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Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmmi10) def /version(1.0-prerelease) def /UnderlinePosition -125.0 def /FamilyName(cmmi10) def end readonly def /UniqueID 4100070 def /PaintType 0 def /FontBBox{-34 -250 1047 749} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmmi10 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 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dup begin /UnderlineThickness 86.9 def /Weight(Medium) def /ItalicAngle 0 def /isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 6 def /FullName(cmsy10) def /version(1.0-prerelease) def /UnderlinePosition -480.0 def /FamilyName(cmsy10) def end readonly def /UniqueID 4100063 def /PaintType 0 def /FontBBox{-28 -960 1122 778} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmsy10 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 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@endspecial 1233 4661 a Fl(Figure)27 b(2.)36 b(Illustration)27 b(for)g(Lemma)h(1.)1872 5280 y(5)p eop %%Page: 6 6 6 5 bop 0 85 a Fn(Lemma)40 b(1.)55 b Fg(Let)35 b Fj(y)t(;)17 b(x)33 b Fg(b)s(e)i(p)s(oin)m(ts)g(of)f Fo(I)-13 b(R)1513 37 y Fi(2)1584 85 y Fg(at)34 b(distance)i Fj(R)f Fg(from)g(eac)m(h)g (other.)49 b(Let)35 b Fj(V)56 b Fg(b)s(e)34 b(a)g(V)-8 b(oronoi)0 218 y(diagram)46 b(based)i(up)s(on)e(p)s(oin)m(ts)h Fj(x;)17 b(x)1400 233 y Fi(1)1438 218 y Fj(;)g(:)g(:)g(:)f(;)h(x)1712 233 y Fh(n)1757 218 y Fg(.)85 b(F)-8 b(or)46 b Fj(r)54 b(<)d(R)q(=)p Fo(2)p Fg(,)e(no)e(b)s(oundary)f(segmen)m(t)i(of)e(the)0 351 y(V)-8 b(oronoi)32 b(cell)h(of)g Fj(x)g Fg(in)m(tersects)h Fj(S)1227 366 y Fh(y)r(;r)1352 351 y Fg(when)g(one)f(of)f(the)h Fj(x)2120 366 y Fh(i)2148 351 y Fg('s)g(falls)g(in)g Fj(S)2624 366 y Fh(y)r(;R)p Fe(\000)p Fi(2)p Fh(r)2851 351 y Fg(.)0 862 y Fn(Pr)n(oof.)56 b Fo(Let)40 b Fj(x)636 877 y Fh(i)705 862 y Fo(b)s(e)h(a)f(p)s(oin)m(t)h(in)g Fj(S)1380 877 y Fh(y)r(;R)p Fe(\000)p Fi(2)p Fh(r)1607 862 y Fo(.)68 b(Let)40 b Fj(\016)46 b Fo(=)41 b Ff(k)p Fj(x)2195 877 y Fh(i)2250 862 y Ff(\000)28 b Fj(y)t Ff(k)p Fo(.)67 b(Let)40 b Fj(y)2781 877 y Fh(i)2850 862 y Fo(=)h(\()p Fj(x)28 b Fo(+)g Fj(x)3247 877 y Fh(i)3275 862 y Fo(\))p Fj(=)p Fo(2,)42 b(and)e(let)0 995 y Fj(z)45 1010 y Fh(i)110 995 y Fo(b)s(e)d(the)h(pro)5 b(jection)38 b(of)f Fj(y)j Fo(to)d(the)h(line)f(through)h Fj(y)1999 1010 y Fh(i)2063 995 y Fo(whic)m(h)h(forms)e(the)h(bisector)g(of)f(p)s(oin)m(t)g Fj(x)h Fo(and)0 1127 y Fj(x)55 1142 y Fh(i)83 1127 y Fo(.)51 b(W)-8 b(e)35 b(need)h(to)f(sho)m(w)h(that)e Ff(k)p Fj(z)1235 1142 y Fh(i)1287 1127 y Ff(\000)24 b Fj(y)t Ff(k)31 b(\025)h Fj(r)s Fo(.)50 b(Let)35 b(\001)d(=)g Ff(k)p Fj(z)2247 1142 y Fh(i)2298 1127 y Ff(\000)24 b Fj(y)t Ff(k)p Fo(.)50 b(F)-8 b(rom)34 b(the)i(\014gure)f(ab)s(o)m(v)m (e,)i(let)e Fj(\015)0 1260 y Fo(b)s(e)h(the)h(angle)566 1246 y Fe(6)p 575 1260 35 3 v 619 1260 a Fj(x)674 1275 y Fh(i)702 1260 y Fj(xy)i Fo(in)d(triangle)g Ff(4)p Fo(\()p Fj(x)1504 1275 y Fh(i)1532 1260 y Fj(xy)t Fo(\),)g(and)g Fj(\022)j Fo(b)s(e)e(angle)2412 1246 y Fe(6)p 2421 1260 V 2465 1260 a Fj(y)t(x)2572 1275 y Fh(i)2599 1260 y Fj(x)f Fo(in)g Ff(4)p Fo(\()p Fj(x)2989 1275 y Fh(i)3017 1260 y Fj(xy)t Fo(\).)53 b(Assume)38 b(that)0 1393 y(0)27 b Ff(\024)i Fj(\022)h Ff(\024)f Fj(\031)t(=)p Fo(2,)j(and)g(note)h(the) g(follo)m(wing)g(t)m(w)m(o)g(prop)s(erties.)1366 1582 y(2\001)22 b(+)g Fj(\016)f Fo(cos\()p Fj(\022)s Fo(\))28 b(=)g Fj(R)17 b Fo(cos)q(\()p Fj(\015)5 b Fo(\))1502 1777 y Fj(\016)20 b Fo(sin)q(\()p Fj(\022)s Fo(\))28 b(=)g Fj(R)17 b Fo(sin)q(\()p Fj(\015)5 b Fo(\))0 1949 y(F)-8 b(rom)32 b(this,)h(w)m(e)h(ha)m(v)m(e)g(the)f(follo)m(wing)g (after)f(squaring)i(the)f(ab)s(o)m(v)m(e)g(and)g(summing:)1216 2138 y(4\001)1346 2096 y Fi(2)1405 2138 y Fo(+)22 b(4\001)p Fj(\016)f Fo(cos)q(\()p Fj(\022)s Fo(\))h Ff(\000)g Fj(R)2148 2096 y Fi(2)2208 2138 y Fo(+)g Fj(\016)2353 2096 y Fi(2)2418 2138 y Fo(=)27 b(0)0 2326 y(This)34 b(expression)h(is)e(quadratic)g(in) g(\001.)44 b(A)m(t)32 b(\001)c(=)g(\()p Fj(R)23 b Ff(\000)g Fj(\016)t Fo(\))p Fj(=)p Fo(2,)32 b(w)m(e)h(ha)m(v)m(e)h(the)f(v)-5 b(alue:)1114 2515 y(\()p Fj(R)23 b Ff(\000)g Fj(\016)t Fo(\))1434 2474 y Fi(2)1471 2515 y Fo(+2\()p Fj(R)g Ff(\000)f Fj(\016)t Fo(\))p Fj(\016)f Fo(cos\()p Fj(\022)s Fo(\))i Ff(\000)f Fj(R)2430 2474 y Fi(2)2490 2515 y Fo(+)g Fj(\016)2635 2474 y Fi(2)1574 2673 y Fo(=)28 b Fj(\016)t Fo(\(2)p Fj(R)23 b Ff(\000)f Fo(2)p Fj(\016)t Fo(\)\(cos)q(\()p Fj(\022)s Fo(\))g Ff(\000)g Fo(1\))1574 2830 y Ff(\024)29 b Fo(0)0 3019 y(when)c(0)j Ff(\024)g Fj(\022)j Ff(\024)d Fj(\031)t(=)p Fo(2.)40 b(The)25 b(p)s(olynomial)f(is)g(minimal)h(when)g (\001)j(=)g Ff(\000)p Fj(\016)21 b Fo(cos\()p Fj(\022)s Fo(\))p Fj(=)p Fo(2,)26 b(and)e(tak)m(es)h(the)f(v)-5 b(alue)0 3152 y Fj(\016)47 3116 y Fi(2)101 3152 y Fo(sin)221 3106 y Fi(2)258 3152 y Fo(\()p Fj(\022)s Fo(\))18 b Ff(\000)h Fj(R)571 3116 y Fi(2)639 3152 y Fo(whic)m(h)33 b(is)e(less)h(than)f (zero.)43 b(Therefore,)32 b(\001)c Ff(\025)h Fo(\()p Fj(R)19 b Ff(\000)g Fj(\016)t Fo(\))p Fj(=)p Fo(2)27 b Ff(\025)h Fo(\()p Fj(R)20 b Ff(\000)f Fo(\()p Fj(R)g Ff(\000)g Fo(2)p Fj(r)s Fo(\)\))p Fj(=)p Fo(2)27 b(=)g Fj(r)0 3285 y Fo(when)35 b(0)30 b Ff(\024)h Fj(\022)i Ff(\024)e Fj(\031)t(=)p Fo(2.)47 b(No)m(w)35 b(consider)g(the)g(case)g (when)g Fj(\031)t(=)p Fo(2)29 b Fj(<)h(\022)k Ff(\024)c Fj(\031)t Fo(.)48 b(Let)34 b Fj(c)2929 3300 y Fh(i)2991 3285 y Fo(b)s(e)g(the)h(in)m(tersection)0 3418 y(of)j(the)i(line)f (through)g Fj(x)g Fo(and)g Fj(x)1202 3433 y Fh(i)1269 3418 y Fo(with)g Fj(S)1557 3433 y Fh(y)r(;R)p Fe(\000)p Fi(2)p Fh(r)1784 3418 y Fo(.)62 b(This)40 b(in)m(tersection)h(m)m(ust)f (b)s(e)f(a)g(c)m(hord)g(of)f Fj(S)3558 3433 y Fh(y)r(;R)p Fe(\000)p Fi(2)p Fh(r)0 3551 y Fo(since)e Fj(\022)f(>)c(\031)t(=)p Fo(2.)49 b(Let)35 b Fj(x)893 3566 y Fh(j)963 3551 y Fo(b)s(e)g(an)f (arbitrary)h(p)s(oin)m(t)g(on)g Fj(c)2089 3566 y Fh(i)2151 3551 y Fo(with)2375 3537 y Fe(6)p 2385 3551 V 2428 3551 a Fj(y)t(x)2535 3566 y Fh(j)2570 3551 y Fj(x)c Ff(\024)h Fj(\031)t(=)p Fo(2.)49 b(Suc)m(h)36 b(a)f(p)s(oin)m(t)g(m)m(ust)0 3683 y(exist)f(since)g Fj(c)512 3698 y Fh(i)572 3683 y Fo(is)f(a)f(c)m(hord.)45 b(But)32 b Ff(k)p Fj(z)1346 3698 y Fh(i)1396 3683 y Ff(\000)23 b Fj(y)t Ff(k)j(\025)j(k)p Fj(z)1825 3698 y Fh(j)1882 3683 y Ff(\000)23 b Fj(y)t Ff(k)j(\025)i Fj(r)s Fo(,)33 b(as)f(required.)p 2861 3629 65 4 v 2861 3688 4 59 v 2923 3688 V 2861 3691 65 4 v 0 4141 a Fn(Lemma)38 b(2.)55 b Fg(Let)32 b Fj(x)c Ff(2)g Fj(C)7 b Fg(,)33 b(and)f(let)g Fj(X)1411 4156 y Fi(1)1481 4141 y Fg(b)s(e)g(a)g(random)g(v)-5 b(ariable)33 b(with)g(densit)m(y)h Fj(f)43 b Fg(as)32 b(in)g(Theorem)i(1.)0 4273 y(Then)g(there)f(exist)h(constan)m(ts)g Fj(r)1211 4288 y Fi(0)1276 4273 y Fj(>)27 b Fo(0)33 b Fg(and)f Fj(\015)h(>)27 b Fo(0)33 b Fg(suc)m(h)h(that)1026 4462 y Fo(inf)1018 4522 y Fh(x)p Fe(2)p Fh(C)1168 4462 y Fk(P)p Ff(fk)p Fj(X)1426 4477 y Fi(1)1485 4462 y Ff(\000)22 b Fj(x)p Ff(k)28 b(\024)g 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%%VMusage: 44913 45310 12 dict begin /FontInfo 11 dict dup begin /UnderlineThickness 49.95 def /Weight(Medium) def /ItalicAngle 0 def /isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 50 def /FullName(cmmi10) def /version(1.0-prerelease) def /UnderlinePosition -125.0 def /FamilyName(cmmi10) def end readonly def /UniqueID 4100070 def /PaintType 0 def /FontBBox{-34 -250 1047 749} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmmi10 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 2 /Theta put dup 3 /Lambda put dup 4 /Xi put dup 5 /Pi put dup 6 /Sigma put dup 7 /Upsilon put dup 8 /Phi put dup 9 /Psi put dup 10 /Omega put dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 14 /delta put dup 15 /epsilon1 put dup 16 /zeta put dup 17 /eta put dup 18 /theta put dup 19 /iota put dup 20 /kappa put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 24 /xi put dup 25 /pi put dup 26 /rho put dup 27 /sigma put dup 28 /tau put dup 29 /upsilon put dup 30 /phi put dup 31 /chi put dup 32 /psi put dup 33 /omega put dup 34 /epsilon put dup 35 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dup begin /UnderlineThickness 86.9 def /Weight(Medium) def /ItalicAngle 0 def /isFixedPitch false def /Notice( Paradissa Fonts Collection. Release I.\012 Copyright \(c\) 1993, Basil K. Malyshev. All Rights Reserved.\012 The digitally encoded machine readable outline data for producing\012 the Typefaces of Computer Modern fonts were designed by Donald E. Knuth.) def /underlinePosition -100 def /underlineThickness 6 def /FullName(cmsy10) def /version(1.0-prerelease) def /UnderlinePosition -480.0 def /FamilyName(cmsy10) def end readonly def /UniqueID 4100063 def /PaintType 0 def /FontBBox{-28 -960 1122 778} def /FontMatrix[0.001 0 0 0.001 0 0] def /FontName/cmsy10 def /FontType 1 def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 4 /divide put dup 5 /diamondmath put dup 6 /plusminus put dup 7 /minusplus put dup 8 /circleplus put dup 9 /circleminus put dup 10 /circlemultiply put dup 11 /circledivide put dup 12 /circledot put dup 13 /circlecopyrt put dup 14 /openbullet put dup 15 /bullet put dup 16 /equivasymptotic put dup 17 /equivalence put dup 18 /reflexsubset put dup 19 /reflexsuperset put dup 20 /lessequal put dup 21 /greaterequal put dup 22 /precedesequal put dup 23 /followsequal put dup 24 /similar put dup 25 /approxequal put dup 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setdash 0 setgray stroke } bind def /S { medline [ 2 1 ] 0 setdash 0 setgray stroke } bind def /xmu 50 def /xsigma 6 def /ymu 300 def /ysigma 6 def /pointsize 2 def /Z 20 def %Z /r 3 def /x [ Z 0 R ] def /X1 [ 0 0 R ] def /NR 450 def /START true def /map { /THETA exch def /CT THETA cos def /ST THETA sin def [ Z CT mul r add CT mul 2 mul Z CT mul r add neg ST mul 2 mul R ] } def % maps theta point newpath 0 1 NR { NR div 360 mul 90 add map aload pop START { moveto /START false def } { lineto } ifelse } for closepath gsave 0.9 setgray eofill grestore 0.5 setlinewidth 0 setgray stroke newpath Z r add 0 R moveto Z 0 R r xsigma mul 0 360 arc 0.5 setlinewidth 0 setgray stroke /z 45 map def /zprime 225 map def /z2 [ z 0 get X1 0 get add 2 div z 1 get X1 1 get add 2 div ] def /zprime2 [ zprime 0 get X1 0 get add 2 div zprime 1 get X1 1 get add 2 div ] def /xprime [ Z r 2 sqrt div add r 2 sqrt div neg R ] def /xprime2 [ Z r 2 sqrt div sub r 2 sqrt div R ] def newpath xprime aload pop moveto z2 aload pop lineto [2 2 ] 0 setdash 0 setgray 0.5 setlinewidth stroke newpath xprime2 aload pop moveto zprime2 aload pop lineto [2 2 ] 0 setdash 0 setgray 0.5 setlinewidth stroke newpath X1 aload pop moveto z aload pop lineto [] 0 setdash 0 setgray 0.5 setlinewidth stroke newpath X1 aload pop moveto x aload pop lineto [] 0 setdash 0 setgray 0.5 setlinewidth stroke [ X1 z zprime z2 zprime2 xprime xprime2 x [ 2 Z mul 2 r mul sub 0 R ] [ 2 Z mul 2 r mul add 0 R ] ] { aload pop () whitepoint } forall MATH X1 aload pop moveto -16 0 rmoveto (X) show 0 -3 rmoveto SMALL (1) show MATH x aload pop moveto 5 3 rmoveto MATH (x) show z aload pop moveto -2 -12 rmoveto MATH (z) show zprime aload pop moveto 5 4 rmoveto MATH (z) show ACCENT show xprime aload pop moveto 5 -3 rmoveto MATH (x) show ACCENT show xprime2 aload pop moveto -3 6 rmoveto MATH (x) show DOUBLEACCENT show z2 aload pop moveto 6 0 rmoveto MATH (y) show ACCENT show zprime2 aload pop moveto 6 0 rmoveto MATH (y) show DOUBLEACCENT show Z r add 0 R moveto 7 5 rmoveto MATH (S) show SMALLMATH 0 -3 rmoveto (x) show THIN (,) show SMALLMATH (r) show Z r sub 2 div 0 R moveto 0 3 rmoveto MATH (Z) show 2 Z mul 2 r mul sub 0 R MATH moveto 4 -5 rmoveto THIN (\(2) show MATH (Z) show MINUS show THIN (2) show MATH (r) show THIN (,0\)) show 2 Z mul 2 r mul add 0 R THIN moveto 4 0 rmoveto THIN (\(2) show MATH (Z) show THIN (+2) show MATH (r) show THIN (,0\)) show X1 aload pop moveto 9 -8.5 rmoveto MATH /theta glyphshow showpage %%EndDocument @endspecial 887 4843 a Fl(Figure)27 b(3.)36 b(Illustration)27 b(for)g(part)g(of)h(the)g(pro)r(of)f(of)g(Lemma)h(3.)1872 5280 y(7)p eop %%Page: 8 8 8 7 bop 0 83 a Fn(Pr)n(oof.)56 b Fo(Let)32 b Fj(A)h Fo(denote)g(the)g (ev)m(en)m(t)i(that)d Fj(X)1711 98 y Fi(1)1781 83 y Fo(has)h(a)f(cell)h (b)s(oundary)g(that)g(in)m(tersects)i Fj(S)3352 98 y Fh(x;r)3446 83 y Fo(.)43 b(De\014ne)0 216 y Fj(Z)35 b Fo(=)27 b Ff(k)p Fj(X)336 231 y Fi(1)395 216 y Ff(\000)c Fj(x)p Ff(k)p Fo(.)44 b(W)-8 b(e)33 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b(some)i(univ)m(ersal)g(constan)m(t)g Fj(c)p Fo(.)e(But)h(w)m(e)h(kno)m (w)f(that)g Fk(E)p Fj(N)2302 3254 y Fi(1)2375 3239 y Ff(\024)f Fj(\014)6 b Fo(2)p Fj(\031)t(r)s(n)35 b Fo(=)h Fj(O)s Fo(\(\(log)17 b Fj(n)p Fo(\))3302 3203 y Fi(2)p Fh(=)p Fi(3)3407 3239 y Fj(n)3465 3203 y Fi(1)p Fh(=)p Fi(3)3569 3239 y Fo(\))38 b(if)f Fj(r)0 3372 y Fo(is)d(pic)m(k)m(ed)h (as)f(in)f(part)g(\(A\).)g(Also,)h Fk(E)p Fj(N)1456 3387 y Fi(2)1522 3372 y Fo(=)28 b Fj(O)s Fo(\()1742 3300 y Ff(p)p 1824 3300 V 1824 3372 a Fj(n)q Fo(\))33 b(as)g(that)g(is)h(what) f(is)h(essen)m(tially)i(sho)m(wn)f(in)e(part)0 3505 y(\(A\).)g(Th)m (us,)h(if)e Fj(N)660 3469 y Fe(00)736 3505 y Fo(denotes)i(the)f(n)m(um) m(b)s(er)h(of)e(con)m(v)m(ex)i(h)m(ull)g(p)s(oin)m(ts,)914 3688 y Fk(E)p Fj(N)1076 3647 y Fe(000)p Fi(2)1199 3688 y Ff(\024)28 b Fk(E)p Fo(\()p Fj(N)1504 3647 y Fe(00)1569 3688 y Fo(+)22 b Fj(N)1755 3647 y Fe(000)1817 3688 y Fo(\))1855 3647 y Fi(2)1199 3845 y Ff(\024)28 b Fk(E)17 b Ff(f)o Fj(I)1487 3860 y Fh(A)1537 3844 y Fc(c)1590 3845 y Fo(\()p Fj(N)1706 3860 y Fi(1)1765 3845 y Fo(+)22 b Fj(N)1941 3860 y Fi(2)1979 3845 y Fo(\))2017 3804 y Fi(2)2054 3845 y Ff(g)g Fo(+)g Fk(E)17 b Ff(f)o Fj(n)2422 3804 y Fi(2)2459 3845 y Fj(I)2502 3860 y Fh(A)2556 3845 y Ff(g)1199 4003 y(\024)28 b Fk(E)17 b Ff(f)o Fo(\()p Fj(N)1560 4018 y Fi(1)1619 4003 y Fo(+)22 b Fj(N)1795 4018 y Fi(2)1833 4003 y Fo(\))1871 3962 y Fi(2)1908 4003 y Ff(g)g Fo(+)g Fj(n)2136 3962 y Fi(2)2173 4003 y Fk(P)17 b Ff(f)o Fj(A)p Ff(g)1199 4161 y(\024)28 b Fj(O)s Fo(\()p Fj(n)p Fo(\))22 b(+)g(4)p Fj(n)1743 4120 y Fi(3)1780 4161 y Fj(e)1825 4120 y Fe(\000)p Fi(\()p Fh(n)p Fe(\000)p Fi(1\))p Fh(\013r)2131 4095 y Fb(3)p Fc(=)p Fb(2)2227 4120 y Fh(=)p Fi(2)1366 4319 y Fo(\(b)m(y)34 b(an)e(inequalit)m(y)j (from)d(part)g(\(A\)\))1199 4476 y(=)27 b Fj(O)s Fo(\()p Fj(n)p Fo(\))1366 4634 y(\(if)33 b Fj(r)d Fo(=)e(\()p Fj(c)17 b Fo(log)g Fj(n=n)p Fo(\))2114 4598 y Fi(2)p Fh(=)p Fi(3)2251 4634 y Fo(and)32 b Fj(c)c Fo(=)g(4)p Fj(=\013)q Fo(\))j Fj(:)0 4817 y Fo(This)j(is)f(what)g(w)m(e)g(set)h (out)e(to)g(sho)m(w.)p 1422 4763 65 4 v 1422 4821 4 59 v 1483 4821 V 1422 4824 65 4 v 1851 5280 a Fl(22)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF