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1、THEERDOSDISCREPANCYPROBLEM˝TERENCETAOAbstract.Weshowthatforanysequencef:NÑt´1,`1utakingvaluesint´1,`1u,thediscrepancyˇˇˇÿnˇˇˇsupˇfpjdqˇn,dPNˇˇj“1offisinfinite.ThisanswersaquestionofErd˝os.InfacttheargumentalsoappliestosequencesftakingvaluesintheunitsphereofarealorcomplexHilbertspace.Theargumentusesth
2、reeingredients.ThefirstisaFourier-analyticreduction,obtainedaspartofthePolymath5projectonthisproblem,whichreducestheproblemtothecasewhenfisre-placedbya(stochastic)completelymultiplicativefunctiong.ThesecondisalogarithmicallyaveragedversionoftheElliottconjec-ture,establishedrecentlybytheauthor,whicheff
3、ectivelyreducestothecasewhengusuallypretendstobeamodulatedDirichletcharacter.Thefinalingredientis(anextensionof)afurtherargu-mentobtainedbythePolymath5projectwhichshowsunboundeddiscrepancyinthiscase.1.IntroductionGivenasequencef:NÑHtakingvaluesinarealorcomplexHilbertspaceH,definethediscrepancyofftobet
4、hequantity››arXiv:1509.05363v1[math.CO]17Sep2015›ÿn›››sup›fpjdq›.n,dPN››j“1HInotherwords,thediscrepancyisthelargestmagnitudeofasumoffalonghomogeneousarithmeticprogressionstd,2d,...,nduinthenaturalnumbersN“t1,2,3,...u.Themainobjectiveofthispaperistoestablishthefollowingresult:Theorem1.1(Erd˝osdiscrep
5、ancyproblem,vector-valuedcase).LetHbearealorcomplexHilbertspace,andletf:NÑHbeafunctionsuchthat}fpnq}H“1foralln.Thenthediscrepancyoffisinfinite.SpecalisingtothecasewhenHisthereals,wethushaveCorollary1.2(Erd˝osdiscrepancyproblem,originalformulation).Ev-eryfunctionf:NÑt´1,`1uhasinfinitediscrepancy.12TERE
6、NCETAOThisanswersaquestionofErd˝os[5],whichwasrecentlythesubjectofthePolymath5project[13];seetherecentreport[6]onthelatterprojectforfurtherdiscussion.Itisinstructivetoconsidersomenear-counterexamplestothesere-sults-thatistosay,functionsthatareofunitmagnitude,ornearlyso,whichhavesurprisinglysmalldisc
7、repancy-toisolatethekeydifficultyoftheproblem.Example1.3(Dirichletcharacter).Letχ:NÑCbeanon-principalDirichletcharacterofperiodq.Thenχiscompletelymultiplicative(thusχpnmq“χpnqχpmqforanyn,mPN)andhasmeanz
