(original) (raw)

%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: nix.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: /usr/local/teTeX/bin/dvips nix -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 1999.12.07:1514 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] 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FC(\000)1966 1780 y FA(X)1976 1975 y FB(i<j)2112 0="" 1="" 2="" 3="" 4="" 41="" 55="" 59="" 66="" 71="" 72="" 74="" 94="" 100="" 111="" 112="" 142="" 157="" 220="" 235="" 244="" 269="" 276="" 292="" 311="" 340="" 368="" 379="" 416="" 425="" 453="" 461="" 472="" 485="" 487="" 489="" 497="" 520="" 533="" 545="" 581="" 588="" 606="" 609="" 647="" 685="" 693="" 702="" 717="" 730="" 757="" 774="" 786="" 822="" 847="" 850="" 872="" 877="" 916="" 952="" 967="" 977="" 980="" 1020="" 1051="" 1073="" 1087="" 1088="" 1141="" 1161="" 1193="" 1205="" 1225="" 1228="" 1278="" 1297="" 1372="" 1374="" 1387="" 1393="" 1395="" 1428="" 1433="" 1451="" 1465="" 1469="" 1484="" 1488="" 1500="" 1501="" 1567="" 1585="" 1586="" 1602="" 1666="" 1682="" 1688="" 1705="" 1721="" 1740="" 1804="" 1808="" 1819="" 1823="" 1866="" 1876="" 1880="" 1888="" 1921="" 1925="" 1940="" 1949="" 1956="" 1978="" 1996="" 2045="" 2084="" 2099="" 2105="" 2116="" 2117="" 2158="" 2172="" 2176="" 2211="" 2217="" 2219="" 2230="" 2237="" 2244="" 2256="" 2257="" 2294="" 2298="" 2313="" 2329="" 2343="" 2357="" 2392="" 2431="" 2456="" 2470="" 2472="" 2477="" 2487="" 2490="" 2497="" 2518="" 2546="" 2550="" 2558="" 2565="" 2568="" 2582="" 2591="" 2598="" 2645="" 2681="" 2683="" 2687="" 2706="" 2707="" 2718="" 2719="" 2726="" 2741="" 2742="" 2767="" 2777="" 2813="" 2838="" 2839="" 2846="" 2855="" 2877="" 2914="" 2947="" 2950="" 2968="" 2983="" 2994="" 3017="" 3066="" 3081="" 3086="" 3104="" 3194="" 3217="" 3253="" 3268="" 3284="" 3286="" 3307="" 3367="" 3373="" 3408="" 3420="" 3423="" 3456="" 3468="" 3469="" 3512="" 3533="" 3549="" 3577="" 3585="" 3630="" 3646="" 3653="" 3687="" 3697="" 3721="" 3723="" 3738="" 3759="" 3817="" 3832="" 3871="" 3893="" 3921="" 3938="" 3946="" 3953="" 3984="" 4032="" 4055="" 4058="" 4069="" 4071="" 4094="" 4097="" 4105="" 4129="" 4146="" 4161="" 4178="" 4193="" 4210="" 4286="" 4299="" 4323="" 4362="" 4363="" 4370="" 4396="" 4399="" 4403="" 4407="" 4414="" 4436="" 4467="" 4527="" 4549="" 4552="" 4567="" 4569="" 4647="" 4662="" 4689="" 4768="" 4775="" 4805="" 4810="" 4820="" 4861="" 4888="" 4922="" 4952="" 4975="" 5009="" 5052="" 5070="" 5096="" 5100="" 5349="" y="" fd(p)13="" b(r)s="" fe(\()p="" fd(a)2330="" fb(i)2379="" fc(\\)20="" b="" fd(a)2528="" fb(j)2564="" fe(\))h(+)f="" fd(:::)244="" fe(assume)j(no)m(w,)i(that)f(w)m(e)f(are)h(giv)m(en)f="" (the)h(probabilities)c(of)j(all)g(the)g="" fd(j)5="" fe(-wise)23="" b(in)m(tersections)g(of)h(the)244="" y(ev)m(en)m(ts,)j(for)d="" fd(j)31="" fe(=")25" b(1)p="" fd(;)15="" b(:::;)g(k)s="" fe(.)41="" b(ho)m(w)25="" b(w)m(ell)e(can)i(w)m(e)f(estimate)h(the)g(probabilit)m(y)d="" (of)i(the)h(union)d="" fc([)p="" fd(a)3411="" fb(i)3463="" fe(?)244="" y(linial)35="" b(and)i(nisan)g([10)q(])h(follo)m(w)m="" (ed)f(b)m(y)h(kahn,)h(linial)c(and)i(samoro)s(dnitsky)f([8])i(dev)m="" (elop)g(a)244="" y(metho)s(d)h(to)h(appro)m(ximate)f="" fd(a)1517="" fb(i)1545="" fe(\))40="" b(from)f(the)g(ab)s(o)m(v)m(e)i(data)f(with)e(an)h="" (additiv)m(e)g(error)g(of)244="" fd(exp)399="" fa(\020)453="" fc(\000)534="" fe(~)524="" y(\012)605="" fa(\000)647="" fd(k)697="" fz(2)736="" fd(="n)836" fa(\001)878="" y(\021)968="" fe(and)c(sho)m(w)g(this)g(error)g(to)i(b)s(e)e(optimal)g(up)f="" (to)j(the)f(logarithmic)e(factor)244="" y(implicit)27="" b(in)693="" y(~)683="" y(\012)j(notation.)380="" y(the)h(inclusion-exclusion)c(form)m(ula)j(is)g(the)h(m\177)-45="" b(obius)30="" b(in)m(v)m(ersion)g(form)m(ula)g(for)g(the)i(lattice)244="" y(of)37="" b(subsets)f(of)h(a)h(\014nite)e(set.)61="" b(one)36="" b(can)i(also)f(ask)g(this)f(question)g(in)g(other)h(settings)g="" (-)g(in)f(a)244="" y(recen)m(t)41="" b(pap)s(er)d(ma)m(vronicolas)i([12)="" q(])g(deals)f(with)g(the)g(question)g(of)h(\\appro)m(ximate)g(m\177)-45="" b(obius)244="" y(in)m(v)m(ersion)30="" b(form)m(ula")h(for)g(the)g="" (lattice)h(of)f(subspaces)g(of)g(a)h(\014nite)e(dimensional)f(v)m="" (ector)j(space)244="" y(o)m(v)m(er)g(a)e(\014nite)g(\014eld.)380="" y(one)37="" b(can)g(try)g(to)h(\014gure)e(out)i(what)f(\\appro)m="" (ximate)g(m\177)-45="" b(obius)36="" b(in)m(v)m(ersion")g(migh)m(t)h(mean)244="" y(for)j(a)g(general)h(p)s(oset.)70="" b(it)40="" b(is)g(easy)g(to)h="" (generalize)g(the)f(question)f(to)i(the)g(class)f(of)g(graded)244="" y(p)s(osets)26="" b(and)g(to)h(sho)m(w,)g(that)g(the)f(b)s(est)g(p)s="" (ossible)e(appro)m(ximation)h(for)h(suc)m(h)g(a)h(p)s(oset)f="" fd(p)40="" fe(equals)244="" y(the)28="" b(distance)g(\(in)f(the)i(suprem)="" m(um)d(norm\))i(b)s(et)m(w)m(een)h(a)g(c)m(haracteristic)g(function)d="" (of)j(the)f(zero)244="" y(elemen)m(t)d(of)g="" fd(p)38="" fe(and)24="" b(a)h(subspace)g(spanned)e(b)m(y)i(c)m(haracteristic)g="" (functions)f(of)h(a)g(certain)g(family)244="" y(of)d(up)m(w)m="" (ards-monotone)h(subsets)f(of)g="" fe(.)38="" b(\(in)22="" b(the)h(sp)s(ecial)d(cases)k(of)e(the)h(lattice)g(of)f(subsets)g(and)="" y(the)k(lattice)h(of)f(subspaces)g(this)f(reduces)g(to)i(the)g="" (results)d(obtained)i(in)f(the)h(ab)s(o)m(v)m(e)i(men)m(tioned)244="" y(pap)s(ers.\))380="" y(f)-8="" b(or)35="" b(the)f(lattice)g(of)g="" (subsets,)g(w)m(e)g(impro)m(v)m(e)f(the)h(additiv)m(e)f(error)g(of)h="" (the)g(appro)m(ximation)244="" y(to)26="" fd(o)s="" fe(\()457="" fc(p)p="" v="" x="" fd(n)o="" fe(\))9="" fc(\001)g="" fe(2)710="" fy(\000)p="" fb(n)814="" fa(\000)965="" fb(n)855="" fz(\()p="" fb(n)p="" fb(k)r="" fz(\))p="" fb(=")p" fz(2)1117="" fa(\001)1159="" fe(,)26="" b(whic)m(h)e(is)g(optimal)f(up)h(to)i(a)f(m)m(ultiplicativ)m(e)e="" (factor)j(of)f="" fe(\()3208="" fd(n)p="" fe(\).)39="" b(a)244="" y(k)m(ey)29="" b(role)e(is)g(pla)m(y)="" m(ed)h(b)m(y)g(three)g(families)e(of)i(orthogonal)h(p)s(olynomials:)36="" b(discrete)29="" b(cheb)m(yshev)244="" y(p)s(olynomials,)f(hahn)i(p)s="" (olynomials)d(and)j(kra)m(w)m(c)m(houk)h(p)s(olynomials.)380="" y(it)c(turns)f(out)h(that)g(this)f(result)g(has)g(some)i(simple)d="" (but)h(somewhat)h(unexp)s(ected)f(connec-)244="" y(tions)k(with)f(co)="" s(ding)g(and)h(graph)g(theory)-8="" b(.)p="" a="" fx(\003)149="" fw(hebrew)28="" b(univ)n(ersit)n(y)e(and)i="" (dima)n(cs)g(cen)n(ter,)f(rutgers)g(univ)n(ersit)n(y)1851="" fv(1)p="" eop="" %%page:="" bop="" fu(1)161="" b(in)l(tro)t(duction)0="" fv(this)26="" b(pap)s(er)g(returns)g(once)h(again)d(to)h(the)i(question)f="" (of)f(appro)m(ximated)g(inclusion-exclusion,)g(raised)0="" y(in)37="" b([10])h(and)h(addressed)h(also)d(in)h(subsequen)m(t)j([8])="" d(and)g([12].)60="" b(w)-8="" b(e)39="" b(refer)g(to)f(these)h(pap)s(ers)g(for)f="" (the)0="" y(computational)29="" b(motiv)-5="" b(ation)28="" b(of)j(the)g="" (problem.)42="" b(here)32="" b(w)m(e)h(start)e(with)g(reviewing)g(the)h="" (approac)m(h)f(of)0="" y([10])k(to)h(the)g(solution)e(of)h(this)h="" (question,)h(con)m(tin)m(ued)f(in)f([8])h(and)g(in)f([12)o(].)53="" b(then)37="" b(w)m(e)g(describ)s(e)f(our)0="" y(results,)d(and)g(discuss)="" h(the)f(w)m(a)m(y)g(they)h(are)e(related)h(to)f(the)h(previous)g(w)m="" (ork.)146="" y(let)26="" b(us)h(\014rst)f(describ)s(e)g(the)g(question:)="" b(the)27="" b(inclusion-exclusion)c(form)m(ula)h(expresses)29="" b(the)d(prob-)0="" y(abilit)m(y)j(of)i(the)h(union)f(of)f(a)i(family)="" c(of)j(sets)i(in)d(terms)i(of)e(the)i(probabilities)d(of)i(their)f(in)m="" (tersections:)893="" ft(p)14="" fv(\()p="" fs([)p="" ft(a)1194="" fb(i)1222="" fv(\))28="" b(=")1391" fr(x)1451="" fb(i)1552="" ft(a)1787="" fb(i)1814="" fv(\))22="" fs(\000)1974="" fr(x)1990="" fb(i<j)2134="" ft(a)2369="" fb(i)2419="" fs(\\)22="" ft(a)2580="" fb(j)2617="" fv(\))g(+)g="" ft(:::)0="" fv(assume)44="" b(no)m(w,)i(that)c(w)m(e)i(are)f(giv)m(en)g(the)g="" (probabilities)d(of)i(all)f(the)i="" ft(j)6="" fv(-wise)43="" b(in)m(tersections)g(of)g(the)0="" y(ev)m(en)m(ts,)35="" b(for)d="" ft(j)i="" fv(=")27" ft(;)17="" fv(.)43="" b(ho)m(w)33="" b(w)m(ell)f(can)h(w)m(e)h(estimate)d(the)i(probabilit)m(y)e="" (of)h(the)h(union)f="" ft(a)3480="" fb(i)3541="" fv(?)146="" y(a)h(precise)g(form)m(ulation)d([10])i(is)g(as)h="" (follo)m(ws:)0="" fq(main)38="" b(question)0="" fv(find)244="" ft(e)6="" ft(k)s(;)17="" b(n)p="" fv(\))27="" fr(\000)894="" fs([)1122="" fb(n)1122="" y(i)p="" fz(="1)1240" ft(a)1313="" fb(i)1341="" fs(\000)h="" fs([)1729="" fb(n)1729="" y(j)t="" ft(b)1929="" fb(j)1966="" fv(\))2004="" fr(\001)3625="" fv(\(1\))0="" y(where)33="" b(the)f(suprem)m(um)g(ranges)h(o)m(v)m(er)f(all)e(families)f="" (of)i(ev)m(en)m(ts)j="" ft(a)2401="" fz(1)2441="" b(:)g(:)g(:)f(;)h(a)2733="" fb(n)2811="" fv(and)32="" ft(b)3074="" fz(1)3114="" b(:)g(:)g(:)f(;)h(b)3407="" fb(n)3485="" fv(in)31="" b(an)m(y)0="" y(probabilit)m(y)g(spaces,)j(that)e(satisfy:)1312="" fs(\\)1540="" fb(i)p="" fy(2)p="" fb(s)1661="" ft(a)1734="" fb(i)1763="" fs(\\)2160="" fb(j)t="" fb(s)2289="" ft(b)2363="" fb(j)2400="" fv(\))0="" y(for)32="" b(all)f="" ft(s)i="" fs(\032)28="" fv([)p="" ft(n)p="" fv(],)33="" b(suc)m(h)h(that)f="" fs(j)p="" ft(s)6="" fs(j)26="" b(\024)j="" ft(k)s="" fv(.)146="" y(the)k(approac)m(h)e="" (suggested)i(in)d([10])h(con)m(tains)h(t)m(w)m(o)g(main)d(steps.)45="" b(in)31="" b(the)h(\014rst)g(step)g(the)g(\\target)0="" y(v)-5="" b(alue")35="" fv(\))36="" b(is)f(sho)m(wn)i(to)f(b)s(e)g(the)g(optim)m(um)e(of)h(a)g="" (certain)h(linear)e(program.)52="" b(p)m(assing)36="" b(to)g(the)0="" y(dual)c(pro)s(duces)h(a)g(follo)m(wing)d(\(rather)i(surprising\))="" g(result:)0="" y(let)42="" ft(\016)227="" fz(0)307="" fv(is)f(a)g(function)g(on)g(0)p="" ft(;)g(:)g(:)g(:)f(;)h(n)41="" fv(de\014ned)i(b)m(y)f="" ft(\016)2031="" fz(0)2070="" ft(i)p="" fv(\))h(=")g" ft(\016)2384="" fz(0)p="" fb(;i)2467="" fv(.)70="" b(\(recall)39="" b(that)i="" fp(kr)-5="" b(one)g(cker's)42="" ft(\016)t="" fp(-)0="" y(function)e="" ft(\016)432="" fb(i;j)553="" fv(is)g(1)g(if)f="" ft(i)j="" ft(j)46="" fv(and)41="" b(is)f(0)g(otherwise.\))68="" b(e)41="" b(w)m(an)m(t)g(to)f(appro)m="" (ximate)g="" ft(\016)3214="" fz(0)3294="" fv(as)g(close)h(as)0="" y(p)s(ossible)33="" b(b)m(y)i(a)f(p)s(olynomial)c(of)k(b)s(ounded)g="" (degree.)49="" b(namely)-8="" b(,)33="" b(giv)m(en)h(0)c="" fs(\024)g="" ft(k)j="" fs(\024)e="" fv(,)j(w)m(e)h(ask:)47="" b(what)34="" b(is)0="" ft(e)p="" fv(\))34="" fb(p)596="" fv(\(max)816="" fb(;:::)n(;n)1091="" ft(p)p="" ft(\016)1442="" fz(0)1481="" fv(\))p="" fv(\))17="" ft(;)36="" fv(where)i(the)f(in\014m)m(um)e(ranges)i(o)m(v)m(er)h(all)c="" (p)s(olynomials)0="" ft(p)e="" fv(of)h(degree)g="" fs(\024)28="" fv(.)44="" b(it)32="" b(turns)i(out)e(that)g="" fs(\001)f="" ft(k)s(;)c(n)p="" ft(;)33="" fv(or)f(in)g(other)h(w)="" m(ords:)244="" fs(\001)g(k)p="" ft(\016)899="" fz(0)960="" fs(\000)h(p)1129="" fb(k)1172="" fs(k)1222="" fy(1)1296="" ft(;)2302="" fv(\(2\))0="" y(where)34="" fs(p)351="" fb(k)426="" fv(is)e(the)h(space)h="" (of)e(p)s(olynomials)e(of)i(degree)h="" ft(k)36="" fv(on)c(0)p="" ft(;)g(:)g(:)g(:)f(;)h(n)p="" y(in)39="" b(the)f(second)h(step)g([10)o(])f(attac)m(ks)h="" (the)g(appro)m(ximation)c(question)k(of)e(\014nding)g="" fv(\))38="" b(using)0="" y(tec)m(hniques)j(from)d(the)h="" fp(the)-5="" b(ory)42="" b(of)e(appr)-5="" b(oximation)40="" b(of)g(functions)h(of)f(the)h(r)-5="" b(e)g(al)41="" b(variable)p="" fv(.)62="" b(it)39="" b(turns)1851="" y(2)p="" fv(out,)29="" b(that)g(suitably)f(scaled)h="" fp(chebyshev)h(p)-5="" b(olynomials)27="" fv(of)i(degree)g="" fv(pro)m(vide)d(a)f(go)s(o)s(d)g(candidate)g(for)0="" y(the)33="" b(appro)m(ximation)d(of)i="" ft(\016)972="" fz(0)1012="" fv(,)h(and)f(this)h(leads)f(to)g(the)h(follo)m(wing)d="" (b)s(ounds)j(on)g="" fv(\):)0="" y(1.)43="" b(f)-8="" b(or)32="" ft(k)f="" ft(o)p="" fv(\()564="" fs(p)p="" fv(\))1371="" fr(\022)1978="" fv(1)22="" fs(\000)2158="" ft(k)2212="" fz(2)p="" ft(n)2262="" fr(\023)2352="" ft(:)0="" fv(2.)43="" b(\()610="" fv(\))1543="" ft(;)0="" fv(3.)43="" b(and)33="" b(for)f="" fs(\035)691="" y(p)p="" ft(n)1234="" y(e)6="" ft(o)1771="" fr(\022)1844="" ft(exp)2010="" fr(\032)2085="" fs(\000)2192="" fv(2)p="" ft(k)p="" ft(n)2324="" fr(\033\023)2489="" fv(while)38="" b(this)g(estimate)g(of)g="" fv(\))39="" b(turns)h(to)e(b)s(e)h(optimal)d(for)i="" ft(k)k="" fs(\024)2603="" fv(,)e(it's)f(optimalit)m(y)c(for)k(the)0="" y(other)31="" b(v)-5="" b(alues)31="" b(of)f="" fv(w)m(as)d(not)g(at)f(all)f="" (clear.)43="" b(fact)f([10])h(sho)m(ws)h(that,)f(at)g(least)f(for)="" g(a)h(certain)f(range)0="" y(of)i="" fv(,)h(the)g(estimate)f(is)g="" (not)g(optimal,)e(b)m(y)j(computing)f="" ft(n)22="" fz(1)p="" a(2)2718="" fo(n)p="" fn(\000)p="" fm(1)2882="" fv(precisely)-8="" b(.)146="" y(the)37="" b(task)f(of)f(impro)m(ving)e(the)j(estimates)f(for)g="" fv(\))35="" b(w)m(as)h(con)m(tin)m(ued)="" g(in)f([8].)52="" b(it)35="" b(w)m(as)i(under-)0="" y(sto)s(o)s(d)f(that)g="" (the)h(\\problem")d(of)i(cheb)m(yshev)k(p)s(olynomials)33="" b(is)j(that)g(they)i(actually)d(do)h(to)s(o)f(m)m(uc)m(h)0="" y(-)g(instead)g(of)g(pro)m(viding)f(a)h(go)s(o)s(d)f(appro)m="" (ximation)f(to)i(zero)h(in)f(the)g(in)m(teger)h(p)s(oin)m(ts)e(1)p="" b(:)g(:)g(:)f(;)h(n)p="" fv(,)36="" b(they)0="" y(appro)m(ximate)26="" b(zero)h(on)g(the)g="" fp(whole)h(interval)f="" fv([1)p="" fv(].)41="" b(a)27="" b(natural)e(attempt)h(to)h(remedy)g(this)g="" (situation)0="" y(is)39="" b(to)h(\\tailor-mak)m(e")c(the)k(appro)m="" (ximating)e(p)s(olynomial)e="" ft(q)43="" fv(b)m(y)e(constraining)e="" ft(q)k="" fv(to)c(ha)m(v)m(e)j(a)d(lot)f(of)0="" y(in)m(teger)32="" b(zero)s(es)h(in)e(the)i(\\righ)m(t")d(places)j(on)e(the)i(in)m(terv)-5="" b(al)31="" b([1)p="" fv(],)32="" b(and)g(then,)h(p)s(ossibly)-8="" b(,)31="" b(m)m(ultiplying)0="" y(the)k(obtained)e(p)s(olynomial)e(b)m="" (y)k(a)f(suitably)g(scaled)g(cheb)m(yshev)k(p)s(olynomial)31="" b(of)i(a)h(small)e(degree)j(-)0="" y(to)h(\\round)g(things)g(out".)54="" b(this)37="" b(approac)m(h)f(leads)g([8])h(to)e(an)i(impro)m(v)m(ed)f="" (estimate)f(on)h="" b(for)0="" ft(k)31="" fs(\035)209="" fv(.)1153="" fs(\024)i="" ft(exp)1762="" fr(\032)1837="" fs(\000)p="" fv(\012)2001="" fr(\022)2084="" ft(k)2138="" fz(2)2195="" fv(log)16="" a(n)2405="" fr(\023)q(\033)2570="" fv(this)29="" b(inequalit)m(y)f(turns)i(out)e(to)h(b)="" s(e)g(rather)g(tigh)m(t.)42="" b(using)28="" b(linear)f(dualit)m(y)h(once)i="" (again)3236="" fz(1)3304="" fv([8])f(pro)m(vide)0="" y(also)j(a)g(lo)m(w)m(er)h(estimate)e(on)i="" fv(\):)1167="" fs(\025)g="" ft(exp)1776="" fr(\032)1851="" ft(o)2022="" fr(\022)2197="" ft(k)2251="" ft(n)17="" fv(log)g="" ft(n)2391="" fr(\023\033)2556="" fv(using)43="" b(the)h(con)m(v)m="" (enien)m(t)971="" y(~)957="" y(\002)f(notation)f(w)m(e)j(actually)d="" (arriv)m(e)h(to)g(a)g(\\tigh)m(t")g(estimate:)64="" fv(\))46="" ft(exp)166="" fr(n)233="" fs(\000)324="" fv(~)310="" y(\002)403="" fr(\020)472="" fb(k)511="" fm(2)p="" fb(n)555="" fr(\021)q(o)698="" ft(:)146="" fv(no)m(w,)27="" b(let)e(us)g(c)m(hange)h="" (direction,)g(and)f(return)g(once)h(more)e(to)g(the)i="" (inclusion-exclusion)d(form)m(ula.)0="" y(it)34="" b(is)g(w)m(ell-kno)m="" (wn)g([19])g(that)g(this)f(form)m(ula)g(is)g(a)h(sp)s(ecial)f(case)i="" (of)f(the)g="" fp(m\177)-50="" b(inversion)f(formula)0="" fv(for)f(partially)e(ordered)k(sets.)51="" b(\(the)35="" b(underlying)f(p)s(oset)h(in)f(our)h(case)h(is)e(the)h(lattice)e(of)h="" (all)f(subsets)0="" y(of)28="" b(an)h="" fv(-set.\))43="" b(therefore)30="" b(one)f(can)g(ask)h([10)o(])f(whether)h(an)f(analog)e="" (of)i(the)g(results)g(describ)s(ed)h(ab)s(o)m(v)m(e)0="" y(exists)41="" b(for)f(other)g(p)s(osets.)68="" b(a)40="" b(step)h(in)f(this)g(direction)f(w)m(as)i(tak)m(en)h(b)m(y)f(ma)m="" (vronicolas)e([12],)j(who,)p="" fl(1)149="" fw(this)28="" b(is)f(this)h(rare)e(case,)h(in)h(whic)n="" (h)f(passing)g(to)g(the)h(dual)f(t)n(wice)h(do)r(esn't)f(lead)g(bac)n="" (k)g(to)h(the)f(original)f(question,)0="" y(since)h(the)h(underlying)="" f(space)g(had)h(b)r(een)g(mean)n(while)f(altered)g(b)n(y)g(considering)="" f(only)i="" fk(symmetric)i(events)d="" fw([10)o(],)h([8].)1851="" fv(3)p="" fv(follo)m(wing)32="" b([2])k(considers)g(the)f="" ft(q)t="" fp(-analo)-5="" b(g)36="" b(of)h(the)g(principle)f(of)h="" (inclusion-exclusion)p="" fv(,)d(whic)m(h)i(means)0="" y(that)41="" b(the)g(lattice)f(of)g(all)f(subsets)k(of)e(an)g="" fv(-set)g(is)g(replaced)g(b)m(y)h(the)f(lattice)f(of)g(all)f="" (subspaces)k(of)0="" y(an)h="" fv(-dimensional)e(v)m(ector)j="" (space)g(o)m(v)m(er)g(a)f(\014nite)g(\014eld)g(with)g="" fv(elemen)m(ts.)79="" b(let)44="" b(us)h(describ)s(e)g(in)0="" y(some)36="" b(detail)f(the)i(framew)m(ork)f(addressed)i(in)e([12)o="" (].)55="" b(let)36="" ft(v)2235="" fb(n)2213="" y(q)2318="" fv(b)s(e)h(an)f="" fv(-dimensional)d(v)m(ector)38="" b(space)0="" y(o)m(v)m(er)g(a)f(\014nite)f(\014eld)h(with)f="" ft(q)41="" fv(elemen)m(ts.)57="" b(let)37="" ft(x)45="" fv(b)s(e)37="" b(a)f(set)i(suc)m(h)g(that)f(with)g(ev)m(ery)h(v)m(ector)g="" ft(v)h="" fs(2)c="" ft(v)3703="" fb(n)3681="" y(q)0="" fv(is)45="" b(asso)s(ciated)h(a)g(subset)h="" fs(a)1076="" fb(v)1167="" fs(\022)k="" ft(x)8="" fv(.)83="" b(assume)47="" b(also)d(that)i(the)g(asso)s(ciation)f(b)s(et)m(w)m(een)j(elemen)m(ts)0="" y(of)c="" ft(x)53="" fv(and)44="" b(subspaces)k(of)c="" ft(v)1118="" fb(n)1097="" y(q)1210="" fv(is)g(regular)g(in)g(the)h(follo)m="" (wing)d(sense:)69="" b(for)45="" b(eac)m(h)g="" ft(x)k="" fs(2)g="" ft(x)j="" fv(the)45="" b(set)0="" ft(v)57="" fb(x)128="" fr(\010)290="" ft(v)31="" fs(2)e="" ft(v)541="" fb(n)520="" y(q)621="" fs(j)j="" ft(x)c="" fs(2)g(a)938="" fb(v)978="" fr(\011)1069="" fv(is)k(either)g(empt)m(y)h(or)f(a)g(subspace)i(of)e="" ft(v)2538="" fb(n)2516="" y(q)2585="" b(an)m(y)h(subspace)h="" ft(t)41="" fs(v)28="" fv(let)g="" ft(s)197="" fb(t)280="" fs(j)17="" b(f)o="" ft(x)28="" ft(x)40="" fs(j)32="" ft(t)42="" ft(v)1097="" fb(x)1141="" fs(g)16="" b(j)p="" fv(.)42="" b(then)30="" b(the)g="" fv(-analog)c(of)j(inclusion-exclusion)e="" (principle)g(is)i(giv)m(en)0="" y(b)m(y)k(the)g(follo)m(wing)d(form)m="" (ula)h([19]:)926="" fs(j)21="" b([)1041="" fb(v)r="" fb(v)1182="" fo(n)1165="" y(q)1251="" fs(a)1331="" fb(v)1372="" fs(j)27="" fr(x)1531="" fb(t)10="" fy(v)p="" fb(v)1693="" fo(n)1678="" y(q)1736="" fv(1\))1938="" y(dim)n="" fb(t)g="" fz(\)+1)2300="" ft(q)2347="" fv(\()2385="" y(dim)o="" fm(\()p="" fo(t)e="" fm(\))2497="" y(2)2640="" fv(\))2682="" ft(s)2742="" fb(t)2797="" fv([12])33="" b(deals)g(with)g(the)h="" (follo)m(wing)d(question:)46="" b(supp)s(ose)34="" b(that)f(w)m(e)i(kno)m(w)f="" ft(s)2718="" fb(t)2774="" fv(,)f(for)g(all)e(subspaces)36="" ft(t)47="" fv(of)0="" ft(v)79="" fb(n)57="" y(q)159="" fv(of)34="" b(dimension)e="" ft(j)k="" b(0)p="" ft(;)g(:)g(:)g(:)f(;)h(k)s="" fv(.)47="" b(ho)m(w)34="" b(w)m(ell)f(can)h(w)m(e)h(estimate)e="" fs(j)23="" b([)2660="" fb(v)2801="" fo(n)2784="" y(q)2870="" fs(a)2950="" fb(v)2991="" fv(?)46="" b(more)35="" b(precisely)-8="" b(,)0="" y(w)m(e)34="" b(can,)f(without)f(loss)g(of)g(generalit)m(y)g([10])g="" (assume)h(that)g="" fv(is)32="" b(a)g(probabilit)m(y)f(space)j(and)="" e(set)244="" ft(e)322="" fb(q)r="" fz(\))415="" ft(sup)946="" fr(\000)991="" fs([)1219="" fb(v)1359="" fo(n)1343="" y(q)1405="" fs(a)1485="" fb(v)1526="" fs(\000)g="" fs([)1913="" fb(v)2053="" fo(n)2037="" y(q)2100="" fs(b)2165="" fb(v)2206="" fv(\))2244="" fr(\001)2307="" ft(;)1291="" fv(\(3\))0="" y(where)41="" b(the)f(suprem)m(um)g(ranges)h(o)m(v)m(er)f(all)e(families)f(of)i(ev)m="" (en)m(ts)j="" fs(fa)2530="" fb(v)2571="" fs(g)2621="" fb(v)2761="" fo(n)2745="" y(q)2847="" fv(and)e="" fs(fb)3159="" fb(v)3200="" fs(g)3250="" fb(v)3391="" fo(n)3374="" y(q)3477="" fv(in)f(an)m(y)0="" y(probabilit)m(y)33="" b(spaces,)k(that)e(satisfy:)48="" ft(s)1457="" fy(a)1451="" fb(t)1549="" ft(s)1722="" fy(b)1716="" fb(t)1809="" fv(for)j(all)f="" ft(t)45="" fs(v)32="" ft(v)2388="" fb(n)2366="" y(q)2435="" fv(,)j(suc)m(h)h(that)f(dim)o(\()p="" ft(t)14="" fv(\))31="" fs(\024)h="" fv(.)50="" b(\(here)0="" ft(s)66="" fy(a)60="" fb(t)159="" fv(and)33="" ft(s)415="" fy(b)409="" fb(t)499="" fv(ha)m(v)m(e)h(the)f(ob)m(vious)g(de\014nition\).)42="" b(what)33="" b(is)f="" ft(e)2204="" fz(\))2297="" fv(\)?)146="" y(the)34="" b(approac)m(h)g(of)f([12)o(])g(follo)m(ws)f(closely)h(that)g(of)f="" ([10].)45="" b(the)34="" b(alue)33="" ft(e)2777="" fz(\))2870="" fv(\))33="" b(is)g(sho)m(wn)h(to)f(b)s(e)0="" y(the)27="" b(optim)m(um)e(of)i(a)g(certain)f(linear)g(program.)40="" b(p)m(assing)27="" b(to)g(the)h(dual)e(pro)s(duces)i(a)f(follo)m(wing)d="" (result:)244="" fs(\001)g="" fv(inf)940="" fb(p)1033="" fr(\022)1143="" fv(max)1107="" fb(;:::)n(;n)1378="" ft(q)1540="" fb(i)1568="" fv(\))g="" ft(\016)1771="" fz(0)1810="" ft(q)1895="" fb(i)1923="" fs(j)1989="" fr(\023)2079="" ft(;)1519="" fv(\(4\))0="" b(the)f(in\014m)m(um)e(ranges)i(o)m(v)m(er)h(all)c(p)s="" (olynomials)g="" ft(p)i="" fv(of)g(degree)i="" fv(,)k(and)h="" ft(\016)2970="" fz(0)3010="" ft(q)3095="" fb(i)3123="" ft(\016)3335="" fb(;i)3418="" y(in)33="" b(other)g(w)m(ords:)244="" ft(\016)992="" fz(0)1053="" fs(\000)h(p)1222="" fb(k)1265="" fs(k)1315="" fy(1)1389="" ft(;)2209="" fv(\(5\))0="" fv(is)e(the)h(space)h(of)e(p)s="" (olynomials)e(of)i(degree)h="" fv(on)c(1)p="" b(q)t(;)g(:)g(:)g(:)f(;)h(q)2754="" fb(n)2800="" y([12])48="" b(then)h(observ)m(es)h(that)d(the)i="" (second)g(step)g(of)e([10])h(fails)e(in)h(this)h(case,)53="" b(namely)47="" b(for)g(an)m(y)0="" y(1)36="" fs(\024)d="" ft(n)h="" fv(a)f(scaled)h(cheb)m(yshev)j(p)s(olynomial)34="" b(of)k(degree)g="" fv(pro)m(vides)e(only)e(a)g(w)m(eak)i(\(b)s="" (ounded)0="" y(b)s(elo)m(w)32="" b(b)m(y)i(a)e(constan)m(t)i="" ft(="">)27 b Fv(0\))32 b(appro)m(ximation)f(of)h Ft(\016)1914 4825 y Fz(0)1953 4810 y Fv(.)146 4980 y(No)m(w)24 b(to)e(the)h (description)f(of)g(our)h(results.)40 b(There)24 b(are)f(t)m(w)m(o)g(p) s(oin)m(ts)f(w)m(e)i(w)m(ould)f(lik)m(e)e(to)i(mak)m(e.)40 b(The)0 5100 y(\014rst)c(is)f(an)g(easy)i(generalization)c(of)i(the)h (notion)e(of)h(\\appro)m(ximate)f(M\177)-49 b(obius)36 b(in)m(v)m(ersion",)g(namely)1851 5349 y(4)p eop %%Page: 5 5 5 4 bop 0 100 a Fv(the)44 b(question)g(ho)m(w)g(w)m(ell)f(M\177)-49 b(obius)43 b(in)m(v)m(ersion)h(w)m(orks)g(giv)m(en)g(partial)d (information,)i(to)g(a)h(general)0 220 y(class)34 b(of)f(graded)h(p)s (osets.)49 b(In)34 b(our)g(opinion)e(the)i(linear)e(programming)f (approac)m(h)k(of)e([10])h(is)f(easier)0 340 y(and)g(more)f(natural)g (to)h(apply)g(at)f(this)h(lev)m(el)g(of)f(generalit)m(y)-8 b(.)44 b(It)33 b(turns)h(out)f(that)g(the)g(b)s(est)h(p)s(ossible)0 461 y(appro)m(ximation)25 b(for)i(suc)m(h)i(a)e(p)s(oset)g Ft(P)41 b Fv(equals)28 b(the)f(distance)h(\(in)e(the)i(suprem)m(um)f (norm\))g(b)s(et)m(w)m(een)i(a)0 581 y(c)m(haracteristic)f(function)g (of)f(the)i(zero)f(elemen)m(t)g(of)g Ft(P)41 b Fv(and)29 b(a)f(subspace)i(spanned)f(b)m(y)g(c)m(haracteristic)0 702 y(functions)37 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b(structure)g(of)e(the)i(space)f Fs(U)3272 700 y FB(k)3355 685 y Fv(seems,)j(in)0 805 y(general,)29 b(to)g(b)s(e)h(rather)f(complex.)42 b(Ho)m(w)m(ev)m(er,)32 b(the)e(function)e Ft(\016)2317 820 y Fz(0)2386 805 y Fv(is)h(quite)g(sp)s(ecial.)41 b(It)30 b(ob)m(viously)f(has)0 926 y(a)34 b(lot)f(of)h(nice)h(prop)s(erties,)g(from)e(whic)m(h)i(w)m (e)g(exploit)e(the)i(follo)m(wing:)45 b Ft(\016)2684 941 y Fz(0)2758 926 y Fv(is)34 b(constan)m(t)h(on)f(the)h(lev)m(el)0 1046 y(sets)f Ft(P)254 1061 y Fz(0)293 1046 y Ft(;)17 b(:)g(:)g(:)f(;)h(P)575 1061 y FB(n)621 1046 y Fv(.)44 b(Let)33 b(us)g(capture)g(this)f(prop)s(ert)m(y)i(of)e Ft(\016)2087 1061 y Fz(0)2159 1046 y Fv(in)g(a)g(de\014nition:)0 1258 y Fq(De\014nition)k(3.4)o(:)45 b Fv(A)33 b(function)f Ft(f)71 b Fv(:)60 b Ft(P)42 b Fs(!)27 b Fj(R)43 b Fv(is)33 b(a)f Fp(symmetric)i(function)f Fv(if)e Ft(f)43 b Fv(is)33 b(constan)m(t)g(on)g(the)0 1378 y(lev)m(el)f(sets)i Ft(P)476 1393 y Fz(0)515 1378 y Ft(;)17 b(:)g(:)g(:)f(;)h(P)797 1393 y 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785 y(-)f(80.)49 988 y([7])49 b(D.)34 b(J.)g(Kleitman,)f Fa(On)h(a)g(com)m(binatorial)d(conjecture)36 b(of)e(Erdo)-5 b(\177)-44 b(s)p Fv(,)36 b(Journal)d(of)h(Com)m(b.)h(Theory)201 1108 y(1,)d(1966,)g(209-214.)49 1312 y([8])49 b(J.)37 b(Kahn,)h(N.)f(Linial)e(and)i(A.)g(Samoro)s(dnitsky)-8 b(,)38 b(Inclusion-exclusion:)52 b(exact)37 b(and)h(appro)m(xi-)201 1432 y(mate,)31 b Fa(Com)m(binatorica)p Fv(,)g(v)m(ol.)h(16\(1996\))f (.)49 1636 y([9])49 b(V.)28 b(Lev)m(ensh)m(tein,)j Fa(Kra)m(w)m(c)m (houk)e(p)s(olynomials)c(and)j(univ)m(ersal)g(b)s(ounds)g(for)g(co)s (des)g(and)g(designs)201 1756 y(in)j(Hamming)f(spaces)p Fv(,)35 b(IEEE)e(T)-8 b(rans.)34 b(Inform.)e(Theory)-8 b(,)33 b(v)m(ol.)g(IT-41,)f(1995,)g(1303-1321.)0 1959 y([10])49 b(N.)34 b(Linial)e(and)j(N.)g(Nisan,)g(Appro)m(ximate)f (inclusion-exclusion,)f Fa(Com)m(binatorica)p Fv(,)g(10\(1990\))201 2080 y(349-365.)0 2283 y([11])49 b(L.)36 b(Lo)m(v)-5 b(asz,)37 b Fa(On)g(the)f(Shannon)h(capacit)m(y)f(of)g(a)g(graph)p 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y([16])49 b(A.)27 b(Sc)m(hrijv)m(er,)j Fa(A)e(comparison)f(of)g(the)h(Delsarte)f(and)h (Lo)m(v)-5 b(asz)29 b(b)s(ounds)p Fv(,)g(IEEE)g(T)-8 b(rans.)29 b(Inform.)201 4022 y(Theory)-8 b(,)33 b(v)m(ol.)f(IT-25,)h (1979,)f(425-429.)0 4226 y([17])49 b(R.)33 b(P)-8 b(.)34 b(Stanley)-8 b(,)33 b Fq(En)m(umerativ)m(e)k(Com)m(binatorics)p Fv(,)32 b(v)m(ol.)h(I,)h(Cam)m(bridge)f(Univ)m(ersit)m(y)h(Press,)201 4346 y(1997.)0 4550 y([18])49 b(G.)32 b(Szego,)h Fq(Orthogonal)k(P)m (olynomials)p Fv(,)30 b(American)i(Mathematical)e(So)s(ciet)m(y)-8 b(,)33 b(1939.)0 4753 y([19])49 b(J.)33 b(H.)g(v)-5 b(an)33 b(Lin)m(t)g(and)g(R.)g(M.)h(Wilson,)e Fq(A)38 b(Course)h(in)e(Com)m (binatorics)p Fv(,)32 b(Cam)m(bridge)g(Uni-)201 4873 y(v)m(ersit)m(y)h(Press,)i(1992.)1826 5349 y(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF </j)2112>