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b(odinger)37 b(op)s(erators.)-24 3225 y(Suc)m(h)j(ma)m(y)d(b)s(e)i(the) f(case)h(with)e(the)i(quasi-ballistic)c(dynamics)i(exhibited)h(b)m(y)h (the)g(Harp)s(er)e(mo)s(del)-24 3345 y(with)c(Liouville)c (incommensuration)i([La].)123 3513 y(The)43 b(Hausdor\013)h(dimension)d (of)h(a)h(measure)g(is)f(related)h(to)f(its)g(lo)m(w)m(er)h(p)s(oin)m (t)m(wise)g(dimensions)-24 3633 y(b)m(y)j(the)g(theory)f(of)g(Rogers)g (and)g(T)-8 b(a)m(ylor)44 b([Ro],)k(whose)e(relev)-5 b(ance)46 b(in)e(the)h(presen)m(t)i(con)m(text)f(w)m(as)-24 3754 y(adv)m(o)s(cated)41 b(b)m(y)h(Last)e([La].)66 b(Our)40 b(presen)m(t)i(result)e(calls)f Ff(upp)-5 b(er)40 b Fm(p)s(oin)m(t)m (wise)g(dimensions)g(in)m(to)f(pla)m(y)-8 b(.)-24 3874 y(Apart)31 b(from)f([G3)o(],)h(these)h(ha)m(v)m(e)g(not)f(y)m(et)h (found)f(their)f(w)m(a)m(y)i(in)e(the)h(theory)h(of)e(quan)m(tum)h (transp)s(ort.)-24 3994 y(In)c(the)g(conclusiv)m(e)g(app)s(endix,)h(w)m (e)f(therefore)g(elab)s(orate)e(on)h(results)h(b)m(y)g(Cutler)g([Cu])g 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y(on)h(the)g(basis)f Fj(B)k Fm(as)d(follo)m(ws:) p 928 973 V 928 1026 a Fk(n)q Fm(\()p Fk(\017;)17 b(T)d Fm(\))55 b(=)g(min)1582 851 y Fe(8)1582 926 y(<)1582 1075 y(:)p 1672 973 V 1672 1026 a Fk(n)28 b Fj(2)g Fd(N)1984 851 y Fe(\014)1984 901 y(\014)1984 951 y(\014)1984 1001 y(\014)1984 1051 y(\014)1984 1100 y(\014)2050 943 y(X)2040 1126 y Fn(n)p Fi(\025)p 2138 1088 43 4 v Fn(n)2197 1026 y Fk(p)2246 1041 y Fn(n)2293 1026 y Fm(\()p Fk(T)14 b Fm(\))27 b Fj(\024)h Fk(\017)2611 851 y Fe(9)2611 926 y(=)2611 1075 y(;)2744 1026 y Fm(,)860 b(\(1\))-24 1287 y(where)34 b(the)f Fk(p)475 1302 y Fn(n)522 1287 y Fm(\()p Fk(T)14 b Fm(\))32 b(is)g(the)h(a)m(v)m(erage)g(probabilit)m(y)e(up)h (to)g(time)f Fk(T)47 b Fm(in)31 b(the)i(basis)f(state)h Fj(j)p Fk(n)p Fj(i)28 b(2)g(B)s Fm(,)33 b(giv)m(en)-24 1408 y(b)m(y:)1177 1673 y Fk(p)1226 1688 y Fn(n)1273 1673 y Fm(\()p Fk(T)14 b Fm(\))55 b(=)1606 1556 y Fe(Z)1689 1582 y Fn(T)1652 1744 y Fr(0)1771 1605 y Fk(dt)p 1771 1650 86 4 v 1779 1741 a(T)1883 1673 y Fj(jh)p Fk(n)p Fj(j)p Fk(e)2081 1632 y Fi(\000)p Fn({H)5 b(t)2252 1673 y Fj(j)p Fk( )t Fj(ij)2414 1632 y Fr(2)2485 1673 y Fm(.)-24 1876 y(Upp)s(er)33 b(and)g(lo)m(w)m(er)g(gro)m(wth)g(exp)s(onen)m(ts)h (of)e(the)h(minimal)c(carriers)j(are)h(de\014ned)h(as:)444 2147 y Fk(\014)505 2105 y Fr(+)499 2171 y(0)564 2147 y Fm(\()p Fk(\017)p Fm(\))55 b(=)h(lim)17 b(sup)919 2225 y Fn(T)10 b Fi(!1)1191 2079 y Fm(log\()p 1355 2026 59 4 v Fk(n)p Fm(\()p Fk(\017;)17 b(T)d Fm(\)\))p 1191 2123 490 4 v 1300 2215 a(log\()p Fk(T)g Fm(\))1723 2147 y(,)228 b Fk(\014)2039 2105 y Fi(\000)2033 2171 y Fr(0)2098 2147 y Fm(\()p Fk(\017)p Fm(\))55 b(=)h(lim)17 b(inf)2439 2206 y Fn(T)10 b Fi(!1)2696 2079 y Fm(log\()p 2860 2026 59 4 v Fk(n)p Fm(\()p Fk(\017;)17 b(T)d Fm(\)\))p 2696 2123 490 4 v 2805 2215 a(log\()p Fk(T)g Fm(\))3229 2147 y(,)375 b(\(2\))-24 2383 y(W)-8 b(e)29 b(also)f(de\014ne)i Fk(\014)671 2342 y Fi(\006)665 2405 y Fr(0)757 2383 y Fm(=)e(lim)996 2398 y Fn(\017)p Fi(!)p Fr(0)1152 2383 y Fk(\014)1213 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b(=)g(lim)17 b(inf)512 5204 y Fn(\017)p Fi(!)p Fr(0)741 5080 y Fm(log\()p Fk(\026)p Fm(\([)p Fk(E)28 b Fj(\000)22 b Fk(\017;)17 b(E)29 b Fm(+)22 b Fk(\017)p Fm(]\)\))p 741 5124 912 4 v 1076 5216 a(log\()p Fk(\017)p Fm(\))1695 5147 y(,)p 1949 5068 51 4 v 227 w Fk(d)2000 5162 y Fn(\026)2047 5147 y Fm(\()p Fk(E)6 b Fm(\))55 b(=)g(lim)17 b(sup)2469 5224 y Fn(\017)p Fi(!)p Fr(0)2713 5080 y Fm(log\()p Fk(\026)p Fm(\([)p Fk(E)27 b Fj(\000)c Fk(\017;)17 b(E)28 b Fm(+)22 b Fk(\017)p Fm(]\)\))p 2713 5124 912 4 v 3048 5216 a(log\()p Fk(\017)p Fm(\))3666 5147 y(,)3631 5324 y(\(4\))-24 5444 y(while)32 b(for)g Fk(E)i Fj(62)28 b Fm(supp)q(\()p Fk(\026)p Fm(\),)k Fk(d)p 975 5457 51 4 v 21 x Fn(\026)1072 5444 y Fm(\()p Fk(E)6 b Fm(\))28 b(=)p 1358 5365 V 28 w Fk(d)1408 5459 y Fn(\026)1455 5444 y Fm(\()p Fk(E)6 b Fm(\))28 b(=)f Fj(1)p Fm(.)1842 5895 y(3)p eop %%Page: 4 4 4 3 bop 123 383 a Fm(The)23 b(\(upp)s(er\))f(Hausdor\013)h(dimension)d (dim)1705 340 y Fr(+)1705 408 y(H)1764 383 y Fm(\()p Fk(\026)p 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Fd(Pro)s(of.)71 b Fm(Let)42 b(us)h(de\014ne)g Fj(j)p Fk( )1051 3142 y Fn(N)1118 3127 y Fj(i)g Fm(=)g Fk(\037)1380 3142 y Fn(A)1433 3153 y Fc(N)r(;\013)1551 3127 y Fm(\()p Fk(H)8 b Fm(\))p Fj(j)p Fk( )t Fj(i)p Fm(,)43 b(so)f Fj(k)17 b(j)p Fk( )2207 3142 y Fn(N)2274 3127 y Fj(ik)2363 3090 y Fr(2)2445 3127 y Fm(=)43 b Fk(b)2605 3142 y Fn(N)2716 3127 y Fk(>)h Fm(0.)71 b(W)-8 b(e)42 b(wish)g(to)f(apply)-24 3247 y(Prop)s(osition)30 b(1,)h(with)g(the)h(follo)m(wing)c(sp)s(eci\014cations:)43 b(replace)31 b Fj(j)p Fk( )t Fj(i)g Fm(b)m(y)h Fj(j)p Fk( )2756 3262 y Fn(N)2823 3247 y Fj(i)f Fm(and)g(c)m(ho)s(ose)h Fj(F)41 b Fm(as)31 b(the)-24 3367 y(in)m(tegers)44 b(smaller)c(than)j (a)g(giv)m(en)g Fk(m)p Fm(,)j(\014nally)41 b(set)j Fk(\017)h Fm(=)g(\()p Fk(b)2199 3382 y Fn(N)2267 3367 y Fk(=)p Fm(9\))2403 3331 y Fr(2)2442 3367 y Fm(.)74 b(Condition)42 b(\(7\),)j(whic)m(h)e(mak)m(es)-24 3488 y(Prop)s(osition)31 b(1)i(applicable,)d(then)j(b)s(ecomes)h(precisely)e Fk(b)2120 3503 y Fn(N)2188 3488 y Fm(2)2237 3452 y Fn(N)7 b Fi(\000)p Fr(1)2422 3488 y Fk(<)27 b Fm(9)p Fk(T)42 b Fj(\024)28 b Fk(b)2819 3503 y Fn(N)2887 3488 y Fm(2)2936 3452 y Fn(N)3003 3488 y Fm(.)123 3608 y(The)41 b(sp)s(ectral)g(measure)f(of)h Fj(j)p Fk( )1304 3623 y Fn(N)1371 3608 y Fj(i)f Fm(is)g Fk(d\026)1666 3623 y Fn(N)1733 3608 y Fm(\()p Fk(E)6 b Fm(\))41 b(=)g Fk(\037)2106 3623 y Fn(A)2159 3634 y Fc(N)r(;\013)2277 3608 y Fm(\()p Fk(E)6 b Fm(\))p Fk(d\026)p Fm(\()p Fk(E)g Fm(\).)67 b(Hence)41 b(the)g(sum)g(on)f(the)-24 3728 y(righ)m(t-hand)24 b(side)h(of)f(\(8\))g(is)g(restricted)h(to)g (the)g(set)g Fj(J)1888 3743 y Fn(N)s(;\013)2040 3728 y Fm(of)f(indices)h Fk(j)33 b Fj(2)28 b Fd(Z)d Fm(for)g(whic)m(h)g Fk(\026)p Fm(\()p Fk(I)3277 3692 y Fn(N)3269 3753 y(j)3343 3728 y Fm(\))j Fk(<)g Fm(2)3562 3692 y Fi(\000)p Fn(N)7 b(\013)3729 3728 y Fm(.)-24 3849 y(The)34 b(sum)e(can)h(b)s(e)g (estimated)f(as)h(follo)m(ws:)689 4103 y Fe(X)676 4287 y Fn(n)p Fi(2F)916 4103 y Fe(X)856 4287 y Fn(j)t Fi(2J)984 4298 y Fc(N)r(;\013)1130 4186 y Fj(jh)p Fk(n)p Fj(j)p Fk(\037)1344 4145 y Fn(N)1344 4211 y(j)1410 4186 y Fj(ij)1477 4145 y Fr(2)1599 4186 y Fj(\024)1773 4103 y Fe(X)1760 4287 y Fn(n)p Fi(2F)1970 4103 y Fe(X)1923 4287 y Fn(j)t Fr(\020)p Fi(J)2024 4298 y Fc(N)r(;\013)2153 4186 y Fj(k)p Fk(\037)2264 4145 y Fn(N)2264 4211 y(j)2331 4186 y Fj(k)2381 4145 y Fr(2)2437 4186 y Fj(k)p Fk(\037)2548 4145 y Fn(N)2548 4211 y(j)2615 4186 y Fm(\()p Fk(H)8 b Fm(\))p Fj(j)p Fk(n)p Fj(ik)2955 4145 y Fr(2)1600 4441 y Fm(=)1773 4358 y Fe(X)1760 4542 y Fn(n)p Fi(2F)1983 4358 y Fe(X)1923 4542 y Fn(j)t Fi(2J)2051 4553 y Fc(N)r(;\013)2180 4441 y Fk(\026)2239 4456 y Fn(N)2306 4441 y Fm(\()p Fk(I)2395 4400 y Fn(N)2387 4465 y(j)2462 4441 y Fm(\))p Fj(k)p Fk(\037)2611 4400 y Fn(N)2611 4465 y(j)2678 4441 y Fm(\()p Fk(H)g Fm(\))p Fj(j)p Fk(n)p Fj(ik)3018 4400 y Fr(2)1599 4690 y Fj(\024)84 b Fk(m)63 b Fm(max)1878 4750 y Fn(j)t Fi(2J)2006 4761 y Fc(N)r(;\013)2135 4690 y Fk(\026)2194 4705 y Fn(N)2261 4690 y Fm(\()p Fk(I)2350 4649 y Fn(N)2342 4715 y(j)2417 4690 y Fm(\))1599 4899 y Fj(\024)84 b Fk(m)17 b Fm(2)1911 4857 y Fi(\000)p Fn(N)7 b(\013)2110 4899 y Fm(.)-24 5116 y(W)-8 b(e)33 b(c)m(ho)s(ose)h Fk(m)f Fm(as)f(follo)m(ws:)1373 5410 y Fk(m)56 b Fm(=)1685 5343 y(1)p 1655 5387 108 4 v 1655 5479 a(4)p Fk(\031)1789 5264 y Fe( )1865 5343 y Fk(b)1906 5358 y Fn(N)p 1865 5387 109 4 v 1895 5479 a Fm(9)1984 5264 y Fe(!)2049 5287 y Fr(3)2106 5410 y Fm(2)2155 5369 y Fn(N)7 b(\013)2299 5410 y Fm(,)1256 b(\(11\))-24 5646 y(and)33 b(denote)g Fk(P)543 5661 y Fn(m)638 5646 y Fm(=)741 5579 y Fe(P)829 5667 y Fn(n)33 b(b)1067 2499 y Fn(N)1135 2484 y Fm(.)52 b(It)36 b(therefore)g(follo)m(ws)e(from)g (the)i(de\014nition)f(of)g(a)g(minimal)d(carrier)-24 2604 y(that)p 193 2551 59 4 v 38 w Fk(n)q Fm(\()p Fk(b)331 2619 y Fn(N)398 2604 y Fk(=)p Fm(2)p Fk(;)17 b(T)d Fm(\))37 b Fj(\025)h Fk(m)p Fm(.)61 b(The)39 b(pro)s(of)e(is)h(concluded)h(on)f (recalling)e(the)j(de\014nition)e(of)h Fk(m)h Fm(and)f Fk(\017)h Fm(as)-24 2724 y(w)m(ell)32 b(as)h(the)g(connection)g(b)s(et) m(w)m(een)h Fk(N)43 b Fm(and)33 b Fk(T)14 b Fm(.)1946 b Fa(2)123 2892 y Fm(A)34 b(link)g(to)g(sp)s(ectral)h(dimensions)e(is)i (pro)m(vided)g(b)m(y)g(\(w)m(e)h(also)e(use)h(lim)17 b(sup)33 b(and)i(lim)17 b(inf)37 b(in)d(their)-24 3012 y(set-theoretic)f(meaning;)e(further)i(w)m(e)h(suppress)h(the)e (subscripts)h Fk(N)k Fj(!)27 b(1)p Fm(\):)661 3353 y(lim)17 b(sup)e Fk(A)1050 3368 y Fn(N)s(;\013)1261 3353 y Fm(=)83 b(lim)17 b(sup)1736 3256 y Fe(n)1791 3353 y Fk(E)34 b Fj(2)28 b Fd(R)17 b Fj(j)g Fk(\026)p Fm(\()p Fk(I)2285 3312 y Fn(N)2277 3377 y(j)t Fr(\()p Fn(E)t Fr(\))2422 3353 y Fm(\))27 b Fk(<)h Fm(2)2640 3312 y Fi(\000)p Fn(N)7 b(\013)2807 3256 y Fe(o)1261 3597 y Fm(=)1420 3451 y Fe(\()1487 3597 y Fk(E)34 b Fj(2)28 b Fd(R)1803 3448 y Fe(\014)1803 3498 y(\014)1803 3548 y(\014)1803 3597 y(\014)1803 3647 y(\014)1848 3597 y Fm(lim)17 b(sup)2173 3517 y Fj(\000)g Fm(log)2393 3541 y Fr(2)2449 3517 y Fk(\026)p Fm(\()p Fk(I)2597 3481 y Fn(N)2589 3544 y(j)t Fr(\()p Fn(E)t Fr(\))2736 3517 y Fm(\))p 2173 3574 601 4 v 2429 3666 a Fk(N)2811 3597 y(>)28 b(\013)2978 3451 y Fe(\))1261 3830 y Fm(=)1420 3734 y Fe(n)1475 3830 y Fk(E)34 b Fj(2)28 b Fd(R)17 b Fj(j)p 1821 3751 51 4 v 17 w Fk(d)1870 3845 y Fn(\026)1917 3830 y Fm(\()p Fk(E)6 b Fm(\))28 b Fk(>)f(\013)2265 3734 y Fe(o)3582 3830 y Fm(\(12\))-24 4050 y(In)33 b(a)g(completely)e(analogous)g(w)m(a)m(y)-8 b(,)34 b(w)m(e)g(get)1079 4291 y(lim)17 b(inf)i Fk(A)1438 4306 y Fn(N)s(;\013)1622 4291 y Fm(=)1753 4194 y Fe(n)1809 4291 y Fk(E)33 b Fj(2)c Fd(R)17 b Fj(j)g Fk(d)p 2155 4304 V 2204 4311 a Fn(\026)2250 4291 y Fm(\()p Fk(E)6 b Fm(\))28 b Fk(>)g(\013)2599 4194 y Fe(o)3582 4291 y Fm(\(13\))123 4477 y(A)m(t)33 b(this)f(p)s(oin)m(t,)h(w)m(e)h(deriv)m (e)f(lo)m(w)m(er)g(b)s(ounds.)45 b(First)32 b(of)h(all,)e(if)g Fk(\013)e(<)f Fm(dim)2761 4492 y Fr(H)2818 4477 y Fm(\()p Fk(\026)p Fm(\),)k(then)i(\(5\))e(and)h(\(13\))-24 4598 y(sho)m(w)39 b(that)e(lim)17 b(inf)i Fk(\026)p Fm(\()p Fk(A)895 4613 y Fn(N)s(;\013)1023 4598 y Fm(\))36 b Fj(\025)g Fk(\026)p Fm(\(lim)17 b(inf)i Fk(A)1666 4613 y Fn(N)s(;\013)1795 4598 y Fm(\))35 b Fk(>)g Fm(0,)j(so)g Fk(b)2259 4613 y Fn(N)2364 4598 y Fm(is)e(larger)g(than)i(some)f Fk(b)3270 4613 y Fr(0)3347 4598 y Fm(for)f(all)f Fk(N)10 b Fm(.)-24 4718 y(Inserting)34 b(suc)m(h)h(a)e Fk(b)731 4733 y Fr(0)804 4718 y Fm(in)f(Prop)s(osition)g(2,)h(w)m(e)h(immediately)d(get)p 2417 4665 59 4 v 33 w Fk(n)p Fm(\()p Fk(b)2554 4733 y Fr(0)2594 4718 y Fk(=)p Fm(2)p Fk(;)17 b(T)d Fm(\))28 b Fj(\025)h Fm(const)18 b Fk(T)3290 4682 y Fn(\013)3372 4718 y Fm(for)33 b(all)e Fk(T)14 b Fm(.)-24 4839 y(This)41 b(pro)m(v)m(es)i(the)e(already)g(kno)m(wn)h(lo)m(w)m(er)f(b)s(ound,)i Fk(\014)2025 4797 y Fi(\000)2019 4860 y Fr(0)2125 4839 y Fj(\025)f Fm(dim)2407 4854 y Fr(H)2464 4839 y Fm(\()p Fk(\026)p Fm(\).)68 b(In)41 b(addition,)g(w)m(e)h(can)f(no)m(w)-24 4959 y(pro)m(v)m(e:)-24 5235 y Fd(Theorem)c(1)49 b Ff(F)-7 b(or)49 b(al)5 b(l)48 b(p)-5 b(ositive)49 b Fk(q)t Ff(,)k(the)c(upp)-5 b(er)49 b(gr)-5 b(owth)49 b(exp)-5 b(onent)49 b(of)g(the)g Fk(q)-13 b Ff(-th)49 b(moment)g(of)g(the)-24 5355 y(p)-5 b(osition)35 b(op)-5 b(er)g(ator)34 b(satis\014es)g Fk(\014)1153 5319 y Fr(+)1147 5380 y Fn(q)1240 5355 y Fj(\025)28 b Fm(dim)1507 5312 y Fr(+)1507 5380 y(P)1566 5355 y Fm(\()p Fk(\026)p Fm(\))p Ff(.)1842 5895 y Fm(6)p eop %%Page: 7 7 7 6 bop -24 383 a Fd(Pro)s(of.)96 b Fm(It)50 b(is)f(ob)m(viously)h (su\016cien)m(t)h(to)f(consider)g(the)g(case)h(dim)2595 340 y Fr(+)2595 408 y(P)2654 383 y Fm(\()p Fk(\026)p Fm(\))56 b Fk(>)i Fm(0.)95 b(Then,)55 b(if)49 b Fk(\013)58 b(<)-24 504 y Fm(dim)139 519 y Fr(P)191 504 y Fm(\()p Fk(\026)p Fm(\),)38 b(it)d(follo)m(ws)h(from)g(the)h(de\014nition)f (\(5\))g(of)h(the)g(pac)m(king)g(dimension)e(and)i(from)f(\(12\))g (that)-24 624 y Fk(\026)p Fm(\(lim)17 b(sup)e Fk(A)462 639 y Fn(N)s(;\013)590 624 y Fm(\))28 b Fk(>)f Fm(0.)41 b(Therefore,)28 b(from)c(the)i(Borel-Can)m(telli)d(lemma)g(w)m(e)k(get) f(that)3145 558 y Fe(P)3233 645 y Fn(N)3316 624 y Fk(\026)p Fm(\()p Fk(A)3486 639 y Fn(N)s(;\013)3614 624 y Fm(\))i(=)-24 744 y Fj(1)p Fm(.)61 b(This)38 b(in)g(turn)g(implies)e(that)i(there)h (is)f(a)g(sequence)j(of)c(in)m(tegers)i Fk(N)2705 759 y Fn(k)2785 744 y Fj(!)e(1)h Fm(suc)m(h)i(that)e Fk(b)3544 759 y Fn(N)3600 771 y Fc(k)3680 744 y Fm(=)-24 865 y Fk(\026)p Fm(\()p Fk(A)146 880 y Fn(N)202 892 y Fc(k)240 880 y Fn(;\013)309 865 y Fm(\))28 b Fk(>)f(N)566 823 y Fi(\000)p Fr(2)556 890 y Fn(k)661 865 y Fm(.)123 985 y(In)33 b(this)f(situation,)f(Prop)s(osition)g(2)h(sa)m(ys)i(that)p 1257 1210 59 4 v 1257 1263 a Fk(n)1331 1117 y Fe( )1407 1196 y Fk(b)1448 1211 y Fn(N)1504 1223 y Fc(k)p 1407 1240 140 4 v 1453 1332 a Fm(2)1557 1263 y Fk(;)17 b(T)1658 1278 y Fn(k)1700 1117 y Fe(!)1821 1263 y Fk(>)1987 1196 y(C)7 b Fm(\()p Fk(\013)q Fm(\))p 1963 1240 264 4 v 1963 1334 a Fk(N)2051 1293 y Fr(6)p Fi(\000)p Fr(2)p Fn(\013)2041 1359 y(k)2253 1263 y Fk(T)2324 1222 y Fn(\013)2310 1288 y(k)2405 1263 y Fm(.)-24 1495 y(In)29 b(this)f(inequalit)m(y)-8 b(,)28 b Fk(k)j Fj(2)d Fd(N)g Fm(is)g(arbitrary)-8 b(,)28 b Fk(C)7 b Fm(\()p Fk(\013)q Fm(\))27 b(is)h(a)g(n)m(umerical)f(factor) h(only)g(dep)s(ending)g(on)h Fk(\013)q Fm(,)f(the)-24 1615 y(sequence)39 b(of)34 b(times)h Fk(T)818 1630 y Fn(k)896 1615 y Fm(satis\014es)h Fk(b)1299 1630 y Fn(N)1355 1642 y Fc(k)1398 1615 y Fm(2)1447 1579 y Fn(N)1503 1591 y Fc(k)1541 1579 y Fi(\000)p Fr(1)1668 1615 y Fk(<)c Fm(9)p Fk(T)1882 1630 y Fn(k)1958 1615 y Fj(\024)h Fk(b)2109 1630 y Fn(N)2165 1642 y Fc(k)2207 1615 y Fm(2)2256 1579 y Fn(N)2312 1591 y Fc(k)2390 1615 y Fm(for)i(all)e Fk(k)s Fm(,)j(and)g Fk(b)3031 1630 y Fn(N)3087 1642 y Fc(k)3162 1615 y Fk(>)c(N)3358 1574 y Fi(\000)p Fr(2)3348 1640 y Fn(k)3453 1615 y Fm(.)52 b(F)-8 b(rom)-24 1736 y(all)28 b(that)h(it)f(follo)m(ws)g(that)i Fk(T)994 1751 y Fn(k)1064 1736 y Fj(!)d(1)i Fm(as)h Fk(k)h Fj(!)c(1)i Fm(and)h(that)f Fk(N)2248 1751 y Fn(k)2318 1736 y Fk(<)f(f)2481 1700 y Fi(\000)p Fr(1)2575 1736 y Fm(\()p Fk(T)2670 1751 y Fn(k)2713 1736 y Fm(\))h(where)i Fk(f)40 b Fm(is)29 b(the)g(function) -24 1856 y Fk(f)11 b Fm(\()p Fk(x)p Fm(\))28 b(=)g(2)347 1820 y Fn(x)390 1856 y Fk(=)p Fm(\(18)17 b Fk(x)647 1820 y Fr(2)686 1856 y Fm(\).)43 b(Hence)34 b(w)m(e)g(get)e(that,)h(for)f (all)e Fk(k)h Fj(2)d Fd(N)p Fm(,)p 1098 2081 59 4 v 1098 2134 a Fk(n)1173 1988 y Fe( )1249 2067 y Fk(b)1290 2082 y Fn(N)1346 2094 y Fc(k)p 1249 2111 140 4 v 1294 2203 a Fm(2)1399 2134 y Fk(;)17 b(T)1500 2149 y Fn(k)1542 1988 y Fe(!)1663 2134 y Fk(>)1987 2067 y(C)7 b Fm(\()p Fk(\013)q Fm(\))p 1805 2111 580 4 v 1805 2203 a(\()p Fk(f)1902 2174 y Fi(\000)p Fr(1)1995 2203 y Fm(\()p Fk(T)2090 2218 y Fn(k)2133 2203 y Fm(\)\))2209 2174 y Fr(6)p Fi(\000)p Fr(2)p Fn(\013)2411 2134 y Fk(T)2482 2093 y Fn(\013)2468 2159 y(k)2563 2134 y Fm(.)-24 2360 y(W)-8 b(e)37 b(denote)g(the)f(righ) m(t-hand)f(side)h(b)m(y)h Fk(C)1528 2375 y Fn(k)1570 2360 y Fm(.)54 b(Since)36 b Fk(\013)e(>)g Fm(0,)i Fk(C)2297 2375 y Fn(k)2376 2360 y Fm(will)e(b)s(e)i(ev)m(en)m(tually)g(larger)f (than)h(1.)-24 2481 y(F)-8 b(rom)37 b(the)i(de\014nition)e(\(1\))g(of)h (a)g(minimal)c(carrier,)39 b(w)m(e)g(obtain)e(that,)i(at)f(all)e(large) h(enough)i Fk(k)s Fm(,)g(the)-24 2601 y(total)31 b(probabilit)m(y)g (supp)s(orted)j(b)m(y)f(basis)g(states)g Fj(j)p Fk(n)p Fj(i)f Fm(with)h Fk(n)27 b(>)h(C)2458 2616 y Fn(k)2523 2601 y Fj(\000)22 b Fm(1)33 b(is)f(larger)f(than)i Fk(b)3347 2616 y Fn(N)3403 2628 y Fc(k)3446 2601 y Fk(=)p Fm(2.)43 b(F)-8 b(or)-24 2722 y(suc)m(h)34 b Fk(k)s Fm('s,)g(the)f(Cheb)m(yshev) j(inequalit)m(y)31 b(yields:)1272 2987 y Fk(M)1366 3002 y Fn(q)1404 2987 y Fm(\()p Fk(T)1499 3002 y Fn(k)1542 2987 y Fm(\))55 b Fj(\025)1778 2919 y Fk(b)1819 2934 y Fn(N)1875 2946 y Fc(k)p 1778 2964 140 4 v 1823 3055 a Fm(2)1944 2987 y(\()p Fk(C)2052 3002 y Fn(k)2117 2987 y Fj(\000)22 b Fm(1\))2303 2938 y Fn(q)2390 2987 y Fm(.)-24 3197 y(Whence,)35 b(replacing)c Fk(C)862 3212 y Fn(k)937 3197 y Fm(and)i(using)f Fk(b)1423 3212 y Fn(N)1479 3224 y Fc(k)1549 3197 y Fk(>)c(N)1741 3155 y Fi(\000)p Fr(2)1731 3222 y Fn(k)1836 3197 y Fm(,)k(w)m(e)i(get)825 3477 y Fk(M)919 3492 y Fn(q)957 3477 y Fm(\()p Fk(T)1052 3492 y Fn(k)1095 3477 y Fm(\))55 b Fk(>)1551 3410 y Fm(1)p 1329 3454 493 4 v 1329 3545 a(2\()p Fk(f)1475 3517 y Fi(\000)p Fr(1)1569 3545 y Fm(\()p Fk(T)1664 3560 y Fn(k)1707 3545 y Fm(\)\))1783 3517 y Fr(2)1849 3331 y Fe( )2046 3410 y Fk(C)7 b Fm(\()p Fk(\013)q Fm(\))p Fk(T)2333 3374 y Fn(\013)2319 3434 y(k)p 1924 3454 580 4 v 1924 3545 a Fm(\()p Fk(f)2021 3517 y Fi(\000)p Fr(1)2115 3545 y Fm(\()p Fk(T)2210 3560 y Fn(k)2253 3545 y Fm(\)\))2329 3517 y Fr(6)p Fi(\000)p Fr(2)p Fn(\013)2536 3477 y Fj(\000)23 b Fm(1)2685 3331 y Fe(!)2750 3354 y Fn(q)2837 3477 y Fm(.)-24 3713 y(Since)49 b(lim)383 3728 y Fn(T)10 b Fi(!1)596 3713 y Fm(log\()p Fk(f)819 3677 y Fi(\000)p Fr(1)913 3713 y Fm(\()p Fk(T)k Fm(\)\))p Fk(=)j Fm(log)n(\()p Fk(T)d Fm(\))55 b(=)h(0,)c(Theorem)d(1)g(follo)m(ws)e(from)h(the)h (de\014nition)f(of)g(the)-24 3833 y(exp)s(onen)m(t)34 b Fk(\014)457 3797 y Fr(+)451 3858 y Fn(q)516 3833 y Fm(.)3138 b Fa(2)-24 4104 y Fd(Remark)37 b(1)49 b Fm(The)36 b(presen)m(t)h(pro)s(of)d(do)s(es)i(not)f(allo)m(w)e(to)i(b)s(ound)g Fk(\014)2447 4063 y Fr(+)2441 4126 y(0)2541 4104 y Fm(b)s(elo)m(w)g(b)m (y)h(means)f(of)g(dim)3535 4062 y Fr(+)3535 4129 y Fn(P)3594 4104 y Fm(\()p Fk(\026)p Fm(\).)-24 4225 y(In)28 b(order)g(to)f(do)h (that)f(one)h(needs)h(to)e(kno)m(w)i(that)e(lim)17 b(sup)e Fk(\026)p Fm(\()p Fk(A)2285 4240 y Fn(N)s(;\013)2413 4225 y Fm(\))27 b Fk(>)h Fm(0)f(.)42 b(W)-8 b(e)28 b(ignore)e(if)h (that)g(follo)m(ws)-24 4345 y(from)36 b(dim)374 4302 y Fr(+)374 4370 y Fn(P)433 4345 y Fm(\()p Fk(\026)p Fm(\))f Fk(>)g(\013)q Fm(.)56 b(Nev)m(ertheless)40 b(w)m(e)e(can)f(pro)m(v)m(e) h(the)g(w)m(eak)m(er)g(result)f(giv)m(en)h(in)e(Prop)s(osition)f(3)-24 4466 y(b)s(elo)m(w.)123 4728 y(F)-8 b(or)31 b(our)i(next)g(result,)g (let)f(us)h(de\014ne)1061 4999 y Fj(D)s Fm(\()p Fk(\026)p Fm(\))55 b(=)g(sup)1521 5068 y Fn(\017)1653 4999 y Fm(lim)17 b(sup)1701 5078 y Fn(N)7 b Fi(!1)1979 4932 y Fm(log)2105 4955 y Fr(2)2144 4932 y Fm(\()p Fj(N)15 b Fm(\()p Fk(\026;)i(N)5 b(;)17 b(\017)p Fm(\)\))p 1979 4976 683 4 v 2276 5067 a Fk(N)3582 4999 y Fm(\(14\))-24 5218 y(where,)32 b(for)d(giv)m(en)h Fk(N)40 b Fm(and)30 b Fk(\017)p Fm(,)h(w)m(e)f(de\014ne)h Fj(N)15 b Fm(\()p Fk(\026;)i(N)5 b(;)17 b(\017)p Fm(\))29 b(as)h(the)g(minimal)25 b(n)m(um)m(b)s(er)30 b(of)g(dy)m(adic)g(in)m (terv)-5 b(als)-24 5338 y(of)42 b(generation)f Fk(N)52 b Fm(whic)m(h)43 b(supp)s(ort)f(more)f(than)h(1)28 b Fj(\000)h Fk(\017)42 b Fm(of)g(the)g(measure)g Fk(\026)p Fm(.)71 b(It)42 b(is)f(sho)m(wn)i(in)e(the)-24 5459 y(app)s(endix)30 b(that)g Fj(D)s Fm(\()p Fk(\026)p Fm(\))f(is)g(bigger)g(than)g(or)h (equal)f(to)h(the)g(b)s(o)m(x-coun)m(ting)f(information)e(dimension)h (of)-24 5579 y Fk(\026)33 b Fm(and)f(smaller)f(than)h(or)h(equal)f(to)g (its)g(pac)m(king)h(dimension.)1842 5895 y(7)p eop %%Page: 8 8 8 7 bop -24 383 a Fd(Prop)s(osition)36 b(3)49 b Fk(\014)748 342 y Fr(+)742 405 y(0)834 383 y Fj(\025)28 b(D)s Fm(\()p Fk(\026)p Fm(\))p Ff(.)-24 659 y Fd(Pro)s(of.)43 b Fm(It)31 b(is)f(enough)h(to)f(sho)m(w)i(that,)f(if)f Fk(\013)h Fm(is)f(less)h(than)g(the)g(expression)h(on)e(the)h(rhs)h(of)e(\(14\),) g(then)-24 779 y(lim)17 b(sup)e Fk(\026)p Fm(\()p Fk(A)462 794 y Fn(N)s(;\013)590 779 y Fm(\))28 b Fk(>)f Fm(0.)43 b(Let)33 b(us)g(c)m(ho)s(ose)h Fk(\017)f Fm(and)f(a)h(sequence)i Fk(N)2313 794 y Fn(k)2388 779 y Fm(so)e(that)1222 1050 y Fk(\013)56 b(<)77 b Fm(lim)1471 1110 y Fn(k)r Fi(!1)1677 983 y Fm(log)1803 1006 y Fr(2)1843 983 y Fm(\()p Fj(N)15 b Fm(\()p Fk(\026;)i(N)2197 998 y Fn(k)2238 983 y Fk(;)g(\017)p Fm(\)\))p 1677 1027 720 4 v 1977 1118 a Fk(N)2055 1133 y Fn(k)2440 1050 y Fm(.)1115 b(\(15\))-24 1270 y(If)24 b(lim)14 b Fk(\026)p Fm(\()p Fk(A)387 1285 y Fn(N)s(;\013)515 1270 y Fm(\))28 b(=)f(0,)f(then)e(the)g(sequence)i(of)e(c)m (haracteristic)f(functions)h Fk(\037)2719 1285 y Fn(A)2772 1296 y Fc(N)2820 1314 y(k)2858 1296 y(;\013)2949 1270 y Fm(con)m(v)m(erges)i(to)d(0)h(in)f Fk(\026)p Fm(-)-24 1391 y(measure.)42 b(So)25 b(there)h(is)e(a)h(sequence)j Fk(N)1401 1406 y Fn(k)1438 1416 y Fc(j)1500 1391 y Fm(suc)m(h)f(that)e Fk(\037)1978 1406 y Fn(A)2031 1417 y Fc(N)2079 1434 y(k)2112 1450 y(j)2148 1417 y(;\013)2241 1391 y Fm(con)m(v)m(erges)i(to)e(0)g Fk(\026)p Fm(-almost)e(ev)m(erywhere.)-24 1543 y(W)-8 b(e)29 b(can)f(then)g(\014nd)g(a)g(compact)f Fk(K)7 b Fm(,)29 b(with)f Fk(\026)p Fm(\()p Fk(K)7 b Fm(\))27 b Fk(>)h Fm(1)12 b Fj(\000)g Fk(\017)p Fm(,)30 b(so)e(that)g Fk(\037)2532 1558 y Fn(A)2585 1569 y Fc(N)2633 1586 y(k)2666 1602 y(j)2702 1569 y(;\013)2798 1543 y Fm(con)m(v)m(erges)i(uniformly) 25 b(to)-24 1689 y(0)34 b(in)g Fk(K)7 b Fm(.)48 b(Hence,)36 b Fk(K)41 b Fm(is)34 b(ev)m(en)m(tually)g(a)g(subset)i(of)e(all)e(the)i Fk(A)2226 1653 y Fn(c)2226 1714 y(N)2282 1726 y Fc(k)2315 1741 y(j)2352 1714 y Fn(;\013)2421 1689 y Fm(.)48 b(Th)m(us,)36 b(for)e(all)e(su\016cien)m(tly)j(large)-24 1829 y Fk(j)6 b Fm(,)32 b Fk(K)39 b Fm(has)33 b(a)e(co)m(v)m(ering)i(b)m(y)f(dy)m (adic)h(in)m(terv)-5 b(als)31 b(of)g(the)h Fk(N)2027 1844 y Fn(k)2064 1854 y Fc(j)2101 1829 y Fm(th)g(generation,)f(ev)m (ery)m(one)k(of)c(whic)m(h)i(is)e(not)-24 1965 y(smaller)k(than)i(2)598 1917 y Fi(\000)p Fn(\013N)754 1929 y Fc(k)787 1945 y(j)864 1965 y Fm(in)f Fk(\026)p Fm(-measure.)55 b(There)37 b(cannot)g(b)s(e)g (more)f(than)g(\(1)25 b Fj(\000)g Fk(\017)p Fm(\)2)3068 1917 y Fn(\013N)3169 1929 y Fc(k)3202 1945 y(j)3280 1965 y Fm(in)m(terv)-5 b(als)36 b(in)-24 2085 y(those)e(co)m(v)m(erings.)44 b(Therefore,)34 b Fj(N)15 b Fm(\()p Fk(\026;)i(N)1475 2100 y Fn(k)1512 2110 y Fc(j)1547 2085 y Fk(;)g(\017)p Fm(\))28 b Fj(\024)g Fm(\(1)22 b Fj(\000)h Fk(\017)p Fm(\)2)2136 2037 y Fn(\013N)2237 2049 y Fc(k)2270 2065 y(j)2311 2085 y Fm(,)32 b(whic)m(h)h(con)m(tradicts)g(\(15\).)329 b Fa(2)-24 2361 y Fd(Remark)37 b(2)49 b Fm(An)38 b(explicit)f(alb)s 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b(in)f(the)g(Borel)g(set)i(\001)e Fj(\032)g Fd(R)g Fm(\(see)i(the)f(app)s(endix)f(for)-24 5456 y(the)32 b(de\014nition\))e(is)h Fd(P)p Fm(-almost)d(surely)k(constan)m(t)g(and) f(th)m(us)h(de\014nes)h(the)f(pac)m(king)f(dimension)f(of)g(the)-24 5577 y(lo)s(cal)k(densit)m(y)j(of)e(states)i(in)e(\001.)53 b(It)36 b(is)f(smaller)f(than)i(or)f(equal)h(to)g(the)g(pac)m(king)g (dimension)e(of)h(the)1842 5895 y(8)p eop %%Page: 9 9 9 8 bop -24 383 a Fm(densit)m(y)39 b(of)e(states)i Fj(N)52 b Fm(whic)m(h)38 b(is)f(de\014ned)i(to)e(b)s(e)h(the)g(disorder)f(a)m (v)m(erage)i(of)e(the)h(sp)s(ectral)g(measure)-24 504 y Fk(\026)35 519 y Fn(!)r(;)p Fi(j)p Fr(0)p Fi(i)220 504 y Fm(of)32 b(the)h(state)g Fj(j)p Fm(0)p Fj(i)f Fm(lo)s(calized)e (at)i(the)h(origin,)e(namely)h(w)m(e)h(ha)m(v)m(e)933 744 y(dim)1096 759 y Fr(P)1148 744 y Fm(\()p Fk(\026)1245 759 y Fn(!)1295 744 y Fk(;)17 b Fm(\001\))55 b Fj(\024)h Fm(dim)1809 759 y Fr(P)1861 744 y Fm(\()p Fj(N)15 b Fk(;)i Fm(\001\))32 b(,)227 b Fd(P)p Fm(-a.s.)32 b(.)123 917 y(Next)h(w)m(e)h(de\014ne)g(the)f(time)e(a)m(v)m(eraged)i(momen)m(ts)g (of)f(the)h(p)s(osition)d(op)s(erator)3060 892 y Fk(~)3039 917 y(X)41 b Fm(on)32 b Fk(`)3337 881 y Fr(2)3377 917 y Fm(\()p Fd(Z)3483 881 y Fn(D)3547 917 y Fm(\))g(b)m(y)310 1233 y(^)274 1259 y Fk(M)368 1274 y Fn(q)r(;)p Fr(\001)p Fn(;!)551 1259 y Fm(\()p Fk(T)14 b Fm(\))55 b(=)884 1141 y Fe(Z)967 1168 y Fn(T)930 1330 y Fr(0)1049 1191 y Fk(dt)p 1049 1235 86 4 v 1057 1327 a(T)1144 1259 y Fj(h)p Fm(0)p Fj(j)17 b Fm(\005)1350 1274 y Fn(!)1400 1259 y Fm(\(\001\))g Fj(j)p Fk(e)1647 1217 y Fn({H)1728 1225 y Fc(!)1772 1217 y Fn(t)1823 1233 y Fk(~)1802 1259 y(X)8 b(e)1936 1217 y Fi(\000)p Fn({H)2072 1225 y Fc(!)2117 1217 y Fn(t)2169 1259 y Fj(\000)2289 1233 y Fk(~)2268 1259 y(X)g Fj(j)2385 1217 y Fn(q)2439 1259 y Fm(\005)2512 1274 y Fn(!)2563 1259 y Fm(\(\001\))17 b Fj(j)p Fm(0)p Fj(i)31 b Fm(,)228 b Fk(q)32 b Fj(6)p Fm(=)27 b(0)33 b(,)-24 1489 y(where)h(\005)331 1504 y Fn(!)382 1489 y Fm(\(\001\))e(is)g(the)h(sp)s(ectal)g(pro)5 b(jection)32 b(of)g Fk(H)1821 1504 y Fn(!)1904 1489 y Fm(to)g(\001.)44 b(The)34 b(corresp)s(onding)e(gro)m(wth)h(exp)s(onen)m (ts)-10 1583 y(^)-24 1609 y Fk(\014)37 1573 y Fi(\006)31 1634 y Fn(q)r(;!)135 1609 y Fm(\(\001\))d(are)g(de\014ned)i(as)e(in)f (\(3\).)42 b(Mo)s(di\014cation)29 b(of)g(the)i(pro)s(of)e(of)g(Theorem) i(1)e(\(for)h(\014xed)h Fk(H)3426 1624 y Fn(!)3476 1609 y Fm(\))f(leads)-24 1730 y(to)923 1970 y(dim)1085 1985 y Fr(P)1137 1970 y Fm(\()p Fk(\026)1234 1985 y Fn(!)1285 1970 y Fk(;)17 b Fm(\001\))55 b Fj(\024)h Fk(D)1750 1944 y Fm(^)1736 1970 y Fk(\014)1797 1929 y Fr(+)1791 1995 y Fn(q)r(;!)1895 1970 y Fm(\(\001\))33 b(,)227 b Fj(8)56 b Fk(!)31 b Fj(2)d Fm(\012)33 b(.)-24 2391 y Fl(App)t(endix:)72 b(v)-9 b(arious)54 b(dimensions)e(of)i(Borel)g(measures.)-24 2610 y Fm(The)37 b(aim)d(of)h(this)g(app)s(endix)h(is)f(to)g(review)i (and)e(extend)j(kno)m(wn)f(results)f(ab)s(out)f(lo)m(w)m(er)h(and)f (upp)s(er)-24 2730 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b(are)h(giv)m(en)g(b)m(y)775 5484 y Fk(d)p 775 5497 51 4 v 825 5504 a Fn(\026)872 5484 y Fm(\()p Fk(E)6 b Fm(\))27 b Fk(<)h(\015)88 b Fj(\))p 1479 5406 84 4 v 83 w Fk(D)1562 5425 y Fn(\015)1562 5508 y(\026)1609 5484 y Fm(\()p Fk(E)6 b Fm(\))28 b(=)f Fj(1)83 b(\))f Fk(d)p 2259 5497 51 4 v 20 x Fn(\026)2357 5484 y Fm(\()p Fk(E)6 b Fm(\))27 b Fj(\024)i Fk(\015)37 b Fm(,)823 b(\(16\))815 5641 y Fk(\015)32 b(<)c(d)p 1002 5654 V 21 x Fn(\026)1099 5641 y Fm(\()p Fk(E)6 b Fm(\))83 b Fj(\))p 1519 5563 84 4 v 83 w Fk(D)1602 5583 y Fn(\015)1602 5666 y(\026)1649 5641 y Fm(\()p Fk(E)6 b Fm(\))28 b(=)f(0)83 b Fj(\))g Fk(\015)32 b Fj(\024)c Fk(d)p 2437 5654 51 4 v 21 x Fn(\026)2535 5641 y Fm(\()p Fk(E)6 b Fm(\))32 b(.)834 b(\(17\))1842 5895 y(9)p eop %%Page: 10 10 10 9 bop -24 383 a Fm(Analogous)42 b(relations)g(hold)g(b)s(et)m(w)m (een)j(upp)s(er)e(p)s(oin)m(t)m(wise)f(dimensions)g(and)h(lo)m(w)m(er)g Fk(\015)5 b Fm(-deriv)-5 b(ativ)m(es.)-24 504 y(Next)33 b(w)m(e)g(denote)f(the)g Fk(\015)5 b 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985 y Fj(\024)d Fk(\016)t(=)p Fm(2.)43 b(A)32 b(premeasure)i(is)e (de\014ned)i(b)m(y)464 1288 y Fk(M)568 1241 y Fn(\015)t(;\016)558 1312 y Fr(P)666 1288 y Fm(\(\001\))56 b(=)f(sup)1173 1114 y Fe(8)1173 1188 y(<)1173 1338 y(:)1272 1205 y(X)1257 1390 y Fn(k)r Fi(2)p Fh(N)1406 1288 y Fm(\(2)p Fk(r)1537 1303 y Fn(k)1579 1288 y Fm(\))1617 1247 y Fn(\015)1689 1114 y Fe(\014)1689 1164 y(\014)1689 1213 y(\014)1689 1263 y(\014)1689 1313 y(\014)1689 1363 y(\014)1761 1288 y Fm(\()p Fk(B)5 b Fm(\()p Fk(E)1988 1303 y Fn(k)2031 1288 y Fk(;)17 b(r)2119 1303 y Fn(k)2162 1288 y Fm(\)\))2238 1303 y Fn(k)r Fi(2)p Fh(N)2446 1288 y Fk(\016)t Fm(-pac)m(king)32 b(of)g(\001)3075 1114 y Fe(9)3075 1188 y(=)3075 1338 y(;)3198 1288 y Fm(.)357 b(\(18\))-24 1538 y(The)34 b Fk(\015)5 b Fm(-pac)m(king)32 b(measure)h(is)f(constructed)i(in)e(t)m (w)m(o)h(steps:)1352 1720 y(~)1316 1745 y Fk(M)1420 1698 y Fn(\015)1410 1770 y Fr(P)1465 1745 y Fm(\(\001\))55 b(=)j(lim)1809 1805 y Fn(\016)r Fi(!)p Fr(0)1965 1745 y Fk(M)2069 1698 y Fn(\015)t(;\016)2059 1770 y Fr(P)2167 1745 y Fm(\(\001\))33 b(,)1198 b(\(19\))615 2083 y Fk(M)719 2035 y Fn(\015)709 2107 y Fr(P)765 2083 y Fm(\(\001\))55 b(=)g(inf)1243 1908 y Fe(8)1243 1983 y(<)1243 2132 y(:)1343 2000 y(X)1327 2183 y Fn(n)p Fi(2)p Fh(N)1533 2057 y Fm(~)1497 2083 y Fk(M)1601 2035 y Fn(\015)1591 2107 y Fr(P)1646 2083 y Fm(\(\001)1765 2098 y Fn(n)1812 2083 y Fm(\))1878 1908 y Fe(\014)1878 1958 y(\014)1878 2008 y(\014)1878 2058 y(\014)1878 2107 y(\014)1878 2157 y(\014)1950 2083 y Fm(\001)2031 2098 y Fn(n)2110 2083 y Fm(Borel,)2443 2000 y Fe([)2413 2183 y Fn(n)p Fi(2)p Fh(N)2583 2083 y Fm(\001)2664 2098 y Fn(n)2739 2083 y Fm(=)27 b(\001)2923 1908 y Fe(9)2923 1983 y(=)2923 2132 y(;)3046 2083 y Fm(.)509 b(\(20\))-24 2339 y(The)25 b(pac)m(king)e(dimension)f(dim)1130 2354 y Fr(P)1182 2339 y Fm(\(\001\))i(of)f(\001)g(is)g(de\014ned)h(as)g (the)f(in\014m)m(um)f(of)h(all)e Fk(\015)28 b Fm(suc)m(h)d(that)e Fk(M)3450 2292 y Fn(\015)3440 2364 y Fr(P)3495 2339 y Fm(\(\001\))28 b(=)-24 2460 y(0.)123 2627 y(F)-8 b(or)34 b(the)i(case)g(of)f(the)h(upp)s(er)g Fk(\015)5 b Fm(-deriv)-5 b(ativ)m(e)34 b(and)h(the)h Fk(\015)5 b Fm(-Hausdor\013)35 b(measure)h(\(the)g(inequalities)-24 2748 y(on)46 b(the)h(left)e (hand-side)g(of)h(\(21\))f(and)h(\()g(22\)\),)j(the)d(follo)m(wing)d (theorem)j(summarizes)f(the)i(main)-24 2868 y(tec)m(hnical)34 b(results)g(of)f([Ro)o(,)h(Chapter)h(3.2)e(and)h(3.3].)46 b(F)-8 b(or)33 b(the)h(case)g(of)f(the)h(lo)m(w)m(er)g Fk(\015)5 b Fm(-deriv)-5 b(ativ)m(e)33 b(and)-24 2989 y(the)c Fk(\015)5 b Fm(-pac)m(king)28 b(dimension,)g(it)f(is)g (strictly)h(sp)s(eaking)g(new,)i(but)f(the)f(pro)s(of)f(uses)j(similar) 25 b(tec)m(hniques)-24 3109 y(as)36 b(in)f([Cu)q(].)53 b(Not)36 b(only)f(the)h(results,)h(but)f(also)f(the)h(pro)s(ofs)g(sho)m (w)g(some)g(kind)g(of)f(dualit)m(y)g(b)s(et)m(w)m(een)-24 3229 y(Hausdor\013)27 b(dimensions)f(and)h(lo)m(w)m(er)g(p)s(oin)m(t)m (wise)f(dimensions)g(on)h(one)f(side)h(and)g(pac)m(king)g(dimensions) -24 3350 y(and)34 b(upp)s(er)h(p)s(oin)m(t)m(wise)e(dimensions)g(on)h (the)g(other:)46 b(for)34 b(the)g(Hausdor\013)g(measure)g(case,)h(the)g (pro)s(of)-24 3470 y(of)i(\(21\))f(is)g(based)i(on)f(a)g(co)m(v)m (ering)g(lemma,)f(and)h(\(22\))f(follo)m(ws)g(directly)g(from)f(the)j (de\014nitions;)g(for)-24 3591 y(the)33 b(pac)m(king)g(measure)g(case,) h(the)f(situation)e(is)h(just)h(the)g(con)m(v)m(erse.)-24 3851 y Fd(Theorem)k(2)49 b Ff(F)-7 b(or)34 b Fk(\025)28 b(>)f Fm(0)35 b Ff(we)f(have)117 4080 y Fk(M)221 4033 y Fn(\015)211 4105 y Fr(H)268 4080 y Fm(\()p Fj(f)p Fk(E)f Fj(2)c Fm(\001)f Fj(j)p 720 4002 84 4 v 27 w Fk(D)804 4022 y Fn(\015)804 4105 y(\026)850 4080 y Fm(\()p Fk(E)6 b Fm(\))28 b Fk(>)f(\025)p Fj(g)p Fm(\))55 b Fj(\024)1478 4013 y Fm(6)1527 3977 y Fn(\015)p 1478 4057 94 4 v 1496 4149 a Fk(\025)1616 4080 y Ff(,)234 b Fk(M)1984 4033 y Fn(\015)1974 4105 y Fr(P)2029 4080 y Fm(\()p Fj(f)p Fk(E)34 b Fj(2)28 b Fm(\001)g Fj(j)f Fk(D)p 2481 4093 84 4 v 2565 4039 a Fn(\015)2565 4105 y(\026)2612 4080 y Fm(\()p Fk(E)6 b Fm(\))27 b Fk(>)h(\025)p Fj(g)p Fm(\))55 b Fj(\024)3239 4013 y Fm(2)3288 3977 y Fn(\015)p 3239 4057 94 4 v 3258 4149 a Fk(\025)3378 4080 y Ff(,)174 b Fm(\(21\))-24 4313 y Ff(and)35 4520 y Fk(\026)p Fm(\()p Fj(f)p Fk(E)34 b Fj(2)28 b Fm(\001)17 b Fj(j)p 525 4441 84 4 v 17 w Fk(D)608 4461 y Fn(\015)608 4544 y(\026)654 4520 y Fm(\()p Fk(E)6 b Fm(\))28 b Fk(<)f(\025)p Fj(g)p Fm(\))44 b Fj(\024)h Fm(2)1299 4478 y Fn(\015)1343 4520 y Fk(\025)17 b(M)1521 4472 y Fn(\015)1511 4544 y Fr(H)1568 4520 y Fm(\(\001\))35 b Ff(,)100 b Fk(\026)p Fm(\()p Fj(f)p Fk(E)33 b Fj(2)28 b Fm(\001)17 b Fj(j)g Fk(D)p 2379 4533 V 2462 4478 a Fn(\015)2462 4544 y(\026)2508 4520 y Fm(\()p Fk(E)6 b Fm(\))28 b Fk(<)f(\025)p Fj(g)p Fm(\))44 b Fj(\024)h Fm(2)3153 4478 y Fi(\000)p Fn(\015)3252 4520 y Fk(\025)17 b(M)3430 4472 y Fn(\015)3420 4544 y Fr(P)3475 4520 y Fm(\(\001\))35 b Ff(.)3582 4640 y Fm(\(22\))-24 4900 y Fd(Pro)s(of.)57 b Fm(The)38 b(\014rst)f(inequalities)e(in)i (\(21\))f(and)h(\(22\))f(b)s(eing)h(pro)m(v)m(en)h(in)e([Ro],)i(w)m(e)g (here)g(only)f(pro)m(v)m(e)-24 5021 y(the)49 b(pac)m(king)f(dimension)e (part.)90 b(Let)48 b Fk(R)1570 4985 y Fn(\015)1569 5045 y(\026)1616 5021 y Fm(\()p Fk(\016)n(;)17 b(\025)p Fm(\))54 b(=)g Fj(f)p Fk(E)60 b Fj(2)54 b Fm(\001)17 b Fj(j)33 b Fm(inf)2597 5036 y Fn(\017<\016)2735 5021 y Fk(\026)p Fm(\([)p Fk(E)38 b Fj(\000)33 b Fk(\017;)17 b(E)39 b Fm(+)32 b Fk(\017)p Fm(]\))p Fk(\017)3524 4985 y Fi(\000)p Fn(\015)3678 5021 y Fj(\025)-24 5141 y Fk(\025)p Fj(g)p Fm(.)92 b(Then)50 b(for)f(an)m(y)g Fk(E)62 b Fj(2)56 b Fk(R)1170 5105 y Fn(\015)1169 5166 y(\026)1215 5141 y Fm(\()p Fk(\016)n(;)17 b(\025)p Fm(\),)53 b Fk(\026)p Fm(\([)p Fk(E)39 b Fj(\000)34 b Fk(\017;)17 b(E)39 b Fm(+)34 b Fk(\017)p Fm(]\))55 b Fj(\025)h Fk(\025\017)2551 5105 y Fn(\015)2707 5141 y Fj(8)g Fk(\017)f(<)h(\016)t Fm(.)92 b(Therefore,)54 b(if)-24 5261 y(\()p Fk(B)5 b Fm(\()p Fk(E)203 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Fm(\(\001\))p Fk(=\025)p Fm(.)-24 5521 y(Moreo)m(v)m(er,)441 5455 y Fe(S)511 5542 y Fn(n)p Fi(2)p Fh(N)685 5521 y Fk(R)760 5485 y Fn(\015)759 5546 y(\026)806 5521 y Fm(\(1)p Fk(=n;)17 b(\025)p Fm(\))40 b(=)h Fj(f)p Fk(E)47 b Fj(2)41 b Fm(\001)h Fj(j)f Fk(D)p 1764 5534 V 1847 5479 a Fn(\015)1847 5546 y(\026)1894 5521 y Fm(\()p Fk(E)6 b Fm(\))41 b Fk(>)g(\025)p Fj(g)f Fm(so)h(that)f Fk(M)2804 5474 y Fn(\015)2794 5545 y Fr(P)2849 5521 y Fm(\()p Fj(f)p Fk(E)47 b Fj(2)42 b Fm(\001)f Fj(j)g Fk(D)p 3355 5534 V 3438 5479 a Fn(\015)3438 5546 y(\026)3485 5521 y Fm(\()p Fk(E)6 b Fm(\))41 b Fk(>)-24 5641 y(\025)p Fj(g)p Fm(\))28 b(=)f(sup)399 5665 y Fn(n)463 5641 y Fk(M)567 5594 y Fn(\015)557 5666 y Fr(P)612 5641 y Fm(\()p Fk(R)725 5605 y Fn(\015)724 5666 y(\026)771 5641 y Fm(\(1)p Fk(=n;)17 b(\025)p Fm(\)\))27 b Fj(\024)h Fm(2)1323 5605 y Fn(\015)1367 5641 y Fk(\026)p Fm(\(\001\))p Fk(=\025)k Fm(due)h(to)f(the)h Fk(\033)t Fm(-additivit)m(y)e(of)h Fk(M)2946 5594 y Fn(\015)2936 5666 y Fr(P)2991 5641 y Fm(.)1817 5895 y(10)p eop %%Page: 11 11 11 10 bop 123 383 a Fm(T)-8 b(o)26 b(pro)m(v)m(e)i(\(22\),)g(w)m(e)g (shall)d(use)j(\(as)e(in)g([Cu)q(]\))h(the)g(fact)f(that)h(an)m(y)g (Borel)f(measure)h(on)g Fd(R)f Fm(p)s(ossesses)-24 504 y(the)42 b(cen)m(tered)h(Vitali)c(co)m(v)m(ering)j(prop)s(ert)m(y)g ([Be].)70 b(A)41 b(cen)m(tered)j(Vitali)38 b(co)m(v)m(ering)k(of)f (\001)g(is)g(a)g(set)h(of)-24 624 y(closed)28 b(balls)e(con)m(taining)f (for)i(an)m(y)h Fk(E)34 b Fj(2)28 b Fm(\001)f(and)h Fk(\016)j(>)d Fm(0)f(some)g(closed)g(ball)e Fk(B)5 b Fm(\()p Fk(E)h(;)17 b(r)s Fm(\))27 b(with)g Fk(r)j Fj(\024)e Fk(\016)t Fm(.)42 b(The)-24 744 y(cen)m(tered)37 b(co)m(v)m(ering)e(prop)s(ert)m(y)g(of)f Fk(\026)g Fm(means)g(that)g(ev)m(ery)j(cen)m(tered)f(Vitali)31 b(co)m(v)m(ering)k(of)f(\001)h(con)m(tains)-24 865 y(a)e(coun)m(table)f (set)i(of)e(disjoin)m(t)f(balls)g Fk(B)1412 880 y Fn(k)1488 865 y Fm(suc)m(h)j(that)e Fk(\026)p Fm(\(\001)p Fj(n)2164 798 y Fe(S)2232 885 y Fn(k)2292 865 y Fk(B)2366 880 y Fn(k)2409 865 y Fm(\))27 b(=)h(0.)123 985 y(Let)34 b(then)h(\001)604 1000 y Fn(n)681 985 y Fj(\032)c(f)p Fk(E)36 b Fj(2)31 b Fm(\001)17 b Fj(j)g Fk(D)p 1187 998 84 4 v 1270 943 a Fn(\015)1270 1010 y(\026)1317 985 y Fm(\()p Fk(E)6 b Fm(\))30 b Fk(<)g(\025)p Fj(g)p Fm(.)48 b(F)-8 b(or)33 b(an)m(y)i Fk(E)i Fj(2)30 b Fm(\001)2437 1000 y Fn(n)2519 985 y Fm(and)k Fk(\016)g(>)d Fm(0,)j(there)h(exists)g Fk(r)e(<)d(\016)-24 1105 y Fm(suc)m(h)h(that)d Fk(\026)p Fm(\([)p Fk(E)21 b Fj(\000)14 b Fk(r)m(;)j(E)k Fm(+)14 b Fk(r)s Fm(]\))28 b Fj(\024)g Fk(\025r)1325 1069 y Fn(\015)1369 1105 y Fm(.)42 b(Hence)30 b(the)f(set)h(of)e(balls)g Fk(B)5 b Fm(\()p Fk(E)h(;)17 b(r)s Fm(\))28 b(suc)m(h)i(that)f Fk(r)h Fj(\024)e Fk(\016)t Fm(,)i Fk(E)k Fj(2)28 b Fm(\001)3709 1120 y Fn(n)-24 1226 y Fm(and)34 b(that)e Fk(\026)p Fm(\([)p Fk(E)d Fj(\000)23 b Fk(r)m(;)17 b(E)28 b Fm(+)23 b Fk(r)s Fm(]\))28 b Fj(\024)h Fk(\025r)1337 1190 y Fn(\015)1414 1226 y Fm(is)k(a)g(cen)m(tered)i(Vitali)30 b(co)m(v)m(ering)j(of)g (\001)2832 1241 y Fn(n)2879 1226 y Fm(.)45 b(Let)34 b(\()p Fk(B)5 b Fm(\()p Fk(E)3354 1241 y Fn(k)3397 1226 y Fk(;)17 b(r)3485 1241 y Fn(k)3527 1226 y Fm(\)\))3603 1241 y Fn(k)r Fi(2)p Fh(N)-24 1346 y Fm(b)s(e)36 b(the)g(asso)s(ciated)f Fk(\016)t Fm(-pac)m(king)g(satisfying)g Fk(\026)p Fm(\(\001)1806 1361 y Fn(n)1853 1346 y Fj(n)1920 1280 y Fe(S)1988 1367 y Fn(k)2048 1346 y Fk(B)5 b Fm(\()p Fk(E)2237 1361 y Fn(k)2280 1346 y Fk(;)17 b(r)2368 1361 y Fn(k)2410 1346 y Fm(\)\))33 b(=)f(0)j(as)h(giv)m(en)g(b)m(y)g(the)g(cen)m(tered)-24 1467 y(Vitali)i(co)m(v)m(ering)i(prop)s(ert)m(y)-8 b(.)67 b(Then)41 b Fk(\026)p Fm(\(\001)1538 1482 y Fn(n)1585 1467 y Fm(\))g Fj(\024)1782 1400 y Fe(P)1870 1487 y Fn(k)1929 1467 y Fk(\026)p Fm(\()p Fk(B)5 b Fm(\()p Fk(E)2215 1482 y Fn(k)2257 1467 y Fk(;)17 b(r)2345 1482 y Fn(k)2388 1467 y Fm(\)\))40 b Fj(\024)h Fk(\025)2696 1400 y Fe(P)2783 1487 y Fn(k)2843 1467 y Fk(r)2890 1419 y Fn(\015)2887 1492 y(k)2934 1467 y Fm(.)66 b(As)41 b(this)f(holds)f(for)-24 1587 y(an)m(y)33 b Fk(\016)e(>)d Fm(0,)j(w)m(e)i(ha)m(v)m(e)f Fk(\026)p Fm(\(\001)989 1602 y Fn(n)1036 1587 y Fm(\))c Fj(\024)g Fm(2)1256 1551 y Fi(\000)p Fn(\015)1355 1587 y Fk(\025)1448 1562 y Fm(~)1412 1587 y Fk(M)1516 1551 y Fn(\015)1506 1612 y Fb(P)1561 1587 y Fm(\(\001)1680 1602 y Fn(n)1728 1587 y Fm(\).)43 b(As)32 b(the)f(decomp)s(osition)f (\001)e(=)3000 1521 y Fe(S)3069 1608 y Fn(n)3132 1587 y Fm(\001)3213 1602 y Fn(n)3292 1587 y Fm(in)j(\(20\))g(can)-24 1707 y(b)s(e)i(c)m(hosen)h(disjoin)m(t,)e(the)h(result)f(follo)m(ws.) 2130 b Fa(2)123 1875 y Fm(Before)45 b(dra)m(wing)f(the)h(for)f(us)h(in) m(teresting)f(consequences)k(of)c(Theorem)h(2,)j(let)c(us)h(in)m(tro)s (duce)-24 1995 y(some)j(further)g(notations.)87 b(W)-8 b(e)48 b(de\014ne)h Fk(d)p 1570 2008 51 4 v 1620 1952 a Fr(+)1620 2020 y Fn(\026)1679 1995 y Fm(\(\001\))k(=)g Fk(\026)16 b Fj(\000)g Fm(ess)h(sup)2470 2019 y Fn(E)t Fi(2)p Fr(\001)2652 1995 y Fk(d)p 2652 2008 V 21 x Fn(\026)2749 1995 y Fm(\()p Fk(E)6 b Fm(\))48 b(and)f Fk(d)p 3155 2008 V -43 x Fi(\000)3206 2020 y Fn(\026)3265 1995 y Fm(\(\001\))53 b(=)g Fk(\026)16 b Fj(\000)-24 2141 y Fm(ess)h(inf)231 2156 y Fn(E)t Fi(2)p Fr(\001)413 2141 y Fk(d)p 413 2154 V 21 x Fn(\026)511 2141 y Fm(\()p Fk(E)6 b Fm(\),)28 b(and)g(similarly)p 1299 2062 V 24 w Fk(d)1350 2081 y Fr(+)1350 2165 y Fn(\026)1409 2141 y Fm(\(\001\))g(and)p 1779 2062 V 28 w Fk(d)1830 2081 y Fi(\000)1830 2165 y Fn(\026)1889 2141 y Fm(\(\001\).)42 b(By)28 b(the)g(ab)s(o)m(v)m(e)h (remarks,)g(these)g(quan)m(tities)-24 2261 y(only)j(dep)s(end)i(on)f (the)g(measure)g(class)f(of)g Fk(\026)h Fm([SB].)123 2382 y(The)41 b(upp)s(er)f(Hausdor\013)h(dimension)e(dim)1701 2339 y Fr(+)1701 2406 y(H)1760 2382 y Fm(\()p Fk(\026;)17 b Fm(\001\))40 b(of)f Fk(\026)h Fm(in)g(\001)g(is)g(de\014ned)h(b)m(y)g (the)g(in\014m)m(um)d(of)-24 2502 y(the)c(Hausdor\013)g(dimensions)e (of)h(all)e(Borel)h(subsets)k(\001)2018 2466 y Fi(0)2070 2502 y Fj(\022)29 b Fm(\001)34 b(satisfying)e Fk(\026)p Fm(\(\001)2904 2466 y Fi(0)2927 2502 y Fm(\))d(=)f Fk(\026)p Fm(\(\001\))34 b([Y)-8 b(o)o(].)46 b(The)-24 2622 y(lo)m(w)m(er)38 b(Hausdor\013)f(dimension)f(dim)1314 2579 y Fi(\000)1314 2647 y Fr(H)1373 2622 y Fm(\()p Fk(\026;)17 b Fm(\001\))37 b(of)g Fk(\026)g Fm(in)f(\001)i(is)f(de\014ned)h(as)g(the)g(suprem)m (um)f(of)g(the)h Fk(\013)q Fm('s)-24 2743 y(suc)m(h)e(that)f(\001)493 2707 y Fi(0)548 2743 y Fj(\032)c Fm(\001,)36 b(dim)964 2758 y Fn(H)1030 2743 y Fm(\(\001)1149 2707 y Fi(0)1172 2743 y Fm(\))31 b Fj(\024)h Fk(\013)j Fm(imply)e Fk(\026)p Fm(\(\001)1901 2707 y Fi(0)1924 2743 y Fm(\))e(=)g(0)p Fk(:)k Fm(The)g(pac)m(king)g(dimensions)f(dim)3437 2700 y Fi(\006)3437 2767 y Fr(P)3496 2743 y Fm(\()p Fk(\026;)17 b Fm(\001\))-24 2863 y(are)33 b(once)g(more)f(de\014ned)i(similarly)-8 b(.)40 b(If)32 b(\001)c(=)g Fd(R)p Fm(,)k(w)m(e)i(further)e(drop)h(the) g(sp)s(eci\014cation.)123 2984 y(F)-8 b(or)35 b(the)i(case)g(of)f(the)g (Hausdor\013)h(dimension)e(and)h(lo)m(w)m(er)h(p)s(oin)m(t)m(wise)f (dimensions,)g(the)h(\014rst)f(of)-24 3104 y(the)h(follo)m(wing)d (corollaries)g(already)i(app)s(ears)h(in)e([Ro].)55 b(A)37 b(v)m(ersion)g(of)f(Corollary)e(2)i(can)h(b)s(e)g(found)-24 3224 y(in)e([Cu)q(].)51 b(Corollaries)33 b(4)i(and)h(5)f(app)s(ear)g (in)g([G2)o(,)g(Co)q(])g(and)g(in)g([G2)o(],)i(resp)s(ectiv)m(ely)-8 b(,)37 b(for)d(the)i(case)g(of)-24 3345 y(the)d(Hausdor\013)g (dimension;)f(here)h(w)m(e)h(giv)m(e)e(a)g(di\013eren)m(t)h(pro)s(of.) -24 3571 y Fd(Corollary)j(1)49 b Fm(dim)742 3586 y Fr(H)799 3571 y Fm(\(\001\))27 b Fk(<)h(d)p 1087 3584 V -43 x Fi(\000)1138 3595 y Fn(\026)1197 3571 y Fm(\(\001\))34 b Ff(implies)g Fk(\026)p Fm(\(\001\))27 b(=)h(0)p Ff(.)44 b Fm(dim)2355 3586 y Fr(P)2407 3571 y Fm(\(\001\))28 b Fk(<)p 2695 3492 V 27 w(d)2746 3511 y Fi(\000)2746 3595 y Fn(\026)2805 3571 y Fm(\(\001\))35 b Ff(implies)e Fk(\026)p Fm(\(\001\))28 b(=)f(0)p Ff(.)-24 3797 y Fd(Corollary)36 b(2)49 b Ff(The)34 b(fol)5 b(lowing)34 b(identities)g(hold:)912 4015 y Fm(dim)1075 4030 y Fr(H)1132 4015 y Fm(\()p Fj(f)p Fk(E)f Fj(2)28 b Fm(\001)g Fj(j)g Fk(d)p 1584 4028 V 1634 4036 a Fn(\026)1681 4015 y Fm(\()p Fk(E)6 b Fm(\))28 b Fj(\024)g Fk(d)p 1968 4028 V 2018 3972 a Fr(+)2018 4039 y Fn(\026)2077 4015 y Fm(\(\001\))p Fj(g)p Fm(\))g(=)g Fk(d)p 2454 4028 V 2504 3972 a Fr(+)2504 4039 y Fn(\026)2563 4015 y Fm(\(\001\))35 b Ff(,)910 4258 y Fm(dim)1072 4273 y Fr(P)1124 4258 y Fm(\()p Fj(f)p Fk(E)f Fj(2)28 b Fm(\001)g Fj(j)p 1576 4179 V 27 w Fk(d)1627 4273 y Fn(\026)1674 4258 y Fm(\()p Fk(E)6 b Fm(\))27 b Fj(\024)p 1960 4179 V 28 w Fk(d)2011 4198 y Fr(+)2011 4283 y Fn(\026)2070 4258 y Fm(\(\001\))p Fj(g)p Fm(\))h(=)p 2446 4179 V 27 w Fk(d)2497 4198 y Fr(+)2497 4283 y Fn(\026)2556 4258 y Fm(\(\001\))35 b Ff(.)-24 4484 y Fd(Corollary)h(3)-24 4604 y Fm(i\))e Ff(L)-5 b(et)35 b Fm(\001)p 244 4617 82 4 v 326 4563 a Fr(+)412 4604 y Fm(=)28 b Fj(f)p Fk(E)34 b Fj(2)28 b Fm(\001)g Fj(j)f Fk(d)p 930 4617 51 4 v 21 x Fn(\026)1027 4604 y Fm(\()p Fk(E)6 b Fm(\))28 b(=)f Fk(d)p 1312 4617 V -43 x Fr(+)1363 4629 y Fn(\026)1422 4604 y Fm(\(\001\))p Fj(g)p Ff(.)45 b(Either)35 b Fk(\026)p Fm(\(\001)p 2100 4617 82 4 v -41 x Fr(+)2240 4604 y Fm(\))27 b(=)h(0)35 b Ff(or)62 b Fm(dim)2809 4619 y Fr(H)2866 4604 y Fm(\(\001)p 2904 4617 V -41 x Fr(+)3044 4604 y Fm(\))27 b(=)h Fk(d)p 3213 4617 51 4 v -43 x Fr(+)3264 4629 y Fn(\026)3323 4604 y Fm(\(\001\))p Ff(.)-24 4750 y Fm(ii\))33 b Ff(L)-5 b(et)p 272 4672 82 4 v 36 w Fm(\001)353 4691 y Fr(+)440 4750 y Fm(=)27 b Fj(f)p Fk(E)34 b Fj(2)28 b Fm(\001)g Fj(j)p 957 4671 51 4 v 27 w Fk(d)1008 4765 y Fn(\026)1054 4750 y Fm(\()p Fk(E)6 b Fm(\))28 b(=)p 1340 4671 V 28 w Fk(d)1390 4690 y Fr(+)1390 4774 y Fn(\026)1449 4750 y Fm(\(\001\))p Fj(g)p Ff(.)45 b(Either)35 b Fk(\026)p Fm(\()p 2127 4672 82 4 v(\001)2208 4691 y Fr(+)2267 4750 y Fm(\))27 b(=)h(0)35 b Ff(or)62 b Fm(dim)2836 4765 y Fr(P)2888 4750 y Fm(\()p 2926 4672 V(\001)3007 4691 y Fr(+)3066 4750 y Fm(\))28 b(=)p 3235 4671 51 4 v 27 w Fk(d)3286 4690 y Fr(+)3286 4774 y Fn(\026)3345 4750 y Fm(\(\001\))p Ff(.)-24 5001 y Fd(Corollary)36 b(4)49 b Fm(dim)742 4958 y Fr(+)742 5026 y(H)801 5001 y Fm(\()p Fk(\026;)17 b Fm(\001\))27 b(=)g Fk(d)p 1191 5014 V -43 x Fr(+)1242 5026 y Fn(\026)1301 5001 y Fm(\(\001\))35 b Ff(and)62 b Fm(dim)1873 4958 y Fr(+)1873 5026 y(P)1932 5001 y Fm(\()p Fk(\026;)17 b Fm(\001\))27 b(=)p 2323 4922 V 28 w Fk(d)2373 4941 y Fr(+)2373 5026 y Fn(\026)2432 5001 y Fm(\(\001\))p Ff(.)-24 5252 y Fd(Corollary)36 b(5)49 b Fm(dim)742 5209 y Fi(\000)742 5277 y Fr(H)801 5252 y Fm(\()p Fk(\026;)17 b Fm(\001\))27 b(=)g Fk(d)p 1191 5265 V -43 x Fi(\000)1242 5277 y Fn(\026)1301 5252 y Fm(\(\001\))35 b Ff(and)62 b Fm(dim)1873 5209 y Fi(\000)1873 5277 y Fr(P)1932 5252 y Fm(\()p Fk(\026;)17 b Fm(\001\))27 b(=)p 2323 5173 V 28 w Fk(d)2373 5193 y Fi(\000)2373 5277 y Fn(\026)2432 5252 y Fm(\(\001\))p Ff(.)123 5525 y Fm(Due)39 b(to)f(the)h(complete)f(symmetry)i(of)e(the)h(results)g(in) g(Theorem)g(2,)h(it)e(is)g(su\016cien)m(t)i(to)f(pro)m(v)m(e)-24 5646 y(the)33 b(corollaries)e(for)h(the)h(Hausdor\013)g(dimension)e (case.)1817 5895 y(11)p eop %%Page: 12 12 12 11 bop -24 383 a Fd(Pro)s(of)33 b Fm(of)g(Corollary)f(1.)46 b(Let)33 b Fk(\015)39 b Fm(b)s(e)33 b(suc)m(h)i(that)e(dim)1941 398 y Fr(H)1998 383 y Fm(\(\001\))c Fk(<)g(\015)34 b(<)29 b(d)p 2479 396 51 4 v -43 x Fi(\000)2530 408 y Fn(\026)2589 383 y Fm(\(\001\).)46 b(Then)34 b Fk(M)3178 336 y Fn(\015)3168 408 y Fr(H)3226 383 y Fm(\(\001\))29 b(=)g(0)k(and)-24 504 y(b)m(y)h(\(17\))e(and)g(\(22\))g(one)h(has)457 744 y Fk(\026)p Fm(\(\001\))28 b(=)f Fk(\026)p Fm(\()p Fj(f)p Fk(E)33 b Fj(2)c Fm(\001)17 b Fj(j)g Fk(\015)31 b(<)d(d)p 1480 757 V 21 x Fn(\026)1577 744 y Fm(\()p Fk(E)6 b Fm(\))p Fj(g)p Fm(\))28 b Fj(\024)g Fk(\026)p Fm(\()p Fj(f)p Fk(E)33 b Fj(2)28 b Fm(\001)17 b Fj(j)p 2441 666 84 4 v 17 w Fk(D)2524 686 y Fn(\015)2524 769 y(\026)2570 744 y Fm(\()p Fk(E)6 b Fm(\))28 b(=)g(0)p Fj(g)p Fm(\))f(=)g(0)33 b(.)3681 899 y Fa(2)-24 1067 y Fd(Pro)s(of)g Fm(of)f(Corollary)f(2.)43 b(First)32 b(note)h(that,)f(for)g(an)m(y)i Fk(d)27 b Fj(2)h Fd(R)p Fm(,)33 b(the)g(set)g(\001)2686 1082 y Fn(d)2755 1067 y Fm(=)27 b Fj(f)p Fk(E)34 b Fj(2)28 b Fm(\001)g Fj(j)g Fk(d)p 3273 1080 51 4 v 3323 1087 a Fn(\026)3370 1067 y Fm(\()p Fk(E)6 b Fm(\))28 b Fk(<)f(d)p Fj(g)-24 1187 y Fm(has)37 b(a)f(Hausdor\013)g(dimension)e(less)j(or)f (equal)f(than)h Fk(d)p Fm(:)50 b(in)36 b(fact,)h(b)m(y)f(equations)h (\(16\))e(and)h(\(21\))g(one)-24 1307 y(has)k Fk(M)261 1271 y Fn(d)251 1332 y Fb(H)302 1307 y Fm(\(\001)421 1322 y Fn(d)462 1307 y Fm(\))f Fj(\024)h Fk(M)760 1271 y Fn(d)750 1332 y Fb(H)801 1307 y Fm(\()p Fj(f)p Fk(E)45 b Fj(2)40 b Fm(\001)f Fj(j)g Fk(D)1383 1271 y Fn(d)1380 1332 y(\026)1466 1307 y Fm(=)g Fj(1g)p Fm(\))g(=)g(0.)63 b(No)m(w)40 b(for)f Fk(n)h Fj(2)f Fd(N)g Fm(let)g Fk(d)2976 1322 y Fn(n)3062 1307 y Fm(=)g Fk(d)p 3177 1320 V 3227 1265 a Fr(+)3227 1332 y Fn(\026)3286 1307 y Fm(\(\001\))27 b(+)g(1)p Fk(=n)p Fm(.)-24 1436 y(Then)50 b(\001)328 1451 y Fr(0)423 1436 y Fm(=)553 1370 y Fe(T)622 1457 y Fn(n)p Fi(2)p Fh(N)797 1436 y Fm(\001)878 1451 y Fn(d)914 1459 y Fc(n)1016 1436 y Fm(=)k Fj(f)p Fk(E)61 b Fj(2)55 b Fm(\001)p Fj(j)p Fk(d)p 1559 1449 V 21 x Fn(\026)1656 1436 y Fm(\()p Fk(E)6 b Fm(\))55 b Fj(\024)h Fk(d)p 1998 1449 V 2048 1393 a Fr(+)2048 1461 y Fn(\026)2107 1436 y Fm(\(\001\))p Fj(g)49 b Fm(satis\014es)g(according)f(to)g(the)h(ab)s (o)m(v)m(e)-24 1565 y(dim)139 1580 y Fr(H)196 1565 y Fm(\(\001)315 1580 y Fr(0)355 1565 y Fm(\))40 b Fj(\024)g Fk(d)p 550 1578 V -43 x Fr(+)601 1590 y Fn(\026)660 1565 y Fm(\(\001\).)66 b(No)m(w)40 b(supp)s(ose)h(there)g(exists)g(a)e Fk(\015)45 b Fm(suc)m(h)d(that)d(dim)2840 1580 y Fr(H)2897 1565 y Fm(\(\001)3016 1580 y Fr(0)3056 1565 y Fm(\))h Fk(<)g(\015)45 b(<)40 b(d)p 3462 1578 V -43 x Fr(+)3513 1590 y Fn(\026)3572 1565 y Fm(\(\001\).)-24 1686 y(Then)34 b Fk(M)335 1638 y Fn(\015)325 1710 y Fr(H)382 1686 y Fm(\(\001)501 1701 y Fr(0)541 1686 y Fm(\))28 b(=)f(0)33 b(and)f(b)s(ecause)i(of)e Fk(\026)p Fm(\(\001)1631 1701 y Fr(0)1671 1686 y Fm(\))27 b(=)h Fk(\026)p Fm(\(\001\),)k(\(17\))g (and)h(\(22\))f(w)m(e)h(ha)m(v)m(e)10 1860 y Fk(\026)p Fm(\()p Fj(f)p Fk(E)g Fj(2)28 b Fm(\001)17 b Fj(j)g Fk(\015)32 b(<)c(d)p 686 1873 V 736 1880 a Fn(\026)783 1860 y Fm(\()p Fk(E)6 b Fm(\))p Fj(g)p Fm(\))44 b(=)g Fk(\026)p Fm(\()p Fj(f)p Fk(E)33 b Fj(2)28 b Fm(\001)1616 1875 y Fr(0)1673 1860 y Fj(j)17 b Fk(\015)32 b(<)27 b(d)p 1904 1873 V 20 x Fn(\026)2001 1860 y Fm(\()p Fk(E)6 b Fm(\))p Fj(g)p Fm(\))44 b Fj(\024)h Fk(\026)p Fm(\()p Fj(f)p Fk(E)33 b Fj(2)28 b Fm(\001)17 b Fj(j)p 2898 1782 84 4 v 17 w Fk(D)2981 1801 y Fn(\015)2981 1884 y(\026)3028 1860 y Fm(\()p Fk(E)6 b Fm(\))27 b(=)h(0)p Fj(g)p Fm(\))44 b(=)g(0)32 b(,)3582 1980 y(\(23\))-24 2100 y(whic)m(h)h(is)g(in)e(con)m (tradiction)h(to)g Fk(\015)h(<)27 b(d)p 1367 2113 51 4 v -42 x Fr(+)1418 2125 y Fn(\026)1477 2100 y Fm(\(\001\))33 b(.)43 b(Therefore)34 b(dim)2341 2115 y Fr(H)2398 2100 y Fm(\(\001)2517 2115 y Fr(0)2557 2100 y Fm(\))28 b(=)f Fk(d)p 2726 2113 V -42 x Fr(+)2777 2125 y Fn(\026)2836 2100 y Fm(\(\001\).)661 b Fa(2)-24 2281 y Fd(Pro)s(of)33 b Fm(of)f(Corollary)f(3.)44 b(If)32 b Fk(E)i Fj(2)28 b Fm(\001)p 1240 2294 82 4 v 1322 2239 a Fr(+)1381 2281 y Fm(,)k(then)p 1663 2202 84 4 v 34 w Fk(D)1747 2222 y Fn(\015)1747 2305 y(\026)1793 2281 y Fm(\()p Fk(E)6 b Fm(\))28 b(=)g(0)k(for)g(all)f Fk(\015)i(>)27 b(d)p 2632 2294 51 4 v -43 x Fr(+)2683 2305 y Fn(\026)2742 2281 y Fm(\(\001\).)44 b(Hence)34 b(\001)p 3260 2294 82 4 v -42 x Fr(+)3428 2281 y Fj(\032)29 b(f)p Fk(E)k Fj(2)-24 2413 y Fm(\001)p -24 2426 V 58 2372 a Fr(+)117 2413 y Fj(j)p 145 2335 84 4 v Fk(D)228 2355 y Fn(\015)228 2438 y(\026)275 2413 y Fm(\()p Fk(E)6 b Fm(\))38 b Fk(<)g(\025)p Fj(g)g Fm(for)g(an)m(y)h Fk(\025)f Fj(\025)h Fm(0.)61 b(No)m(w)39 b(b)m(y)h(\(22\),)g Fk(\026)p Fm(\(\001)p 2127 2426 82 4 v 2207 2372 a Fr(+)2266 2413 y Fm(\))e Fj(\024)h Fm(2)2507 2377 y Fn(\015)2551 2413 y Fk(\025M)2712 2366 y Fn(\015)2702 2438 y Fr(H)2759 2413 y Fm(\(\001)p 2797 2426 V 2879 2372 a Fr(+)2938 2413 y Fm(\).)61 b(Therefore)40 b(either)-24 2542 y Fk(\026)p Fm(\(\001)p 73 2555 V -42 x Fr(+)213 2542 y Fm(\))28 b(=)f(0)33 b(or)f Fk(M)687 2495 y Fn(\015)677 2567 y Fr(H)734 2542 y Fm(\(\001)p 772 2555 V 854 2500 a Fr(+)913 2542 y Fm(\))27 b(=)h Fj(1)k Fm(whic)m(h)h(b)m(y)h(Corollary)c(2)j(implies)57 b(dim)2662 2557 y Fr(H)2719 2542 y Fm(\(\001)p 2757 2555 V -42 x Fr(+)2897 2542 y Fm(\))28 b(=)f Fk(d)p 3066 2555 51 4 v -43 x Fr(+)3117 2567 y Fn(\026)3176 2542 y Fm(\(\001\).)321 b Fa(2)-24 2718 y Fd(Pro)s(of)43 b Fm(of)g(Corollary)e(4.)74 b(W)-8 b(e)44 b(ha)m(v)m(e)g(dim)1581 2733 y Fr(H)1638 2718 y Fm(\()p Fk(\026;)17 b Fm(\001\))45 b Fj(\024)h Fm(dim)2228 2733 y Fr(H)2285 2718 y Fm(\(\001)2404 2733 y Fr(0)2444 2718 y Fm(\))f(=)h Fk(d)p 2649 2731 V 2699 2675 a Fr(+)2699 2743 y Fn(\026)2758 2718 y Fm(\(\001\))d(with)g(\001) 3272 2733 y Fr(0)3355 2718 y Fm(as)g(ab)s(o)m(v)m(e.)-24 2847 y(Supp)s(ose)37 b(a)e Fk(\015)40 b Fm(exists)c(suc)m(h)h(that)e (dim)1408 2862 y Fr(H)1465 2847 y Fm(\()p Fk(\026;)17 b Fm(\001\))32 b Fk(<)g(\015)38 b(<)32 b(d)p 2062 2860 V -43 x Fr(+)2113 2872 y Fn(\026)2172 2847 y Fm(\(\001\).)51 b(Then)37 b(there)f(exists)g(a)f(set)h(\001)3509 2811 y Fi(0)3565 2847 y Fj(\032)d Fm(\001)-24 2968 y(with)g Fk(M)303 2920 y Fn(\015)293 2992 y Fr(H)350 2968 y Fm(\(\001)469 2931 y Fi(0)492 2968 y Fm(\))28 b(=)g(0)k(and)h Fk(\026)p Fm(\(\001)1111 2931 y Fi(0)1134 2968 y Fm(\))27 b(=)h Fk(\026)p Fm(\(\001\).)43 b(As)33 b(in)f(\(23\),)g(this)g(is)h(imp)s (ossible.)840 b Fa(2)-24 3159 y Fd(Pro)s(of)41 b Fm(of)g(Corollary)e (5.)69 b(Corollary)39 b(1)i(directly)f(implies)f Fk(d)p 2225 3172 V 2275 3116 a Fi(\000)2275 3183 y Fn(\026)2334 3159 y Fm(\(\001\))k Fj(\024)p Fm(dim)2775 3118 y Fi(\000)2775 3183 y Fn(H)2841 3159 y Fm(\()p Fk(\026;)17 b Fm(\001\).)68 b(F)-8 b(urthermore,)-24 3279 y(b)m(y)45 b(the)e(de\014nition)g(of)f (essen)m(tial)i(in\014m)m(um)e(there)i(m)m(ust)f(exist)h(a)f(set)h (\001)2750 3243 y Fi(0)2819 3279 y Fj(\022)j Fm(\001)c(with)g Fk(\026)p Fm(\(\001)3478 3243 y Fi(0)3501 3279 y Fm(\))j Fk(>)g Fm(0)-24 3400 y(suc)m(h)f(that)e Fk(d)p 429 3413 V -43 x Fr(+)480 3424 y Fn(\026)539 3400 y Fm(\(\001)658 3363 y Fi(0)681 3400 y Fm(\))j Fj(\025)g Fk(d)p 888 3413 V -43 x Fi(\000)939 3424 y Fn(\026)998 3400 y Fm(\(\001\).)76 b(Then)44 b(Corollary)d(2)i(implies)e(dim)2562 3415 y Fn(H)2628 3400 y Fm(\(\001)2747 3363 y Fi(0)2770 3400 y Fm(\))46 b Fj(\025)g Fk(d)p 2977 3413 V -43 x Fi(\000)3028 3424 y Fn(\026)3087 3400 y Fm(\(\001\).)76 b(Therefore)-24 3528 y Fk(d)p -24 3541 V -42 x Fi(\000)27 3553 y Fn(\026)86 3528 y Fm(\(\001\))28 b Fj(\025)p Fm(dim)512 3487 y Fi(\000)512 3553 y Fn(H)578 3528 y Fm(\()p Fk(\026;)17 b Fm(\001\))32 b(b)m(y)i(the)f(de\014nition)e(of)h(the)h(latter)f(dimension.)1072 b Fa(2)123 3720 y Fm(A)52 b(measure)g Fk(\026)f Fm(is)g(said)g(to)h(ha) m(v)m(e)h Fg(exact)g(\(Hausdor\013)7 b(\))51 b(dimension)g Fk(d)g Fm(in)g(a)h(Borel)f(set)h(\001)g(if)-24 3840 y(dim)140 3799 y Fi(\000)140 3864 y Fn(H)206 3840 y Fm(\()p Fk(\026;)17 b Fm(\001\))50 b(=dim)756 3799 y Fr(+)756 3864 y Fn(H)822 3840 y Fm(\()p Fk(\026;)17 b Fm(\001\),)49 b(that)d(is,)j(if)c(its)h (lo)m(w)m(er)g(p)s(oin)m(t)m(wise)g(dimension)e(is)i Fk(\026)p Fm(-a.e.)83 b(equal)46 b(to)-24 3960 y Fk(d)31 b Fm(in)g(\001.)43 b(In)31 b(parallel,)e(one)i(can)h(sa)m(y)g(that)f Fk(\026)g Fm(has)g Fg(exact)h(\(pac)m(king\))f(dimension)f Fk(d)3008 3924 y Fi(0)3062 3960 y Fm(in)g(\001)i(if)e(its)g(upp)s(er) -24 4081 y(p)s(oin)m(t)m(wise)h(dimension)f(is)h Fk(\026)g Fm(-a.e.)43 b(equal)31 b(to)g Fk(d)1700 4045 y Fi(0)1754 4081 y Fm(in)g(\001.)43 b(In)32 b(ref.)43 b([G2)o(])32 b(an)f Fg(exactly)h(scaling)e Fm(measure)h Fk(\026)-24 4201 y Fm(w)m(as)h(de\014ned)g(as)f(one)h(for)e(whic)m(h)h(the)g(upp)s (er)h(and)f(lo)m(w)m(er)g(lo)s(cal)d(dimensions)i(ha)m(v)m(e)i(a)f (common)e Fk(\026)p Fm(-a.e.)-24 4322 y(constan)m(t)41 b(v)-5 b(alue.)63 b(F)-8 b(or)39 b(exactly)h(scaling)e(measures)i(the)g (Hausdor\013)g(and)g(the)g(pac)m(king)f(dimension)-24 4442 y(coincide,)32 b(but)h(the)g(con)m(v)m(erse)i(is)d(not)h(true.)123 4610 y(In)41 b(the)g(ph)m(ysical)g(literature,)h(b)s(o)m(x-coun)m(ting) f(dimensions)f(ha)m(v)m(e)i(found)f(wide)g(use)h(b)s(ecause)g(of)-24 4730 y(their)e(easy)h(n)m(umerical)d(implemen)m(tation.)62 b(It)40 b(is)f(in)m(teresting)h(to)f(note)h(that)g(pac)m(king)g (dimensions)-24 4850 y(can)e(also)e(b)s(e)h(computed)h(b)m(y)g(a)f(b)s (o)m(x-coun)m(ting)f(pro)s(cedure.)59 b(First)36 b(w)m(e)i(recall)e (that)h(the)g(upp)s(er)h(and)-24 4971 y(lo)m(w)m(er)33 b(b)s(o)m(x-coun)m(ting)f(dimension)g(of)g(a)g(compact)g(set)h Fk(K)40 b Fm(are)33 b(de\014ned)h(as)297 5295 y(dim)460 5252 y Fr(+)460 5320 y(B)519 5295 y Fm(\()p Fk(K)7 b Fm(\))55 b(=)h(lim)17 b(sup)951 5375 y Fn(\016)r Fi(!)p Fr(0)1197 5228 y Fm(log\()p Fk(N)1439 5243 y Fn(\016)1477 5228 y Fm(\()p Fk(K)7 b Fm(\)\))p 1197 5272 484 4 v 1268 5364 a Fj(\000)17 b Fm(log\()p Fk(\016)t Fm(\))1723 5295 y(,)228 b(dim)2141 5252 y Fi(\000)2141 5320 y Fr(B)2200 5295 y Fm(\()p Fk(K)7 b Fm(\))55 b(=)g(lim)17 b(inf)2618 5355 y Fn(\016)r Fi(!)p Fr(0)2849 5228 y Fm(log\()p Fk(N)3091 5243 y Fn(\016)3129 5228 y Fm(\()p Fk(K)7 b Fm(\)\))p 2849 5272 V 2919 5364 a Fj(\000)17 b Fm(log)q(\()p Fk(\016)t Fm(\))3375 5295 y(,)-24 5525 y(where)41 b Fk(N)343 5540 y Fn(\016)381 5525 y Fm(\()p Fk(K)7 b Fm(\))40 b(denotes)h(the)f (minimal)35 b(n)m(um)m(b)s(er)40 b(of)g(closed)f(in)m(terv)-5 b(als)39 b(of)g(size)h Fk(\016)k Fm(needed)d(to)e(co)m(v)m(er)-24 5646 y Fk(K)44 b Fm(\(or)36 b(equiv)-5 b(alen)m(tly)d(,)37 b(b)m(y)h(the)f(b)s(o)m(x-coun)m(ting)f(theorem)h([F)-8 b(a)o(],)38 b(the)f(n)m(um)m(b)s(er)g(of)f(elemen)m(ts)h(of)g(a)f(grid) 1817 5895 y(12)p eop %%Page: 13 13 13 12 bop -24 383 a Fm(co)m(v)m(er)34 b(of)d(size)i Fk(\016)j Fm(whic)m(h)c(o)m(v)m(erlap)g Fk(K)7 b Fm(\).)44 b(The)33 b(upp)s(er)f(and)h(lo)m(w)m(er)f(fractal)e(dimensions)i(of)f Fk(\026)h Fm(in)f(\001)d Fj(\032)g Fd(R)-24 504 y Fm(are)33 b(no)m(w)g(de\014ned)h(as)f(in)f([G2)o(])h(b)m(y)333 744 y(dim)495 701 y Fi(\006)495 769 y Fr(F)555 744 y Fm(\()p Fk(\026;)17 b Fm(\001\))54 b(=)i(sup)1010 820 y Fn(\021)r(>)p Fr(0)1211 744 y Fm(inf)1181 804 y Fn(K)5 b Fi(\032)p Fr(\001)1375 744 y Fj(f)17 b Fm(dim)1604 701 y Fi(\006)1604 769 y Fr(B)1663 744 y Fm(\()p Fk(K)7 b Fm(\))28 b Fj(j)f Fk(\026)p Fm(\()p Fk(K)7 b Fm(\))28 b Fk(>)f(\026)p Fm(\(\001\))22 b Fj(\000)h Fk(\021)36 b Fm(,)d Fk(K)39 b Fm(compact)17 b Fj(g)32 b Fm(.)-24 1135 y Fd(Theorem)37 b(3)49 b Fm(dim)719 1092 y Fr(+)719 1159 y(H)778 1135 y Fm(\()p Fk(\026;)17 b Fm(\001\))27 b Fj(\024)h Fm(dim)1333 1092 y Fi(\000)1333 1159 y Fr(F)1392 1135 y Fm(\()p Fk(\026;)17 b Fm(\001\))27 b Fj(\024)h Fm(dim)1947 1092 y Fr(+)1947 1159 y(F)2006 1135 y Fm(\()p Fk(\026;)17 b Fm(\001\))27 b(=)h(dim)2559 1092 y Fr(+)2559 1159 y(P)2618 1135 y Fm(\()p Fk(\026;)17 b Fm(\001\))28 b Ff(.)-24 1358 y Fd(Pro)s(of.)43 b Fm(T)-8 b(o)32 b(pro)m(v)m(e)g(the) g(\014rst)g(inequalit)m(y)-8 b(,)31 b(w)m(e)h(replace)f(in)g(the)h (de\014nition)e(of)h(dim)3020 1315 y Fi(\000)3020 1382 y Fr(F)3079 1358 y Fm(\()p Fk(\026;)17 b Fm(\001\))31 b(the)g(v)-5 b(alue)-24 1478 y(of)38 b(dim)255 1435 y Fi(\000)255 1503 y Fr(F)314 1478 y Fm(\()p Fk(K)7 b Fm(\))38 b(b)m(y)h(dim)822 1493 y Fr(H)879 1478 y Fm(\()p Fk(K)7 b Fm(\))38 b(and)g(call)e(the)i(result)1929 1452 y(~)1912 1478 y Fk(d)p Fm(.)59 b(Then)39 b(there)g(exist)f(a)g(sequence)i(\()p Fk(K)3417 1493 y Fn(n)3464 1478 y Fm(\))3502 1493 y Fn(n)p Fi(\025)p Fr(1)3677 1478 y Fm(of)-24 1599 y(compact)d(subsets)h(of)e (\001)h(satisfying)f Fk(\026)p Fm(\()p Fk(K)1564 1614 y Fn(n)1610 1599 y Fm(\))f Fk(>)f(\026)p Fm(\(\001\))25 b Fj(\000)g Fm(1)p Fk(=n)37 b Fm(and)f(lim)2658 1614 y Fn(n)p Fi(!1)2863 1599 y Fm(dim)3025 1614 y Fr(H)3082 1599 y Fm(\()p Fk(K)3203 1614 y Fn(n)3250 1599 y Fm(\))f(=)3450 1572 y(~)3433 1599 y Fk(d)p Fm(.)55 b(No)m(w)-24 1653 y Fe(S)46 1740 y Fn(n)109 1719 y Fk(K)192 1734 y Fn(n)268 1719 y Fm(supp)s(orts)29 b Fk(\026)g Fm(in)f(\001)h(and)f(has)h (Hausdor\013)g(dimension)2242 1693 y(~)2225 1719 y Fk(d)f Fm(suc)m(h)i(that)2745 1693 y(~)2728 1719 y Fk(d)d Fj(\025)h Fm(dim)3074 1676 y Fr(+)3074 1744 y(H)3133 1719 y Fm(\()p Fk(\026;)17 b Fm(\001\).)42 b(On)28 b(the)-24 1839 y(other)33 b(hand,)519 1813 y(~)502 1839 y Fk(d)27 b Fj(\024)i Fm(dim)848 1796 y Fi(\000)848 1864 y Fr(F)907 1839 y Fm(\()p Fk(\026;)17 b Fm(\001\))32 b(b)s(ecause)i(dim)1723 1854 y Fr(H)1780 1839 y Fm(\()p Fk(K)7 b Fm(\))27 b Fj(\024)i Fm(dim)2241 1796 y Fi(\000)2241 1864 y Fr(F)2300 1839 y Fm(\()p Fk(K)7 b Fm(\))33 b([F)-8 b(a)o(].)123 1960 y(Next)33 b(let)g(us)g(recall)f (from)f([G2])i(that)f(dim)1706 1917 y Fr(+)1706 1984 y(F)1765 1960 y Fm(\()p Fk(\026;)17 b Fm(\001\))28 b Fj(\024)g Fm(dim)2321 1917 y Fr(+)2321 1984 y(P)2380 1960 y Fm(\()p Fk(\026;)17 b Fm(\001\),)33 b(so)g(w)m(e)h(only)e(need)i (to)f(pro)m(v)m(e)-24 2080 y(the)26 b(con)m(v)m(erse)h(inequalit)m(y)-8 b(.)40 b(Let)25 b Fk(\021)1212 2095 y Fn(n)1287 2080 y Fj(2)j Fd(R)c Fm(and)h Fk(K)1754 2095 y Fn(n)1829 2080 y Fj(\032)j Fm(\001,)f Fk(n)h Fj(2)g Fd(N)p Fm(,)e(b)s(e)f(suc)m(h)i (that)e Fk(\026)p Fm(\()p Fk(K)3112 2095 y Fn(n)3158 2080 y Fm(\))j Fk(>)f(\026)p Fm(\(\001\))7 b Fj(\000)g Fk(\021)3682 2095 y Fn(n)3729 2080 y Fm(,)-24 2200 y(lim)112 2215 y Fn(n)p Fi(!1)317 2200 y Fk(\021)365 2215 y Fn(n)461 2200 y Fm(=)50 b(0)45 b(and)g(sup)1030 2224 y Fn(n)p Fi(\025)p Fr(1)1184 2200 y Fm(dim)1346 2158 y Fr(+)1346 2225 y(B)1406 2200 y Fm(\()p Fk(K)1527 2215 y Fn(n)1573 2200 y Fm(\))50 b(=)f(dim)1949 2215 y Fr(F)1999 2200 y Fm(\()p Fk(\026;)17 b Fm(\001\).)82 b(W)-8 b(e)45 b(set)i Fk(K)2797 2215 y Fi(1)2921 2200 y Fm(=)3046 2134 y Fe(S)3115 2221 y Fn(n)p Fi(\025)p Fr(1)3269 2200 y Fk(K)3352 2215 y Fn(n)3444 2200 y Fm(so)f(that)-24 2321 y Fk(\026)p Fm(\()p Fk(K)156 2336 y Fi(1)231 2321 y Fm(\))39 b(=)h Fk(\026)p Fm(\(\001\).)65 b(Hence,)43 b(b)m(y)d(de\014nition)f(of)g (dim)1922 2278 y Fr(+)1922 2345 y(P)1981 2321 y Fm(\()p Fk(\026;)17 b Fm(\001\))39 b(and)h(coun)m(table)g(stabilit)m(y)e(of)h (pac)m(king)-24 2441 y(dimensions,)140 2777 y(dim)303 2734 y Fr(+)303 2802 y(P)362 2777 y Fm(\()p Fk(\026;)17 b Fm(\001\))55 b Fj(\024)h Fm(dim)972 2792 y Fr(P)1024 2777 y Fm(\()p Fk(K)1145 2792 y Fi(1)1220 2777 y Fm(\))f(=)h(sup)1452 2855 y Fn(n)p Fi(\025)p Fr(1)1625 2777 y Fm(dim)1787 2792 y Fr(P)1839 2777 y Fm(\()p Fk(K)1960 2792 y Fn(n)2007 2777 y Fm(\))g Fj(\024)f Fm(sup)2240 2855 y Fn(n)p Fi(\025)p Fr(1)2413 2777 y Fm(dim)2576 2734 y Fr(+)2576 2802 y(B)2635 2777 y Fm(\()p Fk(K)2756 2792 y Fn(n)2803 2777 y Fm(\))g(=)g(dim)3190 2792 y Fr(F)3240 2777 y Fm(\()p Fk(\026;)17 b Fm(\001\))32 b(,)-24 3060 y(where)i(w)m(e)g(used)g(the)f(fact)f(that)g(dim)1359 3075 y Fr(P)1411 3060 y Fm(\()p Fk(K)7 b Fm(\))28 b Fj(\024)g Fm(dim)1872 3018 y Fr(+)1872 3085 y(B)1931 3060 y Fm(\()p Fk(K)7 b Fm(\))33 b(for)f(an)m(y)h(set)h Fk(K)g Fj(\032)29 b Fd(R)j Fm([F)-8 b(a)o(].)541 b Fa(2)123 3228 y Fm(W)-8 b(e)42 b(\014nally)f(examine)h(the)g(dimension)f Fj(D)s Fm(\()p Fk(\026)p Fm(\))g(in)m(tro)s(duced)h(in)g(\(14\))f(as)h(w)m (ell)g(as)g(b)s(o)m(x-coun)m(ting)-24 3348 y(information)h(dimensions)i (de\014ned)i(next.)83 b(Giv)m(en)46 b Fk(N)60 b Fj(2)51 b Fd(N)p Fm(,)d(the)f(Shannon)f(en)m(trop)m(y)h(\(\\missing)-24 3469 y(information"\))25 b(of)k(the)g(measure)f Fk(\026)h Fm(relativ)m(e)e(to)i(the)g(partition)d(of)i(the)h(real)f(line)f(in)h (dy)m(adic)h(in)m(terv)-5 b(als)-24 3589 y Fk(I)27 3553 y Fn(N)19 3614 y(j)94 3589 y Fm(,)33 b Fk(j)h Fj(2)28 b Fd(Z)p Fm(\),)33 b(of)f(the)h Fk(N)10 b Fm(th)33 b(generation)f(is)g (giv)m(en)h(b)m(y:)1094 3830 y Fk(S)1154 3845 y Fn(N)1222 3830 y Fm(\()p Fk(\026)p Fm(\))55 b(=)g Fj(\000)1641 3747 y Fe(X)1637 3931 y Fn(j)t Fi(2)p Fh(Z)1783 3830 y Fk(\026)p Fm(\()p Fk(I)1931 3789 y Fn(N)1923 3855 y(j)1997 3830 y Fm(\))17 b(log)2178 3854 y Fr(2)2217 3830 y Fm(\()p Fk(\026)p Fm(\()p Fk(I)2403 3789 y Fn(N)2395 3855 y(j)2470 3830 y Fm(\)\))32 b(,)977 b(\(24\))-24 4094 y(where)42 b(b)m(y)e(con)m(v)m(en)m(tion)h(those)g Fk(j)6 b Fm('s)40 b(for)g(whic)m(h)g Fk(\026)p Fm(\()p Fk(I)1908 4058 y Fn(N)1900 4119 y(j)1975 4094 y Fm(\))g(=)g(0)g(do)g(not)f(con)m (tribute)i(in)e(the)h(sum.)65 b(W)-8 b(e)-24 4214 y(consider)33 b(the)g(class)g Fj(E)41 b Fm(of)32 b(measures)h Fk(\026)g Fm(for)f(whic)m(h)h Fk(S)1956 4229 y Fr(1)1995 4214 y Fm(\()p Fk(\026)p Fm(\))f(is)g(\014nite.)43 b(Then)34 b Fk(S)2862 4229 y Fn(N)2929 4214 y Fm(\()p Fk(\026)p Fm(\))e(is)g(also)g(\014nite)g(for)-24 4335 y(all)f Fk(N)43 b Fm(\(see)34 b(b)s(elo)m(w\))e(and)h(the)g(upp)s(er)g(and)f(lo)m(w)m (er)h(information)c(dimensions)j(of)g Fk(\026)g Fm(are)h(de\014ned)h(b) m(y)545 4605 y(dim)708 4562 y Fr(+)708 4630 y Fn(I)767 4605 y Fm(\()p Fk(\026)p Fm(\))55 b(=)g(lim)17 b(sup)1136 4684 y Fn(N)7 b Fi(!1)1414 4538 y Fk(S)1474 4553 y Fn(N)1541 4538 y Fm(\()p Fk(\026)p Fm(\))p 1414 4582 262 4 v 1500 4673 a Fk(N)1718 4605 y Fm(,)228 b(dim)2135 4562 y Fi(\000)2135 4630 y Fn(I)2194 4605 y Fm(\()p Fk(\026)p Fm(\))55 b(=)g(lim)17 b(inf)2548 4664 y Fn(N)7 b Fi(!1)2812 4538 y Fk(S)2872 4553 y Fn(N)2939 4538 y Fm(\()p Fk(\026)p Fm(\))p 2812 4582 V 2899 4673 a Fk(N)3116 4605 y Fm(.)-24 4864 y(Then)34 b(w)m(e)g(ha)m(v)m(e)g(the)f(follo)m(wing:)-24 5135 y Fd(Theorem)k(4)49 b Ff(If)35 b Fk(\026)27 b Fj(2)h(E)9 b Ff(,)34 b(then)h Fm(dim)1345 5092 y Fi(\000)1345 5159 y Fn(H)1412 5135 y Fm(\()p Fk(\026)p Fm(\))27 b Fj(\024)h Fm(dim)1842 5092 y Fi(\000)1842 5159 y Fn(I)1901 5135 y Fm(\()p Fk(\026)p Fm(\))34 b Ff(and)h Fm(dim)2422 5092 y Fr(+)2422 5159 y Fn(I)2481 5135 y Fm(\()p Fk(\026)p Fm(\))27 b Fj(\024)i(D)s Fm(\()p Fk(\026)p Fm(\))d Fj(\024)j Fm(dim)3258 5092 y Fr(+)3258 5159 y Fn(P)3317 5135 y Fm(\()p Fk(\026)p Fm(\))p Ff(.)-24 5405 y Fd(Pro)s(of.)44 b Fm(Let)32 b Fk(A)570 5420 y Fn(N)s(;\013)731 5405 y Fm(b)s(e)h(de\014ned)h(as)f(in)f(Prop)s(osition)f(2.)43 b(It)32 b(is)g(immediate)e(that:)744 5646 y(dim)906 5603 y Fi(\000)906 5670 y Fn(I)965 5646 y Fm(\()p Fk(\026)p Fm(\))55 b Fj(\025)h Fk(\013)34 b Fm(lim)17 b(inf)j Fk(\026)p Fm(\()p Fk(A)1841 5661 y Fn(N)s(;\013)1969 5646 y Fm(\))55 b Fj(\025)h Fk(\013)17 b(\026)p Fm(\(lim)g(inf)i Fk(A)2730 5661 y Fn(N)s(;\013)2859 5646 y Fm(\))32 b(,)1817 5895 y(13)p eop %%Page: 14 14 14 13 bop -24 383 a Fm(whic)m(h)35 b(is)e(not)h(less)g(than)g Fk(\013)g Fm(if)f Fk(\013)d(<)f Fm(dim)1488 340 y Fi(\000)1488 408 y Fn(H)1555 383 y Fm(\()p Fk(\026)p Fm(\))34 b(b)s(ecause)h(of)e (\(13\))g(and)h(of)f(Corollary)f(5.)47 b(This)34 b(pro)m(v)m(es)-24 504 y(the)j(\014rst)g(inequalit)m(y)-8 b(.)53 b(T)-8 b(o)36 b(pro)m(v)m(e)i(the)e(second)i(one,)f(let)f Fj(N)15 b Fm(\()p Fk(\026;)i(N)5 b(;)17 b(\017)p Fm(\))35 b(b)s(e)h(as)h(in)e (equation)h(\(14\),)h(and)-24 624 y(let)h Fk(B)197 639 y Fn(N)302 624 y Fj(\032)f Fd(R)h Fm(b)s(e)h(the)f(union)g(of)g (exactly)g Fj(N)15 b Fm(\()p Fk(\026;)i(N)5 b(;)17 b(\017)p Fm(\))37 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(\(24\))h(in)f(three)i(terms)f Fk(S)1595 1388 y Fn(i)1623 1373 y Fm(\()p Fk(\026;)17 b(N)5 b(;)17 b(\017)p Fm(\),)34 b Fk(i)d Fm(=)f(1)p Fk(;)17 b Fm(2)p Fk(;)g Fm(3,)33 b(whic)m(h)i(result)f(of)f(summing)g(o)m(v)m(er)-24 1493 y(di\013eren)m(t)24 b(sets)g Fk(J)587 1508 y Fn(i)638 1493 y Fm(of)f(indices)f Fk(j)6 b Fm(,)25 b(namely)e Fk(J)1532 1508 y Fr(1)1599 1493 y Fm(=)k Fj(f)p Fk(j)34 b Fj(2)28 b Fd(Z)p Fj(j)p Fk(I)2067 1457 y Fn(N)2059 1518 y(j)2162 1493 y Fj(\032)g Fk(K)2357 1457 y Fn(c)2392 1493 y Fj(g)p Fm(,)c Fk(J)2547 1508 y Fr(2)2614 1493 y Fm(=)k Fj(f)p Fk(j)33 b Fj(2)c Fd(Z)p Fj(j)p Fk(I)3083 1457 y Fn(N)3075 1518 y(j)3153 1493 y Fj(\\)s Fk(B)3296 1508 y Fn(N)3366 1493 y Fj(\\)s Fk(K)35 b Fj(6)p Fm(=)27 b Fj(;g)-24 1613 y Fm(and)40 b Fk(J)227 1628 y Fr(3)306 1613 y Fm(=)g Fj(f)p Fk(j)46 b Fj(2)40 b Fd(Z)p Fj(j)p Fk(I)811 1577 y Fn(N)803 1638 y(j)918 1613 y Fj(\032)h Fk(K)34 b Fj(n)27 b Fk(B)1304 1628 y Fn(N)1371 1613 y Fj(g)p Fm(.)65 b(W)-8 b(e)40 b(denote)g Fk(P)2072 1628 y Fn(i)2100 1613 y Fm(,)i Fk(i)e Fm(=)g(1)p Fk(;)17 b Fm(2)p Fk(;)g Fm(3)38 b(,)k(the)e(total)e(measure)i(of)f(the)-24 1734 y(in)m(terv)-5 b(als)25 b(whose)i(lab)s(el)d(b)s(elongs)h(in)g Fk(J)1374 1749 y Fn(i)1402 1734 y Fm(.)41 b(F)-8 b(rom)25 b(w)m(ell-kno)m(wn)g(prop)s(erties)h(of)f(the)h(conditional)d(en)m (trop)m(y)-24 1854 y(it)32 b(follo)m(ws)f(that)521 2095 y Fk(S)581 2110 y Fr(1)621 2095 y Fm(\()p Fk(N)i Fm(+)22 b(1\))55 b Fj(\024)g Fk(S)1202 2110 y Fr(1)1242 2095 y Fm(\()p Fk(N)10 b Fm(\))22 b(+)g Fk(P)1589 2110 y Fr(1)1645 2095 y Fm(log)1771 2118 y Fr(2)1811 2095 y Fm(\(2\))55 b Fj(\024)h Fk(S)2184 2110 y Fr(1)2223 2095 y Fm(\()p Fk(N)10 b Fm(\))23 b(+)f Fk(\017)55 b Fj(\024)h Fm(\()p Fk(N)33 b Fm(+)22 b(1\))p Fk(\017)32 b Fm(.)-24 2269 y(Again)g(from)f(conditioning)f(w)m(e)k(also)e(get)g(that)516 2510 y Fk(S)576 2525 y Fr(2)615 2510 y Fm(\()p Fk(N)10 b Fm(\))56 b Fj(\024)g Fk(P)1031 2525 y Fr(2)1086 2510 y Fm(log)1212 2533 y Fr(2)1252 2510 y Fm(\(#)p Fk(J)1425 2525 y Fr(2)1464 2510 y Fm(\))23 b Fj(\000)f Fk(P)1687 2525 y Fr(2)1743 2510 y Fm(log)1869 2533 y Fr(2)1909 2510 y Fm(\()p Fk(P)2010 2525 y Fr(2)2049 2510 y Fm(\))55 b Fj(\024)h Fm(log)2401 2533 y Fr(2)2440 2510 y Fm(\()p Fj(N)15 b Fm(\()p Fk(\017;)i(\026;)g(N)10 b Fm(\)\))22 b(+)g Fk(c)32 b Fm(,)-24 2684 y(where)i Fk(c)f Fm(is)f(the)h(maxim)m (um)d(of)i Fj(\000)p Fk(x)17 b Fm(log)1441 2708 y Fr(2)1497 2684 y Fk(x)33 b Fm(in)e(\(0)p Fk(;)17 b Fm(1\).)43 b(In)33 b(the)g(same)f(w)m(a)m(y)-8 b(,)342 2945 y Fk(S)402 2960 y Fr(3)441 2945 y Fm(\()p Fk(N)10 b Fm(\))56 b Fj(\024)g Fk(P)857 2960 y Fr(3)913 2945 y Fm(log)1039 2968 y Fr(2)1078 2945 y Fm(\(#)p Fk(J)1251 2960 y Fr(3)1291 2945 y Fm(\))22 b Fj(\000)g Fk(P)1513 2960 y Fr(3)1569 2945 y Fm(log)1695 2968 y Fr(2)1735 2945 y Fm(\()p Fk(P)1836 2960 y Fr(3)1875 2945 y Fm(\))55 b Fj(\024)h Fk(\017)17 b Fm(log)2283 2968 y Fr(2)2322 2945 y Fm(\()p Fk(N)2438 2960 y Fr(0)2478 2945 y Fm(2)2527 2903 y Fn(N)7 b Fr(+1)2684 2945 y Fm(\))22 b(+)g Fk(\017)17 b 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