陶哲轩获菲尔兹奖论文

arXiv:math/0404188v6[math.NT]23Sep2007THEPRIMESCONTAINARBITRARILYLONGARITHMETICPROGRESSIONSBENGREENANDTERENCETAOAbstract.Weprovethattherearearbitrarilylongarithmeticprogressionsofprimes.Therearethreemajoringredients.ThefirstisSzemer´edi’stheorem,whichassertsthatanysubsetoftheintegersofpositivedensitycontainsprogressionsofarbitrarylength.Thesecond,whichisthemainnewingredientofthispaper,isacertaintrans-ferenceprinciple.ThisallowsustodeducefromSzemer´edi’stheoremthatanysubsetofasufficientlypseudorandomset(ormeasure)ofpositiverelativedensitycontainsprogressionsofarbitrarylength.ThethirdingredientisarecentresultofGoldstonandYıldırım,whichwereproducehere.Usingthis,onemayplace(alargefractionof)theprimesinsideapseudorandomsetof“almostprimes”(ormoreprecisely,apseudorandommeasureconcentratedonalmostprimes)withpositiverelativedensity.1.IntroductionItisawell-knownconjecturethattherearearbitrarilylongarithmeticprogressionsofprimenumbers.Theconjectureisbestdescribedas“classical”,ormaybeeven“folklore”.InDickson’sHistoryitisstatedthataround1770LagrangeandWaringinvestigatedhowlargethecommondifferenceofanarithmeticprogressionofLprimesmustbe,anditishardtoimaginethattheydidnotatleastwonderwhethertheirresultsweresharpforallL.Itisnotsurprisingthattheconjectureshouldhavebeenmade,sinceasimpleheuristicbasedontheprimenumbertheoremwouldsuggestthatthereare≫N2/logkNk-tuplesofprimesp1,...,pkinarithmeticprogression,eachpibeingatmostN.HardyandLittlewood[24],intheirfamouspaperof1923,advancedaverygeneralconjecturewhich,asaspecialcase,containsthehypothesisthatthenumberofsuchk-termprogressionsisasymptoticallyCkN2/logkNforacertainexplicitnumericalfactorCk0(wedonotcomeclosetoestablishingthisconjecturehere,obtaininginsteadalowerbound(γ(k)+o(1))N2/logkNforsomeverysmallγ(k)0).ThefirsttheoreticalprogressontheseconjectureswasmadebyvanderCorput[42](seealso[8])who,in1939,usedVinogradov’smethodofprimenumbersumstoestablishthecasek=3,thatistosaythatthereareinfinitelymanytriplesofprimesinarithmeticprogression.However,thequestionoflongerarithmeticprogressionsseemstohaveremainedcompletelyopen(exceptforupperbounds),evenfork=4.Ontheotherhand,ithasbeenknownforsometimethatbetterresultscanbeobtainedifonereplacestheprimeswithaslightlylargersetofalmostprimes.Themostimpressive1991MathematicsSubjectClassification.11N13,11B25,374A5.WhilethisworkwascarriedoutthefirstauthorwasaPIMSpostdoctoralfellowattheUniversityofBritishColumbia,Vancouver,Canada.ThesecondauthorwasaClayPrizeFellowandwassupportedbyagrantfromthePackardFoundation.12BENGREENANDTERENCETAOsuchresultisduetoHeath-Brown[25].Heshowedthatthereareinfinitelymany4-termprogressionsconsistingofthreeprimesandanumberwhichiseitherprimeoraproductoftwoprimes.Inasomewhatdifferentdirection,letusmentionthebeautifulresultsofBalog[2,3].Amongotherthingsheshowsthatforanymtherearemdistinctprimesp1,...,pmsuchthatalloftheaverages12(pi+pj)areprime.Theproblemoffindinglongarithmeticprogressionsintheprimeshasalsoattractedtheinterestofcomputationalmathematicians.Atthetimeofwritingthelongestknownarithmeticprogressionofprimesisoflength23,andwasfoundin2004byMarkusFrind,PaulUnderwood,andPaulJobling:56211383760397+44546738095860k;k=0,1,...,22.Anearlierarithmeticprogressionofprimesoflength22wasfoundbyMoran,PritchardandThyssen[32]:11410337850553+4609098694200k;k=0,1,...,21.Ourmaintheoremresolvestheaboveconjecture.Theorem1.1.Theprimenumberscontaininfinitelymanyarithmeticprogressionsoflengthkforallk.Infact,wecansaysomethingalittlestronger:Theorem1.2(Szemer´edi’stheoremintheprimes).LetAbeanysubsetoftheprimenumbersofpositiverelativeupperdensity,thuslimsupN→∞π(N)−1|A∩[1,N]|0,whereπ(N)denotesthenumberofprimeslessthanorequaltoN.ThenAcontainsinfinitelymanyarithmeticprogressionsoflengthkforallk.Ifonereplaces“primes”inthestatementofTheorem1.2bythesetofallpositiveintegersZ+,thenthisisafamoustheoremofSzemer´edi[38].Thespecialcasek=3ofTheorem1.2wasrecentlyestablishedbythefirstauthor[21]usingmethodsofFourieranalysis.Incontrast,ourmethodsherehaveamoreergodictheoryflavouranddonotinvolvemuchFourieranalysis(thoughtheargumentdoesrelyonSzemer´edi’stheoremwhichcanbeprovenbyeithercombinatorial,ergodictheory,orFourieranalysisarguments).Wealsoremarkthatiftheprimeswerereplacedbyarandomsubsetoftheintegers,withdensityatleastN−1/2+εoneachinterval[1,N],thenthek=3caseoftheabovetheoremwasestablishedin[30].AcknowledgementsTheauthorswouldliketothankJeanBourgain,EnricoBombieri,TimGowers,BrynaKra,ElonLindenstrauss,ImreRuzsa,RomanSasyk,PeterSarnakandKannanSoundararajanforhelpfulconversations.WeareparticularlyindebtedtoAndrewGranvillefordrawingourattentiontotheworkofGoldstonandYıldırım,andtoDanGoldstonformakingthepreprint[17]available.WearealsoindebtedtoYong-GaoChenandhisstudents,BrynaKra,JamieRadcliffe,LiorSilbermanandMarkWatkinsforcorrectionstoearlierversionsofthemanuscript.Weareparticularlyindebtedtotheanonymousrefereesforaverythoroughreadingandmanyhelpfulcorrectionsandsuggestions,whichhavebeenincorporatedintothisversionofthepaper.Portionsofthisworkwerecompletedwhilethefirstauthorwasvisitin