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1、Ann.Comb.16(2012)121–188AnnalsofCombinatoricsDOI10.1007/s00026-011-0124-3PublishedonlineDecember24,2011©SpringerBaselAG2011TheInverseConjecturefortheGowersNormoverFiniteFieldsinLowCharacteristic∗TerenceTao1andTamarZiegler21UCLADepartmentofMathematics
2、,LosAngeles,CA90095-1596,USAtao@math.ucla.edu2DepartmentofMathematics,Technion-IsraelInstituteofTechnology,Haifa32000,Israeltamarzr@tx.technion.ac.ilReceivedJanuary07,2011MathematicsSubjectClassification:11B30,11T06Abstract.Weestablishtheinverseconjec
3、turefortheGowersnormoverfinitefields,whichasserts(roughlyspeaking)thatifaboundedfunctionf:V→Conafinite-dimensionalvectorspaceVoverafinitefieldFhaslargeGowersuniformitynormfUs+1(V),thenthereexistsa(non-classical)polynomialP:V→Tofdegreeatmostssuchthatfcor
4、relateswiththephasee(P)=e2πiP.Thisconjecturehadalreadybeenestablishedinthe“highcharacteristiccase”,whenthecharacteristicofFisatleastaslargeass.OurproofreliesontheweakformoftheinverseconjectureestablishedearlierbytheauthorsandBergelson[3],togetherwith
5、newresultsonthestructureandequidistributionofnon-classicalpolynomials,inthespiritoftheworkofGreenandthefirstauthor[22]andofKaufmanandLovett[28].Keywords:finitefields,polynomials,Gowersuniformitynorms1.Introduction1.1.TheInverseConjectureLetF=Fpbeafinitefi
6、eldofprimeorderchar(F)=p.Throughoutthispaper,Fwillbeconsideredfixed(e.g.,F=F2orF=F3),andtheterm“vectorspace”willbeshorthandfor“vectorspaceoverF”,andmoregenerallyanylinearalgebraterm(e.g.,span,independence,basis,subspace,lineartransformation,etc.)willb
7、eunderstoodtobeoverthefieldFunlessotherwisestated.IfVisavectorspace,f:V→Cisafunction,andh∈Visashift,wedefinethemultiplicativederivativeΔ•hf:V→CoffbytheformulaΔ•hf:=(Thf)f,∗ThefirstauthorissupportedbyagrantfromtheMacArthurFoundation,andbyNSFgrantCCF-0649
8、473.ThesecondauthorissupportedbyISFgrant557/08,andbyanAlonfellowship.479122T.TaoandT.ZieglerwheretheshiftoperatorThwithshifthisdefinedbyThf(x):=f(x+h).IfVisfinite,andd1isaninteger,wedefinetheGowersuniformitynormfUd(V)bytheformulaE1/2dfUd(V):=h1
