q1qr1thentheestimateofCorollary2becomestrivialTurningtotheupperboundswepresentseveraldi erentconstructionsSomeofthemcanberegardedasre nedandadjustedversionsofpreviouslyknownonesothertoourknow ID: 201248
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2S.KOPPARTY,V.LEV,S.SARAF,ANDM.SUDANWenoticethatTheorem1extends[DKSS,Theorem11]andindeed,thelatterresultisaparticularcaseoftheformer,obtainedforr=1.TheproofofTheorem1usesthepolynomialmethodinthespiritof[DKSS,SS08].Usingtheinequality(1+x)�m1�mx;x0;m1;onereadilyderivesCorollary2.Ifnr1areintegers,qaprimepower,andKFnqaKakeyasetofrankr,thenjKj�1�n(q�1)q�rqn:Tofacilitatecomparisonbetweenestimates,weintroducethefollowingterminology.GiventwoboundsB1andB2forthesmallestsizeofaKakeyasetinFnq(whichareeitherbothupperboundsorbothlowerbounds),wesaythattheseboundsareessentiallyequivalentinsomerangeofnandqifthereisaconstantCsuchthatforallnandqinthisrangewehaveB1CB2;B2CB1;andalsoqn�B1C(qn�B2);qn�B2C(qn�B1):Wewillalsosaythattheestimates,correspondingtothesebounds,areessentiallyequiv-alent.Withthisconvention,itisnotdiculttoverifythatforeveryxed"�0,theestimatesofTheorem1andCorollary2areessentiallyequivalentwhenevern(1�")qr�1.Ifn1+1 q�1qr�1,thentheestimateofCorollary2becomestrivial.Turningtotheupperbounds,wepresentseveraldi erentconstructions.Someofthemcanberegardedasrenedandadjustedversionsofpreviouslyknownones;other,toourknowledge,didnotappearintheliterature,buthavebeen\intheair"forawhile.WerstpresentaKakeyasetconstructiongearedtowardslargeelds.Itisbasedon(i)the\quadraticresidueconstruction"duetoMockenhauptandTao[MT04](witharenementbyDvir,see[SS08]),(ii)the\liftingtechnique"from[EOT],and(iii)the\tensorpowertrick".Ourstartingpointis[SS08,Theorem8],statingthatifn1isanintegerandqaprimepower,thenthereexistsarank-1KakeyasetKFnqsuchthatjKj2�(n�1)qn+O(qn�1);(1)withanabsoluteimplicitconstant.Indeed,theproofin[SS08]yieldstheexplicitestimatejKj(q�q+1 2n�1+qn�1ifqisodd;(q�1)�q 2n�1+qn�1ifqiseven:(2) 4S.KOPPARTY,V.LEV,S.SARAF,ANDM.SUDANCorollary5.Letn�r1beintegersandqaprimepower.ThereexistsaKakeyasetKFnqofrankrsuchthatjKjqn� �qn�(r�1);moreover,ifnrqr�1,theninfactjKjqn� n rqn�(r�1)(withabsoluteimplicitconstants).WeremarkthatCorollaries2and5givenearlymatchingboundsonthesmallestpossiblesizeofaKakeyasetofrankrinFnqinthecasewhererisxed,qgrows,andthedimensionndoesnotgrow\toofast".Thesituationwhereqisboundedandngrowsisquitedi erent:forr=1theO-termin(1)donotallowforconstructingKakeyasetsofsizeo(qn),andforrlargetheestimateofTheorem4isratherweak.Addressingrstthecaser=1,wedevelopfurthertheideabehindtheproofof[SS08,Theorem8]toshowthattheO-termjustmentionedcanbewellcontrolled,makingtheresultnon-trivialintheregimeunderconsideration.Theorem6.Letn1beanintegerandqaprimepower.Thereexistsarank-1KakeyasetKFnqwithjKj8&#x]TJ/;༶ ;.9;Ւ ;&#xTf 1;.42; 31;&#x.682;&#x Td ;&#x[000;&#x]TJ/;༶ ;.9;Ւ ;&#xTf 1;.42; 31;&#x.682;&#x Td ;&#x[000;&#x]TJ ;� -2;.52;&#x Td ;&#x[000;&#x]TJ ;� -2;.52;&#x Td ;&#x[000;:2�1+1 q�1�q+1 2nifqisodd;3 2�1+1 q�1�2q+1 3nifqisanevenpowerof2;3 22(q+p q+1) 3nifqisanoddpowerof2:Theorem6istobecomparedagainstthecaser=1ofTheorem1showingthatifKFnqisarank-1Kakeyaset,thenjKj�q2=(2q�1)n.ForseveralsmallvaluesofqtheestimateofTheorem6canbeimprovedusingacombinationofthe\missingdigitconstruction"andthe\randomrotationtrick"ofwhichwelearnedfromTerryTaowho,inturn,referstoImreRuzsa(personalcommunicationinbothcases).ForaeldF,byFwedenotethesetofnon-zeroelementsofF.Themissingdigitconstructionbyitselfgivesaveryclean,butratherweakestimate.Theorem7.Letn1beanintegerandqaprimepower,andsupposethatfe1;:::;engisalinearbasisofFnq.LetA:=f"1e1++"nen:"1;:::;"n2FqgandB:=f"1e1++"nen:"1;:::;"n2f0;1gg: 6S.KOPPARTY,V.LEV,S.SARAF,ANDM.SUDANCorollary11.Letnr1beintegersandqaprimepower.ThereexistsaKakeyasetKFnqofrankrsuchthatjKjqn(1�q�r)+r+1:ItisnotdiculttoverifythatCorollary11supersedesCorollary5forn(r+2)qr,andthatforngrowing,Theorem10supersedesTheorem4ifrissucientlylargeascomparedtoq(roughly,r&#x-278;Cq=logqwithasuitableconstantC).AslightlymorepreciseversionofCorollary11isthatthereexistsaKakeyasetKFnqofrankrwithjKjqn�bn=qrc+r;thisisessentiallyequivalenttoTheorem10providedthatn(r+1)qr.(Ontheotherhand,Theorem10becomestrivialifnqr.)TheremainderofthepaperismostlydevotedtotheproofsofTheorems1,6,7,and8,andLemma9.Fortheconvenienceofthereaderandself-completeness,wealsoprove(aslightlygeneralizedversionof)Lemma3intheAppendix.Section6containsashortsummaryandconcludingremarks.2.ProofofTheorem1.AsapreparationfortheproofofTheorem1,webrie yreviewsomebasicnotionsandresultsrelatedtothepolynomialmethod;thereaderisreferredto[DKSS]foranin-depthtreatmentandproofs.Fortherestofthissectionweusemultidimensionalformalvariables,whicharetobeunderstoodjustasn-tuplesof\regular"formalvariableswithasuitablen.Thus,forinstance,ifnisapositiveintegerandFisaeld,wecanwriteX=(X1;:::;Xn)andP2F[X],meaningthatPisapolynomialinthenvariablesX1;:::;XnoverF.ByN0wedenotethesetofnon-negativeintegers,andforXasaboveandann-tuplei=(i1;:::;in)2Nn0weletkik:=i1++inandXi:=Xi11Xinn.LetFbeaeld,n1aninteger,andX=(X1;:::;Xn)andY=(Y1;:::;Yn)formalvariables.ToeverypolynomialPinnvariablesoverFandeveryn-tuplei2Nn0therecorrespondsauniquelydenedpolynomialP(i)overFinnvariablessuchthatP(X+Y)=Xi2Nn0P(i)(Y)Xi:ThepolynomialP(i)iscalledtheHassederivativeofPoforderi.Notice,thatP(0)=P(whichfollows,forinstance,bylettingX=(0;:::;0)),andifkik&#x-277;degP,thenP(i)=0.Also,itiseasytocheckthatifPHdenotesthehomogeneouspartofP(meaningthatPHisahomogeneouspolynomialsuchthatdeg(P�PH)degP),and(P(i))HdenotesthehomogeneouspartofP(i),then(P(i))H=(PH)(i). 8S.KOPPARTY,V.LEV,S.SARAF,ANDM.SUDAN(notypo:kentersbothsides!),weshowrstthatm+n�1njKjn+kn;(6)andthenoptimizebymandk.Supposeforacontradictionthat(6)fails;thus,byLemma13,thereexistsanon-zeropolynomialPoverFqofdegreeatmostkinnvariables,vanishingateverypointofKwithmultiplicityatleastm.Writel:=lqm�k q�1mandxi=(i1;:::;in)2Nn0satisfyingw:=kikl.LetQ:=P(i),theithHassederivativeofP.SinceKisaKakeyasetofrankr,foreveryd1;:::;dr2Fnqthereexistsb2Fnqsuchthatb+t1d1++trdr2Kforallt1;:::;tk2Fq;hence,(P;b+t1d1++trdr)m;andtherefore,byLemma12,(Q;b+t1d1++trdr)m�wwhenevert1;:::;tr2Fq.ByLemma14,wehave(Q;b+t1d1++trdr)(Q(b+T1d1++Trdr);(t1;:::;tr));whereQ(b+T1d1++Trdr)isconsideredasapolynomialinthevariablesT1;:::;Tr.Thus,foreveryd1;:::;dr2Fnqthereexistsb2FnqsuchthatQ(b+T1d1++Trdr)vanisheswithmultiplicityatleastm�wateachpoint(t1;:::;tr)2Frq.ComparedwithdegQ(b+T1d1++Trdr)degQk�wq(m�w)(asitfollowsfromwl),inviewofCorollary16thisshowsthatQ(b+T1d1++Trdr)isthezeropolynomial.LetPHandQHdenotethehomogeneouspartsofthepolynomialsPandQ,respec-tively,sothatQ(b+T1d1++Trdr)=0impliesQH(T1d1++Trdr)=0.Thus,(PH)(i)(T1d1++Trdr)=0foralld1;:::;dr2Fnq.Weinterpretthissayingthat(PH)(i),consideredasapolynomialinnvariablesovertheeldofrationalfunctionsFq(T1;:::;Tr),vanishesateverypointofthesetfT1d1++Trdr:d1;:::;dr2Fnqg=Sn;whereS:=f 1T1++ rTr: 1;:::; r2Fqg:ThisshowsthatallHassederivativesofPHoforder,smallerthanl,vanishonSn;inotherwords,PHvanisheswithmultiplicityatleastlateverypointofSn.Since,onthe 10S.KOPPARTY,V.LEV,S.SARAF,ANDM.SUDANcompletingtheproof.TheassertionofTheorem6forqoddfollowsimmediatelyfromLemma17uponchoos-ingF:=Fqandf(x):=x2,andobservingthatthenjIf(t)j=(q+1)=2foreacht2Finviewofx2+tx=(x+t=2)2�t2=4:InthecaseofqeventheassertionfollowseasilybycombiningLemma17withthefollowingtwopropositions.Proposition18.Supposethatqisanevenpowerof2andletf(x):=x3(x2Fq).Thenforeveryt2FqwehavejIf(t)j(2q+1)=3.Proposition19.Supposethatqisanoddpowerof2andletf(x):=xq�2+x2(x2Fq).Thenforeveryt2FqwehavejIf(t)j2(q+p q+1)=3.TocompletetheproofofTheorem6itremainstoprovePropositions18and19.Forthisweneedthefollowingwell-knownfact.Lemma20.Supposethatqisapowerof2,andletTrdenotethetracefunctionfromtheeldFqtoitstwo-elementsubeld.For ;; 2Fqwith 6=0,thenumberofsolutionsoftheequation x2+x+ =0inthevariablex2Fqis8�&#x]TJ ;� -2;.52;&#x Td ;&#x[000;:1if=0;0if6=0andTr( =2)=1;2if6=0andTr( =2)=0:ProofofProposition18.Theassumptionthatqisanevenpowerof2impliesthatq�1isdivisibleby3.Consequently,Fqcontains(q�1)=3+1(2q+1)=3cubes,andweassumebelowthatt6=0.Forx;y2Fqwewritexyifx3+tx=y3+ty.Clearly,thisdenesanequivalencerelationonFq,andjIf(t)jisjustthenumberofequivalenceclasses.Sincetheequationx3+tx=0hasexactlytwosolutions,whichare0andp t,thesetf0;p tgisanequivalenceclass.Fixnowx=2f0;p tgandconsidertheequivalenceclassofx.Forxytoholditisnecessaryandsucientthateithery2+xy+x2=t,orx=y,andthesetwoconditionscannotholdsimultaneouslyinviewofx6=p t.Hence,withTrdenedasinLemma20,andusingtheassertionofthelemma,thenumberofelementsintheequivalenceclassofxis(1ifTr((x2+t)=x2)=1;3ifTr((x2+t)=x2)=0:AsxrunsoverallelementsofFqnf0;p tg,theexpression(x2+t)=x2runsoverallelementsofFqnf0;1g.Sinceqisanevenpowerof2,wehaveTr(1)=Tr(0)=0;thus,thereareq=2�2valuesofx=2f0;p tgwithTr((x2+t)=x2)=0. 12S.KOPPARTY,V.LEV,S.SARAF,ANDM.SUDANToestimateNwenoticethat1 x(x+t)2=1 t2x+1 t2(x+t)+1 t(x+t)2;andthatTr1 t(x+t)2=Tr1 p t(x+t);implyingTr1 x(x+t)2=Tr1 t2x+1 t2+1 p t1 x+t=Trx=p t+1=t x(x+t):Thus,ift=1,thenTr1 x(x+t)2=Tr1 x;showingthatN=#fx2Fqnf0;1g:Tr(1=x)=0g=q=2�1(astheassumptionthatqisanoddpowerof2impliesTr(1)=1),andhencejIf(1)jq�2 3(q=2�1)=2q+2 3by(7).Finally,supposethatt=2f0;1g.ForbrevitywewriteR(x):=x=p t+1=t x(x+t);andlet denotetheadditivecharacteroftheeldFq,denedby (x)=(�1)Tr(x);x2Fq:SinceR(1=p t)=0,wehaveN=1 2Xx2Fqnf0;t;1=p tg�1+ (R(x))=1 2Xx2Fqnf0;tg (R(x))+q 2�2:UsingWeil'sbound(aslaidout,forinstance,in[MM91,Theorem2]),wegetNq 2�2�1 2(2p q+1)=q 2�p q�5 2:Now(7)givesjIf(t)jq�2 3�(q=2)�p q�(5=2)�1=2(q+p q+1) 3; 14S.KOPPARTY,V.LEV,S.SARAF,ANDM.SUDANsothats+l=kands+2lq:(8)Thenq�1 2e(�t)=Xi2[1;k]:mi2mi2l(q�s)=l2=1 2(q�s)q�s l�1=1 2l(q�s)(q�k):Ifwehadkq=2,thiswouldyieldq 2�q�1 21 2l(q�s)q 2;contradicting(8).4.ProofofTheorems7and8.ProofofTheorem7.Givenavectord="1e1++"nenwith"1;:::;"n2Fq,letb:=Xi2[1;n]:"i=0ei:Thus,b2B,anditisreadilyveriedthatfort2Fqwehaveb+td2A.Therefore,thelinethroughbinthedirectiondisentirelycontainedinK.TheassertiononthesizeofKfollowsfromA\B=fe1++eng.ProofofTheorem8.Wenoticethattheassertionistrivialifn=O(q(lnq)3),asinthiscaseforasucientlylargeconstantCwehaveq 22=qn+Cp nlnq=q�qn;consequently,weassumen�32q(lnq)3(9)fortherestoftheproof.Fixalinearbasisfe1;:::;engFnqand,asinTheorem7,letA:=f"1e1++"nen:"1;:::;"n2FqgandB:=f"1e1++"nen:"1;:::;"n2f0;1gg:Givenavectorv="1e1++"nenwith"1;:::;"n2Fqandascalar"2Fq,let"(v)denotethenumberofthoseindicesi2[1;n]with"i=".Set:=2p lnqanddeneD0:=fd2Fnq:"(d)�n=q�(n=q)1=2forall"2Fqg 16S.KOPPARTY,V.LEV,S.SARAF,ANDM.SUDANHencenH(N=n)+(n�N)ln(q�2)=nH(2=q)+n(1�2=q)ln(q�2)+O�(n=q)1=2(lnq)3=2=nlnq�2 qln2+O�(n=q)1=2(lnq)3=2;implyingjA0jq 22=qnexp�O�(n=q)1=2(lnq)3=2:Sinceq=22=q�2forq3,weconcludethatjK0jjA0j+jBjq 22=qnexp�O�(n=q)1=2(lnq)3=2=q 22=qn+O�p nlnq=q:WenowusetherandomrotationtricktoreplaceK0withaslightlylargersetKcontaininglinesinall(notonlypopular)directions.TothisendwechoseatrandomlinearautomorphismsT1;:::;TnofthevectorspaceFnqandsetK:=T1(K0)[[Tn(K0):Thus,KcontainsalineineverydirectionfromthesetD:=T1(D0)[[Tn(D0):Choosingavectord2Fnqnf0gatrandom,foreachxedj2[1;n]theprobabilitythatd=2Tj(D0)isatmost1=q,whencetheprobabilitythatd=2Disatmostq�n.Hence,theprobabilitythatD6=Fnqnf0gissmallerthan1,showingthatT1;:::;TncanbeinstantiatedsothatKisarank-1Kakeyaset.ItremainstonoticethatjKjnjK0j.5.ProofofLemma9.Ifk�n,thentheassertionofthelemmaistrivial;suppose,therefore,thatkn,andletthenm:=bn=kc.FixadecompositionFnq=V0V1Vk,whereV0;V1;:::;VkFnqaresubspaceswithdimVi=mfori=1;:::;k,andforeachi2[0;k]letidenotetheprojectionofFnqontoVialongtheremainderofthedirectsum;thus,v=0(v)+1(v)++k(v)foreveryvectorv2Fnq.Finally,letU:=fu2Fnq:i(u)=0foratleastoneindex1ikg:AsimplecomputationconrmsthatthesizeofUisasclaimed.ToseewhyUcontainsatranslateofeveryk-elementsubsetofFnq,givensuchasubsetfa1;:::;akgweletb:=�1(a1)��k(ak)andobservethat,foreachi2[1;k],i(b+ai)=i(b)+i(ai)=0;whenceb+ai2U. 18S.KOPPARTY,V.LEV,S.SARAF,ANDM.SUDANReferences[ABS]N.Alon,B.Bukh,andB.Sudakov,DiscreteKakeya-typeproblemsandsmallbases,IsraelJ.Math.,toappear.[AS08]N.AlonandJ.H.Spencer,Theprobabilisticmethod,Thirdedition,Wiley-InterscienceSeriesinDiscreteMathematicsandOptimization,JohnWiley&Sons,Inc.,Hoboken,NJ,2008.xviii+352pp.[DKSS]Z.Dvir,S.Kopparty,S.Saraf,andM.Sudan,Extensionstothemethodofmultiplicities,withapplicationstoKakeyasetsandmergers,Submitted.[EOT]J.Ellenberg,R.Oberlin,andT.Tao,TheKakeyasetandmaximalconjecturesforalgebraicvarietiesoverniteelds,Mathematika56(1)(2009),1{25.[MM91]C.J.MorenoandO.Moreno,ExponentialsumsandGoppacodes.I,Proc.Amer.Math.Soc.111(2)(1991),523{531.[MT04]G.MockenhauptandT.Tao,RestrictionandKakeyaphenomenaforniteelds,DukeMath.J.121(1)(2004),35{74.[SS08]S.SarafandM.Sudan,AnimprovedlowerboundonthesizeofKakeyasetsoverniteelds,Anal.PDE1(3)(2008),375{379.ComputerScienceandArtificialIntelligenceLaboratory,MIT,32VassarStreet,Cambridge,MA02139,USAE-mailaddress:swastik@mit.eduDepartmentofMathematics,TheUniversityofHaifaatOranim,Tivon36006,IsraelE-mailaddress:seva@math.haifa.ac.ilComputerScienceandArtificialIntelligenceLaboratory,MIT,32VassarStreet,Cambridge,MA02139,USAE-mailaddress:shibs@mit.eduMicrosoftResearch,OneMemorialDrive,Cambridge,MA02142,USAE-mailaddress:madhu@mit.edu