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This page intentionally left blank CAMBRIDGESTUDIESIN ADVANCEDMATHEMATICS105 EDITORIALBOARD B.BOLLOBAS,W.FULTON,A.KATOK,F.KIRWAN, P.SARNAK,B.SIMON,B.TOTARO ADDITIVECOMBINATORICS Additivecombinatoricsisthetheoryofcountingadditivestructuresinsets.Thistheory hasseenexcitingdevelopmentsanddramaticchangesindirectioninrecentyears,thanks toitsconnectionswithareassuchasnumbertheory,ergodictheoryandgraphtheory.This graduateleveltextbookwillallowstudentsandresearcherseasyentryintothisfascinating field.Here,forthefirsttime,theauthorsbringtogether,inaself-containedandsystematic manner,themanydifferenttoolsandideasthatareusedinthemoderntheory,presenting theminanaccessible,coherent,andintuitivelyclearmanner,andprovidingimmediate applicationstoproblemsinadditivecombinatorics.Thepowerofthesetoolsiswell demonstratedinthepresentationofrecentadvancessuchastheGreen-Taotheoremon arithmeticprogressionsandErdo˝sdistanceproblems,andthedevelopingfieldof sum-productestimates.Thetextissupplementedbyalargenumberofexercisesandnew material. Terence Tao isaprofessorintheDepartmentofMathematicsattheUniversityof California,LosAngeles. Van Vu isaprofessorintheDepartmentofMathematicsatRutgersUniversity, NewJersey. CAMBRIDGESTUDIESINADVANCEDMATHEMATICS EditorialBoard: B.Bolloba´s,W.Fulton,A.Katok,F.Kirwan,P.Sarnak,B.Simon,B.Totaro Authetitle,listedbelowcanbeobtainedfromgoodbooksellersorfromCambridgeUniversityPress foracompletelistingvisitwww.cambridge.org/uk/series/&Series.asp?code=CSAM. 49 R.StanleyEnumerativecombinatoricsI 50 I.PorteousCliffordalgebrasandtheclassicalgroups 51 M.AudinSpinningtops 52 V.JurdjevicGeometriccontroltheory 53 H.VolkleinGroupsasGaloisgroups 54 J.LePotierLecturesonvectorbundles 55 D.BumpAutomorphicformsandrepresentations 56 G.LaumonCohomologyofDrinfeldmodularvarietiesII 57 D.M.Clark&B.A.DaveyNaturaldualitiesfortheworkingalgebraist 58 J.McClearyAuser’sguidetospectralsequencesII 59 P.TaylorPracticalfoundationsofmathematics 60 M.P.Brodmann&R.Y.SharpLocalcohomology 61 J.D.Dixonetal.Analyticpro-Pgroups 62 R.StanleyEnumerativecombinatoricsII 63 R.M.DudleyUniformcentrallimittheorems 64 J.Jost&X.Li-JostCalculusofvariations 65 A.J.Berrick&M.E.KeatingAnintroductiontoringsandmodules 66 S.MorosawaHolomorphicdynamics 67 A.J.Berrick&M.E.KeatingCategoriesandmoduleswithK-theoryinview 68 K.SatoLevyprocessesandinfinitelydivisibledistributions 69 H.HidaModularformsandGaloiscohomology 70 R.Iorio&V.IorioFourieranalysisandpartialdifferentialequations 71 R.BleiAnalysisinintegerandfractionaldimensions 72 F.Borceaux&G.JanelidzeGaloistheories 73 B.BollobasRandomgraphs 74 R.M.DudleyRealanalysisandprobability 75 T.Sheil-SmallComplexpolynomials 76 C.VoisinHodgetheoryandcomplexalgebraicgeometryI 77 C.VoisinHodgetheoryandcomplexalgebraicgeometryII 78 V.PaulsenCompletelyboundedmapsandoperatoralgebras 79 F.Gesztesy&H.HoldenSolitonEquationsandtheirAlgebro-GeometricSolutionsVolume1 81 ShigeruMukaiAnIntroductiontoInvariantsandModuli 82 G.TourlakisLecturesinlogicandsettheoryI 83 G.TourlakisLecturesinlogicandsettheoryII 84 R.A.BaileyAssociationSchemes 85 JamesCarlson,StefanMu¨ller-Stach,&ChrisPetersPeriodMappingsandPeriodDomains 86 J.J.Duistermaat&J.A.C.KolkMultidimensionalRealAnalysisI 87 J.J.Duistermaat&J.A.C.KolkMultidimensionalRealAnalysisII 89 M.Golumbic&A.N.TrenkToleranceGraphs 90 L.H.HarperGlobalMethodsforCombinatorialIsoperimetricProblems 91 I.Moerdijk&J.MrcunIntroductiontoFoliationsandLieGroupoids 92 JańosKolla´r,KarenE.Smith,&AlessioCortiRationalandNearlyRationalVarieties 93 DavidApplebaumLe´vyProcessesandStochasticCalculus 95 MartinSchechterAnIntroductiontoNonlinearAnalysis Additive Combinatorics TERENCE TAO, VAN VU cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press TheEdinburghBuilding,Cambridgecb22ru,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridg e.org /9780521853866 © Cambridge University Press 2006 Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. Firstpublishedinprintformat 2006 isbn-13 978-0-511-24530-5eBook(EBL) isbn-10 0-511-24530-0 eBook(EBL) isbn-13 978-0-521-85386-6hardback isbn-10 0-521-85386-9 hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyofurls forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. Toourfamilies Contents Prologue pagexi 1 Theprobabilisticmethod 1 1.1 Thefirstmomentmethod 2 1.2 Thesecondmomentmethod 6 1.3 Theexponentialmomentmethod 9 1.4 Correlationinequalities 19 1.5 TheLova´szlocallemma 23 1.6 Janson’sinequality 27 1.7 Concentrationofpolynomials 33 1.8 Thinbasesofhigherorder 37 1.9 ThinWaringbases 42 1.10 Appendix:thedistributionoftheprimes 45 2 Sumsetestimates 51 2.1 Sumsets 54 2.2 Doublingconstants 57 2.3 Ruzsadistanceandadditiveenergy 59 2.4 Coveringlemmas 69 2.5 TheBalog–Szemere´di–Gowerstheorem 78 2.6 Symmetrysetsandimbalancedpartialsumsets 83 2.7 Non-commutativeanalogs 92 2.8 Elementarysum-productestimates 99 3 Additivegeometry 112 3.1 Additivegroups 113 3.2 Progressions 119 3.3 Convexbodies 122 vii viii Contents 3.4 TheBrunn–Minkowskiinequality 127 3.5 Intersectingaconvexsetwithalattice 130 3.6 Progressionsandproperprogressions 143 4 Fourier-analyticmethods 149 4.1 Basictheory 150 4.2 Lp theory 156 4.3 Linearbias 160 4.4 Bohrsets 165 4.5 (cid:2)(p)constants, B [g]sets,anddissociatedsets 172 h 4.6 Thespectrumofanadditiveset 181 4.7 Progressionsinsumsets 189 5 Inversesumsettheorems 198 5.1 Minimalsizeofsumsetsandthee-transform 198 5.2 Sumsetsinvectorspaces 211 5.3 Freimanhomomorphisms 220 5.4 Torsionandtorsion-freeinversetheorems 227 5.5 Universalambientgroups 233 5.6 Freiman’stheoreminanarbitrarygroup 239 6 Graph-theoreticmethods 246 6.1 BasicNotions 247 6.2 Independentsets,sum-freesubsets,andSidonsets 248 6.3 Ramseytheory 254 6.4 ProofoftheBalog–Szemere´di–Gowerstheorem 261 6.5 Plu¨nnecke’stheorem 267 7 TheLittlewood–Offordproblem 276 7.1 Thecombinatorialapproach 277 7.2 TheFourier-analyticapproach 281 7.3 TheEsseénconcentrationinequality 290 7.4 InverseLittlewood–Offordresults 292 7.5 RandomBernoullimatrices 297 7.6 ThequadraticLittlewood–Offordproblem 304 8 Incidencegeometry 308 8.1 Thecrossingnumberofagraph 308 8.2 TheSzemere´di–Trottertheorem 311 8.3 Thesum-productprobleminR 315 8.4 Celldecompositionsandthedistinctdistancesproblem 319 8.5 Thesum-productprobleminotherfields 325