 b(j)s Fm(\))14 b Fk(\025)g Fl(\021)779 b Fm(\(52\))950 2699 y(21)p eop %%Page: 22 22 22 21 bop 0 26 a Fe(Then,)17 b(for)f(any)g Fm(\003)e Fk(\032)f Fm(\004)p Fe(,)j(for)g(any)g Fm(\()p Fl(i;)8 b(j)s Fm(\))16 b Fe(in)h Fm(\004)p Fe(,)f(ther)n(e)g(exists)h(a)g(r)n(andom)e(variable)i Fl(C)1572 3 y Fh(\003)p Fj(;N)1568 38 y(i;j)1640 26 y Fm(\()p Fl(J)5 b Fm(\))16 b Fe(such)g(that)h(:)625 148 y Fl(\013)656 128 y Fh(\003)656 161 y Fj(i;j)709 148 y Fk(\024)d Fl(C)801 125 y Fh(\003)p Fj(;N)797 160 y(i;j)868 148 y Fm(\()p Fl(J)5 b Fm(\))p Fl(m)981 155 y Fh(\004)1006 148 y Fm(\()p Fl(N)g Fm(\))1088 128 y Fg(\000)1120 114 y Fc(1)p 1121 120 16 2 v 1121 141 a(2)1143 148 y Fl(\013)1174 123 y Fg(j)p Fj(i)p Fg(\000)p Fj(j)r Fg(j\000)p Fj(N)1174 159 y Fh(0)1311 148 y Fl(;)0 258 y Fe(with)18 b Fl(\013)137 265 y Fh(0)170 258 y Fm(=)c(\(exp)p Fk(f)p Fm(4)p Fl(J)392 265 y Fh(0)412 258 y Fk(g)d(\000)f Fm(1\))540 240 y Fj(\021)562 258 y Fe(.)73 319 y(F)l(urthermor)n(e,)19 b(if)f(the)i(inter)n(action)g(satis\014es)38 b(I)-12 b(H1)19 b(and)38 b(I)-12 b(H2)18 b(with)i Fl(J)1378 326 y Fh(0)1416 319 y Fe(smal)r(l)g(enough,)h(ther)n(e)e(exist)0 379 y(an)i(inte)n(ger)g (numb)n(er)g Fl(N)453 386 y Fh(0)473 379 y Fm(\()p Fl(\021)r(;)8 b(J)567 386 y Fh(0)586 379 y Fm(\))21 b Fe(and)g(a)f(\014nite)i(c)n(onstant)g Fl(C)t Fm(\()p Fl(\021)r(;)8 b(J)1229 386 y Fh(0)1247 379 y Fm(\))21 b Fe(such)g(that,)g(for)f(any)h(set)g Fm(\004)f Fe(which)0 439 y(satis\014es)e(\(52\))f(with)h(an)g(inte)n(ger)g Fl(N)h Fk(\025)14 b Fl(N)777 446 y Fh(0)796 439 y Fm(\()p Fl(\021)r(;)8 b(J)890 446 y Fh(0)910 439 y Fm(\))p Fe(,)17 b(we)h(have)g(:)645 559 y Fm(sup)644 599 y Fh(\003)p Fg(\032)p Fh(\004)760 559 y Fm(sup)728 603 y Fh(\()p Fj(i;j)r Fh(\))p Fg(2)821 595 y Fh(\026)818 603 y(\004)842 609 y Ff(R)874 559 y Fl(I)-13 b(E)r Fm([)p Fl(C)974 536 y Fh(\003)p Fj(;N)970 571 y(i;j)1041 559 y Fm(])14 b Fl(<)g(C)t Fm(\()p Fl(\021)r(;)8 b(J)1254 566 y Fh(0)1272 559 y Fm(\))p Fl(:)0 709 y Fm(Let)15 b(us)g(assume)g(for)g(a)g (while)f(that)i(Prop)q(ert)o(y)f(4.6)g(is)g(true)f(and)i(c)o(hec)o(k)d(that)i (Lemma)e(4.5)i(and)h(Prop)q(ert)o(y)0 769 y(4.6)j(giv)o(e)e(the)h(sw)o (eeping)g(out)h(relations)f(stated)h(in)f(Theorem)f(4.2.)28 b(W)l(e)19 b(\014rst)f(pro)o(v)o(e)g(Theorem)f(4.2.2\),)0 829 y(namely)d(w)o(e)i(sho)o(w)h(that)g(:)0 943 y Fb(Corollary)h(4.7)24 b Fe(If)17 b(the)h(inter)n(action)h(satis\014es)36 b(I)-12 b(H1)17 b(and)35 b(I)-12 b(H2)17 b(with)h Fl(J)1341 950 y Fh(0)1378 943 y Fe(smal)r(l)h(enough,)g(ther)n(e)e(exist)i(a)0 1003 y(\014nite)e(p)n (ositive)e(c)n(onstants)h Fl(\034)21 b Fe(and)15 b Fl(\013)p Fe(,)h(for)f(which)h(we)g(have)g Fl(\013)e(<)g Fm(1)h Fe(and)h Fm(\()p Fl(\013;)8 b(\034)e Fm(\))15 b Fe(go)g(to)h(zer)n(o)e(when)j Fl(J)1827 1010 y Fh(0)1862 1003 y Fe(go)n(es)0 1064 y(to)f(zer)n(o,)g(such)g (that,)h(for)e(almost)i(every)f(c)n(oupling)i Fl(J)5 b Fe(,)16 b(ther)n(e)g(exists)h(an)f(inte)n(ger)h(numb)n(er)g Fl(N)5 b Fm(\()p Fl(J)g Fm(\))p Fe(,)16 b(so)g(that)0 1124 y(for)h(any)g(inte)n(ger) h Fl(n)c Fk(\025)g Fl(N)5 b Fm(\()p Fl(J)g Fm(\))p Fe(,)17 b(any)g Fl(k)f Fk(2)e Fm([\()p Fl(\034)6 b Fm(\(log)i Fl(n)p Fm(\))952 1106 y Fj(d)p Fg(\000)p Fh(1)1028 1124 y Fm(+)j(2)p Fl(R)p Fm(\))p Fl(Z)-21 b(Z)5 b Fm(])1221 1106 y Fj(d)1251 1124 y Fk(\\)1300 1111 y Fm(\026)1295 1124 y(\003)1329 1106 y Fj(R)1329 1136 y(n)1358 1124 y Fe(,)17 b(al)r(l)i Fl(i)13 b Fk(2)h Fl(@)1563 1131 y Fj(R)1592 1124 y Fl(Y)1620 1133 y Fj(\034)t Fh(\(log)5 b Fj(n)p Fh(\))1740 1124 y Ff(d)p Fd(\000)p Fc(1)t Fm(\()p Fl(k)r Fm(\))p Fe(,)18 b(al)r(l)0 1184 y Fl(j)f Fk(2)d Fl(Y)112 1194 y Fj(\034)t Fh(\(log)5 b Fj(n)p Fh(\))232 1184 y Ff(d)p Fd(\000)p Fc(1)t Fm(\()p Fl(k)r Fm(\))18 b Fe(we)g(have)754 1310 y Fl(\013)785 1273 y Fj(Y)805 1286 y Ff(\034)s Fc(\(log)8 b Ff(n)p Fc(\))911 1279 y Ff(d)p Fd(\000)p Fc(1)971 1273 y Fh(\()p Fj(k)q Fh(\))785 1321 y Fj(i;j)1033 1310 y Fk(\024)14 b Fl(\013)1117 1289 y Fg(j)p Fj(i)p Fg(\000)p Fj(j)r Fg(j)0 1397 y Fe(pr)n(ovide)n(d)i(that)i Fk(j)p Fl(i)11 b Fk(\000)f Fl(j)s Fk(j)k(\025)g Fl(\034)6 b Fm(\(log)j Fl(n)p Fm(\))652 1379 y Fj(d)p Fg(\000)p Fh(1)717 1397 y Fe(.)0 1511 y(Pr)n(o)n(of.)p 0 1527 129 2 v 16 w Fm(Let)17 b Fl(\021)i Fm(and)f Fl(\034)23 b Fm(b)q(e)18 b(p)q(ositiv)o(e)e(real)h(n)o(um)o(b)q(ers.)23 b(F)l(urthermore,)15 b(assume)i(that)g Fl(J)1581 1518 y Fh(0)1618 1511 y Fm(is)g(small)f(enough)0 1571 y(so)h(that)g Fl(e)189 1553 y Fh(4)p Fj(J)226 1558 y Fc(0)256 1571 y Fk(\000)11 b Fm(1)k Fl(<)g Fm(1)i(and)g(c)o(ho)q(ose)g Fl(\013)e Fk(2)p Fm(])p Fl(\013)813 1578 y Fh(0)847 1571 y Fm(=)f(\()p Fl(e)941 1553 y Fh(4)p Fj(J)978 1558 y Fc(0)1008 1571 y Fk(\000)d Fm(1\))1101 1553 y Fj(\021)1122 1571 y Fl(;)d Fm(1[.)22 b(T)l(o)c(simplify)13 b(the)k(notation)g(let)f(us)h(set)0 1638 y Fl(Y)28 1645 y Fj(n)52 1638 y Fm(\()p Fl(k)r Fm(\))f(for)h Fl(Y)236 1648 y Fj(\034)t Fh(\(log)5 b Fj(n)p Fh(\))356 1638 y Ff(d)p Fd(\000)p Fc(1)t Fm(\()p Fl(k)r Fm(\))16 b(and)h Fl(C)630 1620 y Fj(k)626 1650 y(i;j)666 1638 y Fm(\()p Fl(r)707 1645 y Fh(0)727 1638 y Fl(;)8 b(J)d Fm(\))16 b(for)g Fl(C)929 1613 y Fj(Y)949 1617 y Ff(n)971 1613 y Fh(\()p Fj(k)q Fh(\))p Fj(;r)1044 1618 y Fc(0)1066 1613 y Fh(log)6 b Fj(n)925 1650 y(i;j)1141 1638 y Fm(\()p Fl(J)f Fm(\).)21 b(Then)16 b(according)h(to)g(Borel)e(Can)o(telli)0 1698 y(lemma,)e(it)i(is)i(enough)g(to)f(sho)o(w)h(that)g(the)f(follo)o(wing)g (sum)f(is)h(\014nite)g(:)35 1868 y Fl(S)h Fm(:=)147 1827 y Fi(X)166 1914 y Fj(n)215 1868 y Fl(P)262 1820 y Fi(\020)287 1868 y Fk([)320 1879 y Fj(k)q Fg(2)p Fh([\(2)p Fj(R)p Fh(+)p Fj(\034)t Fh(\(log)t Fj(n)p Fh(\))578 1869 y Ff(d)p Fd(\000)p Fc(1)635 1879 y Fh(\))p Fj(Z)-23 b(Z)r Fh(])686 1869 y Ff(d)704 1879 y Fg(\\)731 1870 y Fh(\026)728 1879 y(\003)752 1869 y Ff(R)752 1887 y(n)779 1868 y Fk(f9)p Fm(\()p Fl(i;)8 b(j)s Fm(\))p Fk(j)p Fl(\013)977 1843 y Fj(Y)997 1847 y Ff(n)1018 1843 y Fh(\()p Fj(k)q Fh(\))977 1880 y Fj(i;j)1081 1868 y Fl(>)14 b(\013)1164 1848 y Fg(j)p Fj(i)p Fg(\000)p Fj(j)r Fg(j)1258 1868 y Fm(and)i Fk(j)p Fl(i)11 b Fk(\000)g Fl(j)s Fk(j)i(\025)h Fl(\034)6 b Fm(\(log)j Fl(n)p Fm(\))1712 1848 y Fj(d)p Fg(\000)p Fh(1)1777 1868 y Fk(g)1802 1820 y Fi(\021)1862 1868 y Fm(\(53\))0 2010 y(F)l(or)17 b(this,)f(let)h(us)g(notice)f(that)h(Prop)q(ert)o(y)g(4.6)h (\()e(with)h Fl(N)j Fm(=)15 b Fl(r)1149 2017 y Fh(0)1177 2010 y Fm(log)10 b Fl(n)17 b Fm(\))f(implies)f(that,)i(for)g(an)o(y)g(p)q(ositiv)o (e)0 2070 y(in)o(teger)e(n)o(um)o(b)q(er)g Fl(r)361 2077 y Fh(0)380 2070 y Fm(,)h(for)h(an)o(y)f(\014nite)g(constan)o(t)g Fl(C)k Fm(and)d(an)o(y)f Fl(n)g Fm(large)h(enough)g(:)87 2192 y Fi(n)114 2240 y Fk([)147 2250 y Fj(k)q Fg(2)p Fh([\(2)p Fj(R)p Fh(+)p Fj(\034)t Fh(\(log)t Fj(n)p Fh(\))405 2241 y Ff(d)p Fd(\000)p Fc(1)463 2250 y Fh(\))p Fj(Z)-23 b(Z)r Fh(])514 2241 y Ff(d)532 2250 y Fg(\\)559 2242 y Fh(\026)556 2250 y(\003)580 2241 y Ff(R)580 2259 y(n)607 2240 y Fk(f)p Fl(\013)663 2215 y Fj(Y)683 2219 y Ff(n)705 2215 y Fh(\()p Fj(k)q Fh(\))663 2251 y Fj(i;j)768 2240 y Fl(>)14 b(\013)851 2219 y Fg(j)p Fj(i)p Fg(\000)p Fj(j)r Fg(j)928 2240 y Fl(;)8 b Fk(j)p Fl(i)j Fk(\000)f Fl(j)s Fk(j)k(\025)g Fl(\034)6 b Fm(\(log)j Fl(n)p Fm(\))1310 2219 y Fj(d)p Fg(\000)p Fh(1)1375 2240 y Fk(g)1400 2192 y Fi(o)1441 2240 y Fk(\032)14 b Fl(A)1531 2247 y Fj(n)1565 2240 y Fk([)d Fl(B)1646 2247 y Fj(n)1681 2240 y Fk([)g Fl(C)1760 2247 y Fj(n)1862 2240 y Fm(\(54\))0 2350 y(where)391 2410 y Fl(A)428 2417 y Fj(n)465 2410 y Fk(\021)j(f9)p Fm(\()p Fl(i;)8 b(j)s Fm(\))13 b Fk(2)h Fm(\003)765 2417 y Fj(n)804 2410 y Fk(j)i Fl(\025)p Fm(\()p Fl(i;)8 b(j)s Fm(\))14 b Fk(\024)g Fl(\021)k Fm(and)f Fk(j)p Fl(i)10 b Fk(\000)h Fl(j)s Fk(j)j(\025)f Fl(r)1382 2417 y Fh(0)1410 2410 y Fm(log)d Fl(n)p Fk(g)e Fl(;)20 2516 y(B)57 2523 y Fj(n)94 2516 y Fk(\021)146 2455 y Fi(\032)178 2516 y Fk([)211 2526 y Fj(k)q Fg(2)p Fh([\(2)p Fj(R)p Fh(+)p Fj(\034)t Fh(\(log)t Fj(n)p Fh(\))469 2517 y Ff(d)p Fd(\000)p Fc(1)526 2526 y Fh(\))p Fj(Z)-23 b(Z)r Fh(])577 2517 y Ff(d)p Fd(\000)p Ff(l)629 2526 y Fg(\\)656 2518 y Fh(\026)653 2526 y(\003)677 2517 y Ff(R)677 2535 y(n)704 2516 y Fk(f)p Fl(C)768 2495 y Fj(k)764 2528 y(i;j)804 2516 y Fm(\()p Fl(r)845 2523 y Fh(0)865 2516 y Fl(;)8 b(J)d Fm(\))13 b Fl(>)h(C)1042 2495 y Fg(\000)p Fh(\(log)5 b Fj(n)p Fh(\))1169 2484 y Ff(d)p Fd(\000)p Fc(1)1229 2516 y Fm(\()1262 2482 y Fl(\013)p 1253 2504 51 2 v 1253 2550 a(\013)1284 2557 y Fh(0)1308 2516 y Fm(\))1327 2495 y Fg(j)p Fj(i)p Fg(\000)p Fj(j)r Fg(j)1405 2516 y Fl(;)j Fk(j)p Fl(i)i Fk(\000)h Fl(j)s Fk(j)i(\025)h Fl(\034)6 b Fm(\(log)j Fl(n)p Fm(\))1786 2495 y Fj(d)p Fg(\000)p Fh(1)1851 2516 y Fk(g)1876 2455 y Fi(\033)1916 2516 y Fl(;)950 2699 y Fm(22)p eop %%Page: 23 23 23 22 bop 0 26 a Fm(and)17 b(:)448 86 y Fl(C)483 93 y Fj(n)520 86 y Fk(\021)573 38 y Fi(n)600 86 y Fl(m)643 96 y Fh([)p Fg(\000)p Fj(n;n)p Fh(])742 87 y Ff(d)763 86 y Fm(\()p Fl(r)804 93 y Fh(0)832 86 y Fm(log)9 b Fl(n)p Fm(\))14 b Fk(\024)g Fl(\013)1049 63 y Fg(\000)p Fh(2)p Fj(r)1110 68 y Fc(0)1133 63 y Fh(log)6 b Fj(n)1049 97 y Fh(0)1208 86 y Fl(C)1247 66 y Fg(\000)p Fh(2\(log)f Fj(n)p Fh(\))1392 54 y Ff(d)p Fd(\000)p Fc(1)1451 38 y Fi(o)1487 86 y Fl(;)0 170 y Fm(for)17 b(some)e(\014nite)g(constan)o(t)i Fl(C)t Fm(.)k(Hence,)552 271 y Fl(S)c Fk(\024)651 230 y Fi(X)670 317 y Fj(n)719 271 y Fl(P)7 b Fm(\()p Fl(A)813 278 y Fj(n)837 271 y Fm(\))k(+)916 230 y Fi(X)935 317 y Fj(n)984 271 y Fl(P)c Fm(\()p Fl(B)1078 278 y Fj(n)1102 271 y Fm(\))k(+)1181 230 y Fi(X)1200 317 y Fj(n)1249 271 y Fl(P)c Fm(\()p Fl(C)1341 278 y Fj(n)1365 271 y Fm(\))p Fl(:)464 b Fm(\(55\))0 404 y(But,)16 b(for)h(an)o(y)g Fl(r)301 411 y Fh(0)335 404 y Fl(>)g Fm(\026)-26 b Fl(r)410 411 y Fh(0)430 404 y Fm(\()p Fl(p)473 411 y Fh(1)493 404 y Fl(;)8 b(\021)r(;)g(d)p Fm(\),)16 b(Lemma)f(4.5)i(implies)d(that)1163 371 y Fi(P)1207 414 y Fj(n)1239 404 y Fl(P)7 b Fm(\()p Fl(A)1333 411 y Fj(n)1356 404 y Fm(\))17 b(is)g(\014nite.)22 b(Let)17 b(us)g(no)o(w)g(fo)q(cus)0 464 y(on)23 b(the)f(second)h(term)e(in)h(the)g (righ)o(t)g(hand)i(side)e(of)h(\(55\))g(and)g(notice)f(that)h(the)f(n)o(um)o (b)q(er)f(of)i(cub)q(es)0 540 y Fl(Y)28 547 y Fj(n)52 540 y Fm(\()p Fl(k)r Fm(\)'s)15 b(is)g(less)g(than)h(2)439 522 y Fj(d)p Fg(\000)p Fh(1)513 492 y Fi(\020)537 540 y Fl(n=\034)6 b Fm(\(log)k Fl(n)p Fm(\))756 522 y Fj(d)p Fg(\000)p Fh(1)821 492 y Fi(\021)846 504 y Fj(d)866 540 y Fm(,)15 b(that)h(their)e(v)o(olume)f (is)i(\(2\()p Fl(\034)6 b Fm(\(log)k Fl(n)p Fm(\))1558 522 y Fj(d)p Fg(\000)p Fh(1)1632 540 y Fm(+)f Fl(R)p Fm(\))h(+)e(1\))1834 522 y Fj(d)1870 540 y Fm(and)0 615 y(the)14 b(v)o(olume)e(of)j(their)f(b)q (oundary)i(is)e(of)h(order)f(\(log)c Fl(n)p Fm(\))1000 597 y Fh(\()p Fj(d)p Fg(\000)p Fh(1\))1091 586 y Fc(2)1110 615 y Fm(.)20 b(Therefore,)14 b(Cheb)o(yshev)g(inequalit)o(y)f(giv)o(es)0 676 y(:)431 777 y Fl(P)j Fm(\()p Fl(B)534 784 y Fj(n)557 777 y Fm(\))42 b Fk(\024)57 b Fm(const.)p Fl(C)t Fm(\()p Fl(\021)r(;)8 b(J)972 784 y Fh(0)991 777 y Fm(\))p Fl(n)1039 756 y Fh(2)p Fj(d)1077 777 y Fl(e)1100 756 y Fh(\(log)d Fj(C)r Fh(+)p Fj(\034)10 b Fh(log)1302 738 y Ff(\013)1322 745 y Fc(0)p 1302 749 38 2 v 1310 769 a Ff(\013)1344 756 y Fh(\)\(log)5 b Fj(n)p Fh(\))1458 745 y Ff(d)p Fd(\000)p Fc(1)0 878 y Fm(Th)o(us,)16 b(if)g(\()p Fl(C)q(;)8 b(\034)s(;)g(\013)p Fm(\))16 b(is)g(c)o(hosen)g(so)h(that)g(log)9 b Fl(C)14 b Fm(+)d Fl(\034)j Fm(log)1023 859 y Fj(\013)1046 864 y Fc(0)p 1023 867 40 2 v 1031 896 a Fj(\013)1082 878 y Fl(<)f Fm(0,)j(w)o(e)g(get)h(that)f(:)814 945 y Fi(X)833 1032 y Fj(n)882 987 y Fl(P)g Fm(\()p Fl(B)985 994 y Fj(n)1009 987 y Fm(\))d Fl(<)h Fk(1)719 b Fm(\(56\))0 1119 y(W)l(e)19 b(\014nally)f(sho)o (w)h(that)469 1086 y Fi(P)513 1130 y Fj(n)544 1119 y Fl(P)d Fm(\()p Fl(C)645 1126 y Fj(n)668 1119 y Fm(\))j(is)g(\014nite)f(if)g Fl(C)23 b Fm(is)18 b(large)h(enough.)30 b(This)19 b(can)g(b)q(e)g(pro)o(v)o (en)f(using)h(a)0 1179 y(standard)e(sp)q(ectral)g(gap)g(lo)o(w)o(er)e(b)q (ound.)23 b(Indeed,)15 b(it)g(is)h(kno)o(wn)h(that,)f(for)h(an)o(y)f(set)g Fl(Y)27 b Fm(w)o(e)16 b(ha)o(v)o(e)g(:)488 1339 y Fl(m)p Fm(\()p Fl(Y)11 b Fm(\))j Fk(\025)674 1254 y Fi(0)674 1328 y(@)711 1339 y Fk(j)p Fl(Y)d Fk(j)d Fm(sup)791 1378 y Fj(i)p Fg(2)p Fj(Y)868 1339 y Fm(exp)o Fk(f)p Fm(8)1068 1298 y Fi(X)1000 1393 y Fj(X)s Fg(\\)p Fj(V)1083 1383 y Ff(i)1096 1393 y Fh(\()p Fj(Y)g Fh(\))p Fg(6)p Fh(=)p Fg(;)1205 1339 y Fk(jj)p Fm(\010)1268 1346 y Fj(X)1302 1339 y Fk(jj)1330 1346 y Fj(u)1352 1339 y Fk(g)1377 1254 y Fi(1)1377 1328 y(A)1413 1265 y Fg(\000)p Fh(1)1862 1339 y Fm(\(57\))0 1473 y(where)16 b Fl(V)180 1455 y Fj(i)194 1473 y Fm(\()p Fl(Y)11 b Fm(\))j(=)g Fk(f)p Fl(j)j Fk(2)d Fl(Y)25 b Fm(:)13 b Fk(j)p Fl(j)560 1480 y Fh(1)591 1473 y Fk(\000)e Fl(i)658 1480 y Fh(1)677 1473 y Fk(j)j(\024)f Fl(R)p Fk(g)p Fm(.)22 b(This)16 b(result)g(implies)e(that)81 1574 y Fl(m)124 1584 y Fh([)p Fg(\000)p Fj(n;n)p Fh(])223 1574 y Ff(d)t Fm(\()p Fl(r)284 1581 y Fh(0)312 1574 y Fm(log)c Fl(n)p Fm(\))42 b(=)89 b(inf)554 1608 y Fj(i)p Fg(2)p Fh([)p Fg(\000)p Fj(n;n)p Fh(])689 1598 y Ff(d)774 1574 y Fm(inf)715 1608 y Fj(X)s Fg(\032)p Fh([)p Fg(\000)p Fj(n;n)p Fh(])873 1598 y Ff(d)900 1574 y Fl(m)p Fm(\()p Fl(X)15 b Fk(\\)d Fl(B)s Fm(\()p Fl(i;)c(r)1182 1581 y Fh(0)1209 1574 y Fm(log)h Fl(n)p Fm(\)\))473 1732 y Fk(\025)554 1647 y Fi(0)554 1721 y(@)590 1732 y Fm(\(2)p Fl(r)655 1739 y Fh(0)683 1732 y Fm(log)g Fl(n)j Fm(+)f(1\))887 1712 y Fj(d)968 1732 y Fm(sup)916 1776 y Fj(i;j)r Fg(2)p Fh([)p Fg(\000)p Fj(n;n)p Fh(])1077 1766 y Ff(d)1103 1732 y Fm(exp)p Fk(f)p Fm(8)1386 1691 y Fi(X)1235 1786 y Fj(X)s Fg(\\)p Fj(V)1319 1776 y Ff(i)1332 1786 y Fh(\()p Fj(B)r Fh(\()p Fj(j;r)1429 1791 y Fc(0)1451 1786 y Fh(log)5 b Fj(n)p Fh(\)\))p Fg(6)p Fh(=)p Fg(;)1604 1732 y Fk(jj)p Fm(\010)1667 1739 y Fj(X)1701 1732 y Fk(jj)1729 1739 y Fj(u)1751 1732 y Fk(g)1776 1647 y Fi(1)1776 1721 y(A)1812 1658 y Fg(\000)p Fh(1)0 1916 y Fm(Th)o(us,)16 b(Cheb)o(yshev)g(inequalit)o(y) e(implies)f(that)k(:)142 2018 y Fl(P)f Fm(\()p Fl(C)243 2025 y Fj(n)266 2018 y Fm(\))42 b Fk(\021)f Fl(P)453 1970 y Fi(\020)478 2018 y Fl(m)521 2027 y Fh([)p Fg(\000)p Fj(n;n)p Fh(])620 2018 y Ff(d)5 b Fm(\()p Fl(r)682 2025 y Fh(0)710 2018 y Fm(log)k Fl(n)p Fm(\))14 b Fk(\024)g Fl(\013)927 1994 y Fg(\000)p Fh(2)p Fj(r)988 1999 y Fc(0)1011 1994 y Fh(log)5 b Fj(n)927 2029 y Fh(0)1086 2018 y Fl(C)1125 1997 y Fg(\000)p Fh(2\(log)g Fj(n)p Fh(\))1270 1986 y Ff(d)p Fd(\000)p Fc(1)1329 1970 y Fi(\021)327 2125 y Fk(\024)41 b Fm(const.\(log)9 b Fl(n)p Fm(\))670 2104 y Fj(d)690 2125 y Fl(n)719 2104 y Fh(2)p Fj(d)757 2125 y Fl(C)796 2104 y Fg(\000)p Fh(2\(log)c Fj(n)p Fh(\))941 2093 y Ff(d)p Fd(\000)p Fc(1)1001 2125 y Fl(I)-13 b(E)1057 2077 y Fi(h)1076 2125 y Fm(exp)p Fk(f)p Fm(8)p Fk(jj)1233 2112 y Fm(~)1228 2125 y(\010)1263 2104 y Fh(\()p Fj(j)r Fh(\))1309 2125 y Fk(jj)1337 2132 y Fh(1)1356 2125 y Fm(\))p Fk(g)1400 2077 y Fi(i)1420 2088 y Fh(\(2)p Fj(R)p Fh(+1\)\(2)p Fj(r)1586 2093 y Fc(0)1607 2088 y Fh(log)6 b Fj(n)p Fh(+1\))1739 2077 y Ff(d)p Fd(\000)p Fc(1)0 2299 y Fm(where)22 b(w)o(e)g(used)g(in)g(the)g(last)g(line)g(the)g (fact)g(that)h Fk(j)p Fl(V)1049 2281 y Fj(i;n)1094 2299 y Fk(j)h(\024)g Fm(\(2)p Fl(R)16 b Fm(+)f(1\)\(2)p Fl(r)1452 2306 y Fh(0)1481 2299 y Fm(log)9 b Fl(n)15 b Fm(+)g(1\))1692 2281 y Fj(d)p Fg(\000)p Fh(1)1780 2299 y Fm(and)23 b(the)0 2359 y(in)o(v)m(ariance)13 b(b)o(y)g(translation)h(of)g(the)g(la)o(w)f(of)h Fk(jj)843 2347 y Fm(~)838 2359 y(\010)873 2341 y Fh(\()p Fj(j)r Fh(\))919 2359 y Fk(jj)947 2366 y Fh(1)966 2359 y Fm(.)21 b(As)13 b(a)h(consequence,)f (it)g(is)h(clear)f(that)h(if)f Fl(C)k Fm(is)d(large)0 2426 y(enough)j(so)g(that)g Fk(\000)p Fm(2)8 b(log)h Fl(C)15 b Fm(+)c(\(2)p Fl(R)h Fm(+)f(1\)\(2)p Fl(r)824 2433 y Fh(0)844 2426 y Fm(\))863 2408 y Fj(d)p Fg(\000)p Fh(1)937 2426 y Fm(log)e Fl(I)-13 b(E)1064 2378 y Fi(h)1084 2426 y Fm(exp)o Fk(f)p Fm(8)p Fk(jj)1240 2413 y Fm(~)1235 2426 y(\010)1270 2408 y Fh(\()p Fj(j)r Fh(\))1316 2426 y Fk(jj)1344 2433 y Fh(1)1364 2426 y Fm(\))p Fk(g)1408 2378 y Fi(i)1441 2426 y Fl(<)14 b Fm(0,)i(w)o(e)g(get)g(:)804 2500 y Fi(X)824 2587 y Fj(n)873 2542 y Fl(P)f Fm(\()p Fl(C)973 2549 y Fj(n)997 2542 y Fm(\))e Fl(<)h Fk(1)p Fl(:)950 2699 y Fm(23)p eop %%Page: 24 24 24 23 bop 0 26 a Fm(As)16 b(a)g(conclusion,)g(for)g(an)o(y)g Fl(\021)f Fk(2)p Fm(]0)p Fl(;)8 b Fm(1[,)16 b(w)o(e)g(can)g(c)o(ho)q(ose)g Fl(r)1065 33 y Fh(0)1099 26 y Fl(>)g Fm(\026)-27 b Fl(r)1172 33 y Fh(0)1192 26 y Fm(\()p Fl(p)1235 33 y Fh(1)1255 26 y Fl(;)8 b(\021)r(;)g(d)p Fm(\))16 b(so)h(that)1550 -7 y Fi(P)1594 37 y Fj(n)1626 26 y Fl(P)7 b Fm(\()p Fl(A)1720 33 y Fj(n)1743 26 y Fm(\))16 b(is)g(\014nite,)0 86 y(then)h(a)g(constan)o(t)h Fl(C)i Fm(large)d(enough)h(so)f(that)861 53 y Fi(P)905 97 y Fj(n)937 86 y Fl(P)f Fm(\()p Fl(C)1038 93 y Fj(n)1061 86 y Fm(\))h(is)g(\014nite)f(and)h(\014nally)g(t)o(w)o(o)g(constan)o(ts)g Fl(\013)f(<)e Fm(1)0 147 y(and)h Fl(\034)20 b(>)14 b Fm(0)h(so)g(that)g Fl(P)7 b Fm(\()p Fl(B)481 154 y Fj(n)505 147 y Fm(\))14 b(is)h(\014nite.)20 b(Then,)14 b(w)o(e)h(get)f(\(53\),)h(i.e.)20 b(Corollary)15 b(4.7.)21 b(Moreo)o(v)o(er,)13 b(it)h(is)h(clear)0 207 y(that)i(w)o(e)e(can)i (tak)o(e)f Fl(\013)g Fm(and)h Fl(\034)22 b Fm(as)17 b(small)e(as)h(w)o(e)g(w) o(an)o(t)g(when)h Fl(\013)1155 214 y Fh(0)1191 207 y Fm(go)q(es)g(to)g(zero,) e(i.e.)20 b Fl(J)1585 214 y Fh(0)1621 207 y Fm(do)q(es.)1910 267 y Fk(})0 352 y Fm(Similarly)l(,)13 b(w)o(e)j(can)g(pro)o(v)o(e)f(the)h (second)h(part)g(of)f(Theorem)f(4.2,)h(i.e.)k(:)0 466 y Fb(Lemma)c(4.8)24 b Fe(If)29 b(I)-12 b(H1)14 b(and)29 b(I)-12 b(H2)14 b(ar)n(e)g(satis\014e)n (d)h(with)g Fl(J)1021 473 y Fh(0)1055 466 y Fe(smal)r(l)h(enough,)g(ther)n(e) f(exists)g(a)g(\014nite)h(c)n(onstant)0 526 y Fl(\034)27 508 y Fg(0)38 526 y Fe(,)h(such)g(that,)g(for)g Fl(I)-13 b(E)19 b Fe(almost)e(every)g Fl(J)5 b Fe(,)17 b(ther)n(e)f(exists)i(an)f(inte)n(ger) h(numb)n(er)1453 514 y Fm(~)1439 526 y Fl(N)5 b Fm(\()p Fl(J)g Fm(\))16 b Fe(for)g(which)i(we)f(have)0 587 y(for)h(any)h(inte)n(ger)h Fl(n)d Fk(\025)f Fl(N)5 b Fm(\()p Fl(J)g Fm(\))p Fe(,)19 b(for)f(any)h Fl(l)e Fk(2)g Fm([0)p Fl(;)8 b(d)k Fk(\000)g Fm(1])p Fe(,)19 b(any)g Fl(k)g Fk(2)d(f)p Fm(0)p Fk(g)1324 569 y Fj(l)1337 587 y Fk(\002)p Fm([\(2)p Fl(R)d Fm(+)f Fl(\034)6 b Fm(\(log)j Fl(n)p Fm(\))1698 569 y Fj(d)p Fg(\000)p Fh(1)1763 587 y Fm(\))p Fl(Z)-21 b(Z)t Fm(])1845 569 y Fj(d)p Fg(\000)p Fj(l)1916 587 y Fk(\\)0 647 y Fm([)p Fk(\000)p Fl(n;)8 b(n)p Fm(])147 629 y Fj(d)166 647 y Fe(,)18 b(any)f Fl(i)c Fk(2)h Fl(@)395 654 y Fj(R)424 647 y Fl(Y)463 629 y Fj(l)452 662 y(\034)t Fh(\(log)5 b Fj(n)p Fh(\))572 653 y Ff(d)p Fd(\000)p Fc(1)632 647 y Fm(\()p Fl(k)r Fm(\))p Fe(,)17 b(the)h(fol)r(lowing)i(b)n(ound)584 755 y Fi(X)502 853 y Fj(j)r Fg(2)p Fj(Y)570 841 y Ff(l)562 872 y(\034)s Fc(\(log)7 b Ff(n)p Fc(\))667 865 y Ff(d)p Fd(\000)p Fc(1)735 796 y Fl(\013)766 753 y Fj(Y)795 742 y Ff(l)786 773 y(\034)s Fc(\(log)h Ff(n)p Fc(\))892 766 y Ff(d)p Fd(\000)p Fc(1)952 753 y Fh(\()p Fj(k)q Fh(\))766 808 y Fj(i;j)1014 796 y Fk(\024)14 b Fm(exp)o Fk(f)p Fl(\034)1193 776 y Fg(0)1205 796 y Fm(\(log)9 b Fl(n)p Fm(\))1343 776 y Fj(d)p Fg(\000)p Fh(1)1408 796 y Fk(g)p Fl(:)0 975 y Fe(Pr)n(o)n(of.)p 0 992 129 2 v 73 1041 a Fm(Let)20 b(us)g(\014rst)g(de\014ne,)f(for)h(an)g(in)o (teger)f Fl(n)g Fm(in)h Fl(I)-13 b(N)5 b Fm(,)20 b(a)g(p)q(oin)o(t)f Fl(i)g Fk(2)h Fl(Z)-21 b(Z)1314 1021 y Fj(d)1354 1041 y Fm(and)20 b(a)g(\014nite)f(set)h Fl(Y)30 b Fm(of)20 b 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Fg(j)p Fh(+)726 1294 y Fi(P)771 1304 y Ff(l)p Fd(\000)p Fc(1)771 1337 y Ff(p)p Fc(=1)833 1323 y Fg(j)p Fj(k)861 1327 y Ff(p)879 1323 y Fg(\000)p Fj(k)924 1328 y Ff(p)p Fc(+1)982 1323 y Fg(j)502 1370 y Fh(0)994 1299 y Fi(\023)1862 1359 y Fm(\(59\))73 1542 y(W)l(e)20 b(no)o(w)h(b)q(ound)h(the) e(exp)q(ectation)g(of)g Fl(C)875 1518 y Fh(\003)p Fj(;N)871 1553 y(i;j)943 1542 y Fm(\()p Fl(J)5 b Fm(\))20 b(b)o(y)f(b)q(ounding)j(the)e (exp)q(ectation)g(of)h(the)40 b(righ)o(t)0 1602 y(hand)22 b(side)f(of)g (\(59\).)37 b(T)l(o)22 b(do)g(so,)h(w)o(e)e(need)f(a)i(probabilit)o(y)e (estimate,)h(i.e.)35 b(w)o(e)21 b(ha)o(v)o(e)f(to)i(b)q(ound)g(the)0 1662 y(exp)q(ectation)16 b(of)g(terms)f(of)i(the)f(form)f(:)564 1754 y Fj(l)543 1766 y Fi(Y)538 1857 y Fj(p)p Fh(=1)609 1808 y Fk(jjr)679 1815 y Fj(k)697 1819 y Ff(p)722 1795 y Fm(~)717 1808 y(\010)752 1787 y Fh(\()p Fj(k)784 1791 y Ff(p)802 1787 y Fh(\))818 1808 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)946 1815 y Fh(1)973 1808 y Fm(exp)p Fk(f)p Fm(3)p 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Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)918 2266 y Fh(1)946 2259 y Fm(exp)o Fk(f)p Fm(3)p Fk(jj)1102 2246 y Fm(~)1097 2259 y(\010)1132 2238 y Fh(\()p Fj(k)1164 2242 y Ff(p)1182 2238 y Fh(\))1198 2259 y Fm(\()p Fl(J)o(;)g(:)p Fm(\))p Fk(jj)1326 2266 y Fh(1)1345 2259 y Fk(g)1370 2174 y Fi(3)1370 2248 y(5)1411 2259 y Fk(\024)14 b Fl(a\017)1510 2238 y Fj(l)1862 2259 y Fm(\(60\))0 2414 y(Before)i(turning)i(to)f(the)g(pro)q(of)h(of)f(Lemma)e(4.9,)j(w)o(e)e (\014rst)i(end)f(the)f(pro)q(of)j(of)e(Prop)q(ert)o(y)g(4.6.)24 b(F)l(or)17 b(this,)0 2474 y(let)h(us)h(notice)f(that)h(eac)o(h)f(elemen)o(t) d(of)k Fl(G)782 2456 y Fh(\003)782 2487 y Fj(l;N)837 2474 y Fm(\()p Fl(i)p Fm(\))f(is)g(made)g(of)g(di\013eren)o(t)g(p)q(oin)o(ts)h(so)g (that)g(w)o(e)f(can)h(apply)950 2699 y(25)p eop %%Page: 26 26 26 25 bop 0 26 a Fm(Lemma)14 b(4.9)j(in)f(\(59\))h(and)f(\014nd)h(already)f (that)h(:)166 180 y Fl(I)-13 b(E)s Fm([)p Fl(C)267 157 y Fh(\003)p Fj(;N)263 192 y(i;j)334 180 y Fm(\()p Fl(J)5 b Fm(\)])41 b Fk(\024)g Fl(C)t(c\017)619 120 y Fi(\022)649 180 y Fm(1)11 b(+)733 122 y Fg(j)p Fh(\003)p Fg(j\000)p Fh(1)748 139 y Fi(X)750 231 y Fj(l)p Fh(=1)831 180 y Fl(a\017)877 160 y Fj(l)889 180 y Fl(c)910 160 y Fj(l)1046 139 y Fi(X)931 236 y Fg(f)p Fj(k)967 241 y Fc(1)985 236 y Fj(;)p Fg(\001\001\001)o Fj(;k)1052 242 y Ff(l)1064 236 y Fg(g2)p Fj(G)1133 225 y Fc(\003)1133 249 y Ff(l;N)1181 236 y Fh(\()p Fj(i)p Fh(\))1229 180 y Fl(\013)1260 144 y Fg(j)p Fj(i)p Fg(\000)p Fj(k)1327 149 y Fc(1)1345 144 y Fg(j\000j)p Fj(j)r Fg(\000)p Fj(i)p Fg(j)p Fh(+)1484 115 y Fi(P)1529 125 y Ff(l)p Fd(\000)p Fc(1)1529 158 y Ff(p)p Fc(=1)1591 144 y Fg(j)p Fj(k)1619 148 y Ff(p)1638 144 y Fg(\000)p Fj(k)1683 149 y Ff(p)p Fc(+1)1740 144 y Fg(j)1260 191 y Fh(0)1752 120 y Fi(\023)1862 180 y Fm(\(61\))0 385 y(But,)k(since)h Fk(j)p Fl(i)10 b Fk(\000)h Fl(k)346 392 y Fh(1)366 385 y Fk(j)j(\025)g(j)p Fl(j)f Fk(\000)e Fl(i)p Fk(j)p Fm(,)16 b(it)f(is)h(clear)g(that)h(:)183 507 y Fi(X)69 604 y Fg(f)p Fj(k)105 609 y Fc(1)122 604 y Fj(;)p Fg(\001\001\001)o Fj(;k)189 610 y Ff(l)201 604 y Fg(g2)p Fj(G)270 593 y Fc(\003)270 617 y Ff(l;N)318 604 y Fh(\()p Fj(i)p Fh(\))366 548 y Fl(\013)397 512 y Fg(j)p Fj(i)p Fg(\000)p Fj(k)464 517 y Fc(1)482 512 y Fg(j\000j)p Fj(j)r Fg(\000)p Fj(i)p Fg(j)p Fh(+)621 482 y Fi(P)666 493 y Ff(l)p Fd(\000)p Fc(1)666 526 y Ff(p)p Fc(=1)729 512 y Fg(j)p Fj(k)757 516 y Ff(p)775 512 y Fg(\000)p Fj(k)820 517 y Ff(p)p Fc(+1)877 512 y Fg(j)397 559 y Fh(0)903 548 y Fk(\024)956 463 y Fi(0)956 537 y(@)995 507 y(X)992 598 y Fj(n)p Fg(\025)p Fh(0)1058 548 y Fm(\(2)p Fl(l)12 b Fm(+)f(1\))1220 527 y Fj(d)p Fg(\000)p Fh(1)1286 548 y Fl(\013)1317 527 y Fj(l)1330 463 y Fi(1)1330 537 y(A)1375 463 y(0)1375 537 y(@)1422 507 y(X)1411 598 y Fj(n)p Fg(\025)p Fj(N)1492 548 y Fm(\(2)p Fl(l)h Fm(+)f(1\))1654 527 y Fj(d)p Fg(\000)p Fh(1)1720 548 y Fl(\013)1751 527 y Fj(l)1764 463 y Fi(1)1764 537 y(A)1800 474 y Fj(l)p Fg(\000)p Fh(1)1867 548 y Fl(;)0 721 y Fm(and,)20 b(if)e Fl(\013)i Fm(is)f(smaller)e(than)i(1)h(and)f Fl(N)24 b Fm(large)19 b(enough)h(so)g(that)f Fl(c\017)1266 688 y Fi(P)1310 731 y Fj(n)p Fg(\025)p Fj(N)1393 721 y Fm(\(2)p Fl(l)13 b Fm(+)g(1\))1558 703 y Fj(d)p Fg(\000)p Fh(1)1624 721 y Fl(\013)1655 703 y Fj(l)1687 721 y Fl(<)18 b Fm(1,)i(w)o(e)f(can)0 781 y(conclude)d(that)g(:)405 970 y Fl(I)-13 b(E)s Fm([)p Fl(C)506 946 y Fh(\003)p Fj(;N)502 981 y(i;j)573 970 y Fm(\()p Fl(J)5 b Fm(\)])41 b Fk(\024)g Fl(A\017)843 897 y Fi( )875 970 y Fm(1)12 b(+)1209 936 y(1)p Fl(\017)p 965 958 533 2 v 965 1004 a Fm(1)f Fk(\000)g Fl(c\017)1099 971 y Fi(P)1143 1014 y Fj(n)p Fg(\025)p Fj(N)1225 1004 y Fm(\(2)p Fl(l)h Fm(+)f(1\))1387 990 y Fj(d)p Fg(\000)p Fh(1)1453 1004 y Fl(\013)1484 990 y Fj(l)1502 897 y Fi(!)1862 970 y Fm(\(62\))0 1152 y(where)16 b Fl(A)d Fm(=)h Fl(C)t(c\017)331 1104 y Fi(\020)355 1119 y(P)399 1163 y Fj(n)p Fg(\025)p Fh(0)468 1152 y Fm(\(2)p Fl(l)e Fm(+)f(1\))630 1134 y Fj(d)p Fg(\000)p Fh(1)695 1152 y Fl(\013)726 1134 y Fj(l)740 1104 y Fi(\021)765 1152 y Fm(.)73 1238 y Fe(Pr)n(o)n(of)16 b(of)h(L)n(emma)g(4.9.)p 73 1254 441 2 v 73 1298 a Fm(W)l(e)c(w)o(an)o(t)f (to)h(get)g(a)g(b)q(ound)h(indep)q(enden)o(tly)e(of)h(the)f(c)o(hoice)g(of)h (our)g(distinct)f(p)q(oin)o(ts)h Fk(f)p Fl(i;)8 b(j;)g(k)1756 1305 y Fh(1)1775 1298 y Fl(;)g Fk(\001)g(\001)g(\001)h Fl(;)f(k)1911 1305 y Fj(l)1924 1298 y Fk(g)0 1358 y Fm(of)17 b(the)f(follo)o(wing)f(quan)o (tit)o(y)h(:)436 1503 y Fl(A)e Fm(:)p Fk(\021)f Fl(I)-13 b(E)601 1443 y Fi(\024)657 1450 y Fj(l)636 1462 y Fi(Y)631 1553 y Fj(p)p Fh(=1)702 1503 y Fk(jjr)772 1510 y Fj(k)790 1514 y Ff(p)815 1491 y Fm(~)810 1503 y(\010)845 1483 y Fh(\()p Fj(k)877 1487 y Ff(p)895 1483 y Fh(\))911 1503 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1039 1510 y Fh(1)1066 1503 y Fm(exp)p Fk(f)p Fm(3)p Fk(jj)1223 1491 y Fm(~)1218 1503 y(\010)1253 1483 y Fh(\()p Fj(k)1285 1487 y Ff(p)1303 1483 y Fh(\))1319 1503 y Fm(\()p Fl(J)o(;)g(:)p Fm(\))p Fk(jj)1447 1510 y Fh(1)1466 1503 y Fk(g)1491 1443 y Fi(\025)0 1640 y Fm(But)16 b(using)g(the)g (de\014nition)g(of)h Fk(jjr)651 1647 y Fj(i)670 1627 y Fm(~)664 1640 y(\010)699 1622 y Fh(\()p Fj(i)p Fh(\))741 1640 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)869 1647 y Fh(1)904 1640 y Fm(and)17 b(expanding)f(the)g(pro)q(duct,)h(w)o(e)e(\014nd)i(:)294 1841 y Fl(A)42 b Fm(=)476 1800 y Fi(X)452 1892 y Fj(X)481 1896 y Ff(p)499 1892 y Fg(3)p Fj(k)541 1896 y Ff(p)567 1841 y Fl(I)-13 b(E)615 1781 y Fi(\024)671 1787 y Fj(l)650 1800 y Fi(Y)645 1890 y Fj(p)p Fh(=1)717 1841 y Fk(jj)p Fm(\010)780 1848 y Fj(X)809 1852 y Ff(p)829 1841 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)957 1848 y Fj(u)987 1841 y Fm(exp)f(3)1101 1754 y Fi(8)1102 1791 y(<)1102 1866 y(:)1165 1787 y Fj(l)1140 1800 y Fi(X)1139 1890 y Fj(p)p Fh(=1)1234 1800 y Fi(X)1210 1892 y Fj(X)1239 1896 y Ff(p)1257 1892 y Fg(3)p Fj(k)1299 1896 y Ff(p)1326 1841 y Fk(jj)p Fm(\010)1389 1848 y Fj(X)1418 1852 y Ff(p)1438 1841 y Fm(\()p Fl(J)o(;)h(:)p Fm(\))p Fk(jj)1566 1848 y Fj(u)1587 1754 y Fi(9)1587 1791 y(=)1587 1866 y(;)1632 1781 y(\025)0 2032 y Fm(Then,)16 b(b)o(y)g(Cauc)o(h)o(y)f(Sc)o(h)o(w)o(arz)h(inequalit)o(y) l(,)d(w)o(e)j(get)h(that)f(:)105 2250 y Fl(A)41 b Fk(\024)287 2208 y Fi(X)263 2300 y Fj(X)292 2304 y Ff(p)310 2300 y Fg(3)p Fj(k)352 2304 y Ff(p)379 2164 y Fi(0)379 2239 y(@)415 2250 y Fl(I)-13 b(E)471 2164 y Fi(2)471 2239 y(4)525 2196 y Fj(l)504 2208 y Fi(Y)499 2299 y Fj(p)p Fh(=1)570 2250 y Fk(jjr)640 2257 y Fj(i)653 2250 y Fm(\010)688 2257 y Fj(X)717 2261 y Ff(p)737 2250 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)865 2229 y Fh(2)865 2262 y Fj(u)887 2164 y Fi(3)887 2239 y(5)915 2164 y(1)915 2239 y(A)956 2162 y Fc(1)p 956 2168 16 2 v 956 2189 a(2)987 2164 y Fi(0)987 2239 y(@)1023 2250 y Fl(I)-13 b(E)1079 2164 y Fi(2)1079 2239 y(4)1107 2250 y Fm(exp)8 b(6)1222 2162 y Fi(8)1222 2200 y(<)1222 2274 y(:)1285 2196 y Fj(l)1261 2208 y Fi(X)1259 2299 y Fj(p)p Fh(=1)1354 2208 y Fi(X)1330 2300 y Fj(X)1359 2304 y Ff(p)1378 2300 y Fg(3)p Fj(k)1420 2304 y Ff(p)1446 2250 y Fk(jj)p Fm(\010)1509 2257 y Fj(X)1538 2261 y Ff(p)1558 2250 y Fm(\()p Fl(J)o(;)g(:)p Fm(\))p Fk(jj)1686 2257 y Fj(u)1708 2162 y Fi(9)1708 2200 y(=)1708 2274 y(;)1744 2164 y(3)1744 2239 y(5)1772 2164 y(1)1772 2239 y(A)1813 2162 y Fc(1)p 1813 2168 V 1813 2189 a(2)1862 2250 y Fm(\(63\))0 2440 y(T)l(o)15 b(b)q(ound)h(the)e(terms)g(in)g(the)29 b(righ)o(t)14 b(hand)i(side)e(of)h (\(63\),)g(w)o(e)g(\014rst)g(mak)o(e)d(the)j(basic)f(observ)m(ation)i(that,)0 2500 y(since)e Fk(f)p Fl(k)168 2507 y Fh(1)188 2500 y Fl(;)8 b Fk(\001)g(\001)g(\001)g Fl(;)g(k)323 2507 y Fj(l)336 2500 y Fk(g)15 b Fm(are)f(di\013eren)o(t)g(p)q(oin)o(ts,)g(eac)o(h)g(set)h Fl(X)k Fm(in)14 b(the)g(ab)q(o)o(v)o(e)g(sum)g(can)g(app)q(ear)i(at)f(most)e Fk(j)p Fl(X)t Fk(j)i Fm(\()950 2699 y(26)p eop %%Page: 27 27 27 26 bop 0 26 a Fk(\024)14 b Fm(\(2)p Fl(R)7 b Fm(+)g(1\))228 8 y Fj(d)262 26 y Fm(\))14 b(times.)19 b(Th)o(us,)14 b(if)g Fl(C)t Fm(\()p Fl(R;)8 b(d)p Fm(\))14 b(is)g(the)g(n)o(um)o(b)q(er)e(of)j (sets)f(con)o(taining)g(a)g(p)q(oin)o(t)h Fl(i)e Fm(and)i(included)0 86 y(in)h Fl(B)s Fm(\()p Fl(i;)8 b(R)p Fm(\),)15 b(w)o(e)h(get)g(:)161 240 y Fl(I)-13 b(E)217 155 y Fi(2)217 229 y(4)244 240 y Fm(exp)8 b(6)359 153 y Fi(8)360 190 y(<)360 265 y(:)423 186 y Fj(l)398 199 y Fi(X)397 289 y Fj(p)p Fh(=1)492 199 y Fi(X)468 291 y Fj(X)497 295 y Ff(p)515 291 y Fg(3)p Fj(k)557 295 y Ff(p)583 240 y Fk(jj)p Fm(\010)646 247 y Fj(X)675 251 y Ff(p)695 240 y Fm(\()p Fl(J)o(;)g(:)p Fm(\))p Fk(jj)823 247 y Fj(u)845 153 y Fi(9)845 190 y(=)845 265 y(;)882 155 y(3)882 229 y(5)924 240 y Fk(\024)976 167 y Fi( )1009 240 y Fm(sup)1030 279 y Fj(X)1091 240 y Fl(I)-13 b(E)s Fm([exp)o Fk(f)p Fm(6)p Fk(j)p Fl(X)t Fk(jjj)p Fm(\010)1411 247 y Fj(X)1445 240 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1573 247 y Fj(u)1594 240 y Fk(g)p Fm(])1633 167 y Fi(!)1666 178 y Fj(C)r Fh(\()p Fj(R;d)p Fh(\))p Fj(l)0 392 y Fm(Similarly)l(,)13 b(w)o(e)j(ha)o(v)o(e)f(:)265 544 y Fl(I)-13 b(E)321 458 y Fi(2)321 533 y(4)375 490 y Fj(l)353 502 y Fi(Y)349 593 y Fj(p)p Fh(=1)420 544 y Fk(jjr)490 551 y Fj(i)503 544 y Fm(\010)538 551 y Fj(X)567 555 y Ff(p)587 544 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)715 523 y Fh(2)715 556 y Fj(u)737 458 y Fi(3)737 533 y(5)779 544 y Fk(\024)831 471 y Fi( )864 544 y Fm(sup)885 583 y Fj(X)1010 544 y Fm(max)946 578 y Fj(k)q Fg(\024)p Fh(\(2)p Fj(R)p Fh(+1\))1110 568 y Ff(d)p Fc(+1)1174 544 y Fl(I)-13 b(E)1230 496 y Fi(h)1250 544 y Fk(jjr)1320 551 y Fj(i)1333 544 y Fm(\010)1368 551 y Fj(X)1402 544 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1530 523 y Fj(k)1530 556 y(u)1552 496 y Fi(i)1576 494 y Fc(1)p 1576 500 16 2 v 1576 520 a(2)1599 471 y Fi(!)1632 482 y Fh(2)p Fj(l)1671 544 y Fl(:)0 675 y Fm(Hence,)15 b(\(63\))h(can)h(b)q(e)f(b)q(ounded)h(b)o(y)f(:)0 826 y Fl(A)41 b Fk(\024)g Fl(B)206 753 y Fi( )239 826 y Fl(C)t Fm(\()p Fl(R;)8 b(d)p Fm(\))g(sup)429 865 y Fj(X)555 826 y Fm(max)490 859 y Fj(k)q Fg(\024)p Fh(\(2)p Fj(R)p Fh(+1\))654 850 y Ff(d)p Fc(+1)719 778 y Fi(\020)744 826 y Fl(I)-13 b(E)800 778 y Fi(h)819 826 y Fk(jjr)889 833 y Fj(i)903 826 y Fm(\010)938 833 y Fj(X)972 826 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1100 805 y Fj(k)1100 838 y(u)1121 778 y Fi(i\021)1171 776 y Fc(1)p 1171 782 V 1171 802 a(2)1201 826 y Fm(sup)1222 865 y Fj(X)1283 826 y Fm(\()p Fl(I)-13 b(E)s Fm([exp)o Fk(f)p Fm(6)p Fk(j)p Fl(X)t Fk(jjj)p Fm(\010)1622 833 y Fj(X)1656 826 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1784 833 y Fj(u)1805 826 y Fk(g)p Fm(]\))1868 785 y Ff(C)r Fc(\()p Ff(R;d)p Fc(\))p 1868 794 97 2 v 1908 815 a(2)1971 753 y Fi(!)2004 764 y Fj(l)0 1008 y Fm(where)16 b Fl(B)j Fm(denotes)d(some)f(\014nite)h(constan)o(t.)22 b(But,)15 b(when)33 b(I)-12 b(H1)15 b(is)h(satis\014ed,)g(it)g(is)g(ob)o(vious)g(that) 605 1139 y(sup)626 1178 y Fj(X)729 1139 y Fm(max)687 1173 y Fj(k)q Fg(\024)p Fh(\(2)p Fj(R)p Fh(\))806 1163 y Ff(d)p Fc(+1)870 1091 y Fi(\020)895 1139 y Fl(I)-13 b(E)951 1091 y Fi(h)971 1139 y Fk(jjr)1041 1146 y Fj(i)1054 1139 y Fm(\010)1089 1146 y Fj(X)1123 1139 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1251 1119 y Fj(k)1251 1151 y(u)1272 1091 y Fi(i\021)1322 1089 y Fc(1)p 1322 1095 16 2 v 1322 1115 a(2)0 1251 y Fm(go)q(es)17 b(to)g(zero)f(when)g Fl(p)421 1258 y Fh(1)458 1251 y Fm(and)h Fl(J)580 1258 y Fh(0)615 1251 y Fm(do,)g(therefore)131 1431 y Fl(C)t Fm(\()p Fl(R;)8 b(d)p Fm(\))g(sup)321 1471 y Fj(X)424 1431 y Fm(max)382 1465 y Fj(k)q Fg(\024)p Fh(\(2)p Fj(R)p Fh(\))501 1456 y Ff(d)p Fc(+1)566 1383 y Fi(\020)590 1431 y Fl(I)-13 b(E)647 1383 y Fi(h)666 1431 y Fk(jjr)736 1438 y Fj(i)749 1431 y Fm(\010)784 1438 y Fj(X)818 1431 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)946 1411 y Fj(k)946 1444 y(u)968 1383 y Fi(i\021)1017 1381 y Fc(1)p 1017 1387 V 1017 1408 a(2)1048 1431 y Fm(sup)1069 1471 y Fj(X)1130 1431 y Fm(\()p Fl(I)-13 b(E)r Fm([exp)p Fk(f)p Fm(6)p Fk(j)p Fl(X)t Fk(jjj)p Fm(\010)1469 1438 y Fj(X)1503 1431 y Fm(\()p Fl(J)o(;)8 b(:)p Fm(\))p Fk(jj)1631 1438 y Fj(u)1652 1431 y Fk(g)p Fm(]\))1715 1390 y Ff(C)r Fc(\()p Ff(R;d)p Fc(\))p 1715 1400 97 2 v 1755 1420 a(2)0 1566 y Fm(go)q(es)17 b(to)g(zero)f(with)g Fl(p)405 1573 y Fh(1)441 1566 y Fm(and)h Fl(J)563 1573 y Fh(0)583 1566 y Fm(,)f(whic)o(h)f(ac)o(hiev)o(es)g(the)h(pro)q(of)h(of)g(Lemma)d(4.9.) 1910 1626 y Fk(})0 1711 y Fb(Ac)n(kno)n(wledgemen)n(t)j(:)73 1772 y Fm(Alice)12 b(Guionnet)i(wishes)g(to)h(thank)f(the)g(Europ)q(ean)h (Comm)o(unit)o(y)c(for)j(giving)g(her)g(the)f(opp)q(ortunit)o(y)0 1832 y(to)k(visit)e(Imp)q(erial)f(College)h(and)i(supp)q(orting)g(this)f(w)o (ork)h(b)o(y)e(EEC)i(gran)o(t)f(SC1)h(-)g(CT92)g(-)g(784,)f(as)h(w)o(ell)0 1892 y(as)g(the)f(Couran)o(t)h(Institute)e(for)i(w)o(elcoming)d(her)i(for)g (most)g(of)h(the)f(researc)o(h)f(p)q(erio)q(d.)0 2058 y Fn(References)24 2168 y Fm([1])24 b(AIZENMAN)13 b(M.)i(;)g(HOLLEY)h(R.)f(;)g(Rapid)h(Con)o(v)o (ergence)e(to)i(Equilibrium)d(of)i(Sto)q(c)o(hastic)h(Ising)100 2228 y(Mo)q(dels)h(in)e(the)h(Dobrushin-Shlosman)h(Regime.)100 2309 y(pp.)j(1{11)i(in)e(P)o(ercolation)g(Theorey)g(and)h(Ergo)q(dic)g 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(olev)i(inequalities)d(and)i(sto)q(c)o(hastic)g(Ising)g(mo)q(d-)100 2457 y(els.)100 2538 y(J.)k(Stat.)g(Ph)o(ys.)g(V)l(ol)g(46,)g(1159{1194)k (\(1987\))950 2699 y(28)p eop %%Page: 29 29 29 28 bop 0 26 a Fm([15])24 b(KESTEN)17 b(H.)e(;)h(Asp)q(ect)g(of)g(\014rst)h (passage)g(p)q(ercolation.)100 107 y(Ecole)f(d'ete)f(de)h(St)h(Flour,)e(LNM)h (1180,)i(p.)d(125{264)k(\(1986)f(\).)0 207 y([16])24 b(LAR)o(OCHE)16 b(E.;)g(Hyp)q(ercon)o(tractivit)o(\023)-23 b(e)13 b(p)q(our)18 b(des)e(syst)o(\022)-23 b(emes)15 b(de)h(spin)g(de)g(p)q(ort)o(\023)-23 b(ee)16 b(in\014nie)100 287 y(Probab.)h(Theor.)f(Relat.)g(Fields)f(V)l(ol)h (101,)g(89{132)j(\(1995\).)0 387 y([17])24 b(SHENG)19 b(LIN)g(LU;)g(HORNG)f (-)i(TZER)f(Y)l(A)o(U;)e(Sp)q(ectral)i(Gap)h(and)g(Logarithmic)e(Sob)q(olev)i (In-)100 448 y(equalit)o(y)15 b(for)h(Ka)o(w)o(asaki)h(and)g(Glaub)q(er)f (Dynamics.)100 528 y(Comm)o(un.)e(Math.)i(Ph)o(ys.)f(V)l(ol)h(156,)h(399{433) i(\(1993\))0 628 y([18])24 b(MAJEWSKI)13 b(A.W.)f(;)h(ZEGARLINSKI)g(B;)f 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