Dynamics and Analytic Number Theory PDF
Preview Dynamics and Analytic Number Theory
LONDONMATHEMATICALSOCIETYLECTURENOTESERIES ManagingEditor:ProfessorM.Reid,MathematicsInstitute, UniversityofWarwick,CoventryCV47AL,UnitedKingdom Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressat http://www.cambridge.org/mathematics 314 Spectralgeneralizationsoflinegraphs,D.CVETKOVIC´,P.ROWLINSON&S.SIMIC´ 315 Structuredringspectra,A.BAKER&B.RICHTER(eds) 316 Linearlogicincomputerscience,T.EHRHARD,P.RUET,J.-Y.GIRARD&P.SCOTT(eds) 317 Advancesinellipticcurvecryptography,I.F.BLAKE,G.SEROUSSI&N.P.SMART(eds) 318 Perturbationoftheboundaryinboundary-valueproblemsofpartialdifferentialequations,D.HENRY 319 DoubleaffineHeckealgebras,I.CHEREDNIK 320 L-functionsandGaloisrepresentations,D.BURNS,K.BUZZARD&J.NEKOVÁRˇ(eds) 321 Surveysinmodernmathematics,V.PRASOLOV&Y.ILYASHENKO(eds) 322 Recentperspectivesinrandommatrixtheoryandnumbertheory,F.MEZZADRI&N.C.SNAITH(eds) 323 Poissongeometry,deformationquantisationandgrouprepresentations,S.GUTTetal(eds) 324 Singularitiesandcomputeralgebra,C.LOSSEN&G.PFISTER(eds) 325 LecturesontheRicciflow,P.TOPPING 326 ModularrepresentationsoffinitegroupsofLietype,J.E.HUMPHREYS 327 Surveysincombinatorics2005,B.S.WEBB(ed) 328 Fundamentalsofhyperbolicmanifolds,R.CANARY,D.EPSTEIN&A.MARDEN(eds) 329 SpacesofKleiniangroups,Y.MINSKY,M.SAKUMA&C.SERIES(eds) 330 Noncommutativelocalizationinalgebraandtopology,A.RANICKI(ed) 331 Foundationsofcomputationalmathematics,Santander2005,L.MPARDO,A.PINKUS,E.SÜLI&M.J.TODD (eds) 332 Handbookoftiltingtheory,L.ANGELERIHÜGEL,D.HAPPEL&H.KRAUSE(eds) 333 Syntheticdifferentialgeometry(2ndEdition),A.KOCK 334 TheNavier–Stokesequations,N.RILEY&P.DRAZIN 335 Lecturesonthecombinatoricsoffreeprobability,A.NICA&R.SPEICHER 336 Integralclosureofideals,rings,andmodules,I.SWANSON&C.HUNEKE 337 MethodsinBanachspacetheory,J.M.F.CASTILLO&W.B.JOHNSON(eds) 338 Surveysingeometryandnumbertheory,N.YOUNG(ed) 339 GroupsStAndrews2005I,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds) 340 GroupsStAndrews2005II,C.M.CAMPBELL,M.R.QUICK,E.F.ROBERTSON&G.C.SMITH(eds) 341 Ranksofellipticcurvesandrandommatrixtheory,J.B.CONREY,D.W.FARMER,F.MEZZADRI&N.C. SNAITH(eds) 342 Ellipticcohomology,H.R.MILLER&D.C.RAVENEL(eds) 343 AlgebraiccyclesandmotivesI,J.NAGEL&C.PETERS(eds) 344 AlgebraiccyclesandmotivesII,J.NAGEL&C.PETERS(eds) 345 Algebraicandanalyticgeometry,A.NEEMAN 346 Surveysincombinatorics2007,A.HILTON&J.TALBOT(eds) 347 Surveysincontemporarymathematics,N.YOUNG&Y.CHOI(eds) 348 Transcendentaldynamicsandcomplexanalysis,P.J.RIPPON&G.M.STALLARD(eds) 349 ModeltheorywithapplicationstoalgebraandanalysisI,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY& A.WILKIE(eds) 350 ModeltheorywithapplicationstoalgebraandanalysisII,Z.CHATZIDAKIS,D.MACPHERSON,A.PILLAY& A.WILKIE(eds) 351 FinitevonNeumannalgebrasandmasas,A.M.SINCLAIR&R.R.SMITH 352 Numbertheoryandpolynomials,J.MCKEE&C.SMYTH(eds) 353 Trendsinstochasticanalysis,J.BLATH,P.MÖRTERS&M.SCHEUTZOW(eds) 354 Groupsandanalysis,K.TENT(ed) 355 Non-equilibriumstatisticalmechanicsandturbulence,J.CARDY,G.FALKOVICH&K.GAWEDZKI 356 EllipticcurvesandbigGaloisrepresentations,D.DELBOURGO 357 Algebraictheoryofdifferentialequations,M.A.H.MACCALLUM&A.V.MIKHAILOV(eds) 358 Geometricandcohomologicalmethodsingrouptheory,M.R.BRIDSON,P.H.KROPHOLLER&I.J.LEARY(eds) 359 Modulispacesandvectorbundles,L.BRAMBILA-PAZ,S.B.BRADLOW,O.GARCÍA-PRADA&S. RAMANAN(eds) 360 Zariskigeometries,B.ZILBER 361 Words:Notesonverbalwidthingroups,D.SEGAL 362 Differentialtensoralgebrasandtheirmodulecategories,R.BAUTISTA,L.SALMERÓN&R.ZUAZUA 363 Foundationsofcomputationalmathematics,HongKong2008,F.CUCKER,A.PINKUS&M.J.TODD(eds) 364 Partialdifferentialequationsandfluidmechanics,J.C.ROBINSON&J.L.RODRIGO(eds) 365 Surveysincombinatorics2009,S.HUCZYNSKA,J.D.MITCHELL&C.M.RONEY-DOUGAL(eds) 366 Highlyoscillatoryproblems,B.ENGQUIST,A.FOKAS,E.HAIRER&A.ISERLES(eds) 367 Randommatrices:Highdimensionalphenomena,G.BLOWER 368 GeometryofRiemannsurfaces,F.P.GARDINER,G.GONZÁLEZ-DIEZ&C.KOUROUNIOTIS(eds) 369 Epidemicsandrumoursincomplexnetworks,M.DRAIEF&L.MASSOULIÉ 370 Theoryofp-adicdistributions,S.ALBEVERIO,A.YU.KHRENNIKOV&V.M.SHELKOVICH 371 Conformalfractals,F.PRZYTYCKI&M.URBAN´SKI 372 Moonshine:Thefirstquartercenturyandbeyond,J.LEPOWSKY,J.MCKAY&M.P.TUITE(eds) 373 Smoothness,regularityandcompleteintersection,J.MAJADAS&A.G.RODICIO 374 Geometricanalysisofhyperbolicdifferentialequations:Anintroduction,S.ALINHAC 375 Triangulatedcategories,T.HOLM,P.JØRGENSEN&R.ROUQUIER(eds) 376 Permutationpatterns,S.LINTON,N.RUŠKUC&V.VATTER(eds) 377 AnintroductiontoGaloiscohomologyanditsapplications,G.BERHUY 378 Probabilityandmathematicalgenetics,N.H.BINGHAM&C.M.GOLDIE(eds) 379 Finiteandalgorithmicmodeltheory,J.ESPARZA,C.MICHAUX&C.STEINHORN(eds) 380 Realandcomplexsingularities,M.MANOEL,M.C.ROMEROFUSTER&C.T.CWALL(eds) 381 Symmetriesandintegrabilityofdifferenceequations,D.LEVI,P.OLVER,Z.THOMOVA&P.WINTERNITZ (eds) 382 Forcingwithrandomvariablesandproofcomplexity,J.KRAJÍCˇEK 383 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryI,R.CLUCKERS,J. NICAISE&J.SEBAG(eds) 384 Motivicintegrationanditsinteractionswithmodeltheoryandnon-ArchimedeangeometryII,R.CLUCKERS, J.NICAISE&J.SEBAG(eds) 385 EntropyofhiddenMarkovprocessesandconnectionstodynamicalsystems,B.MARCUS,K.PETERSEN& T.WEISSMAN(eds) 386 Independence-friendlylogic,A.L.MANN,G.SANDU&M.SEVENSTER 387 GroupsStAndrews2009inBathI,C.M.CAMPBELLetal(eds) 388 GroupsStAndrews2009inBathII,C.M.CAMPBELLetal(eds) 389 Randomfieldsonthesphere,D.MARINUCCI&G.PECCATI 390 Localizationinperiodicpotentials,D.E.PELINOVSKY 391 Fusionsystemsinalgebraandtopology,M.ASCHBACHER,R.KESSAR&B.OLIVER 392 Surveysincombinatorics2011,R.CHAPMAN(ed) 393 Non-abelianfundamentalgroupsandIwasawatheory,J.COATESetal(eds) 394 Variationalproblemsindifferentialgeometry,R.BIELAWSKI,K.HOUSTON&M.SPEIGHT(eds) 395 Howgroupsgrow,A.MANN 396 Arithmeticdifferentialoperatorsoverthep-adicintegers,C.C.RALPH&S.R.SIMANCA 397 Hyperbolicgeometryandapplicationsinquantumchaosandcosmology,J.BOLTE&F.STEINER(eds) 398 Mathematicalmodelsincontactmechanics,M.SOFONEA&A.MATEI 399 Circuitdoublecoverofgraphs,C.-Q.ZHANG 400 Densespherepackings:ablueprintforformalproofs,T.HALES 401 AdoubleHallalgebraapproachtoaffinequantumSchur–Weyltheory,B.DENG,J.DU&Q.FU 402 Mathematicalaspectsoffluidmechanics,J.C.ROBINSON,J.L.RODRIGO&W.SADOWSKI(eds) 403 Foundationsofcomputationalmathematics,Budapest2011,F.CUCKER,T.KRICK,A.PINKUS& A.SZANTO(eds) 404 Operatormethodsforboundaryvalueproblems,S.HASSI,H.S.V.DESNOO&F.H.SZAFRANIEC(eds) 405 Torsors,étalehomotopyandapplicationstorationalpoints,A.N.SKOROBOGATOV(ed) 406 Appalachiansettheory,J.CUMMINGS&E.SCHIMMERLING(eds) 407 Themaximalsubgroupsofthelow-dimensionalfiniteclassicalgroups,J.N.BRAY,D.F.HOLT& C.M.RONEY-DOUGAL 408 Complexityscience:theWarwickmaster’scourse,R.BALL,V.KOLOKOLTSOV&R.S.MACKAY(eds) 409 Surveysincombinatorics2013,S.R.BLACKBURN,S.GERKE&M.WILDON(eds) 410 Representationtheoryandharmonicanalysisofwreathproductsoffinitegroups, T.CECCHERINI-SILBERSTEIN,F.SCARABOTTI&F.TOLLI 411 Modulispaces,L.BRAMBILA-PAZ,O.GARCÍA-PRADA,P.NEWSTEAD&R.P.THOMAS(eds) 412 Automorphismsandequivalencerelationsintopologicaldynamics,D.B.ELLIS&R.ELLIS 413 Optimaltransportation,Y.OLLIVIER,H.PAJOT&C.VILLANI(eds) 414 AutomorphicformsandGaloisrepresentationsI,F.DIAMOND,P.L.KASSAEI&M.KIM(eds) 415 AutomorphicformsandGaloisrepresentationsII,F.DIAMOND,P.L.KASSAEI&M.KIM(eds) 416 Reversibilityindynamicsandgrouptheory,A.G.O’FARRELL&I.SHORT 417 Recentadvancesinalgebraicgeometry,C.D.HACON,M.MUSTAT¸A˘&M.POPA(eds) 418 TheBloch–KatoconjecturefortheRiemannzetafunction,J.COATES,A.RAGHURAM,A.SAIKIA& R.SUJATHA(eds) 419 TheCauchyproblemfornon-Lipschitzsemi-linearparabolicpartialdifferentialequations,J.C.MEYER& D.J.NEEDHAM 420 Arithmeticandgeometry,L.DIEULEFAITetal(eds) 421 O-minimalityandDiophantinegeometry,G.O.JONES&A.J.WILKIE(eds) 422 GroupsStAndrews2013,C.M.CAMPBELLetal(eds) 423 Inequalitiesforgrapheigenvalues,Z.STANIC´ 424 Surveysincombinatorics2015,A.CZUMAJetal(eds) 425 Geometry,topologyanddynamicsinnegativecurvature,C.S.ARAVINDA,F.T.FARRELL&J.-F.LAFONT (eds) 426 Lecturesonthetheoryofwaterwaves,T.BRIDGES,M.GROVES&D.NICHOLLS(eds) 427 RecentadvancesinHodgetheory,M.KERR&G.PEARLSTEIN(eds) 428 GeometryinaFréchetcontext,C.T.J.DODSON,G.GALANIS&E.VASSILIOU 429 Sheavesandfunctionsmodulop,L.TAELMAN 430 RecentprogressinthetheoryoftheEulerandNavier-Stokesequations,J.C.ROBINSON,J.L.RODRIGO, W.SADOWSKI&A.VIDAL-LÓPEZ(eds) 431 Harmonicandsubharmonicfunctiontheoryontherealhyperbolicball,M.STOLL 432 Topicsingraphautomorphismsandreconstruction(2ndEdition),J.LAURI&R.SCAPELLATO 433 RegularandirregularholonomicD-modules,M.KASHIWARA&P.SCHAPIRA 434 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems(2ndEdition),K.TAIRA 435 GradedringsandgradedGrothendieckgroups,R.HAZRAT 436 Groups,graphsandrandomwalks,T.CECCHERINI-SILBERSTEIN,M.SALVATORI&E.SAVA-HUSS(eds) 437 Dynamicsandanalyticnumbertheory,D.BADZIAHIN,A.GORODNIK&N.PEYERIMHOFF(eds) LondonMathematicalSocietyLectureNoteSeries:437 Dynamics and Analytic Number Theory Proceedings of the Durham Easter School 2014 Editedby DZMITRY BADZIAHIN UniversityofDurham ALEXANDER GORODNIK UniversityofBristol NORBERT PEYERIMHOFF UniversityofDurham UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781107552371 (cid:2)c CambridgeUniversityPress2016 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2016 PrintedintheUnitedKingdombyClays,StIvesplc AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloging-in-PublicationData Names:DurhamEasterSchool(2014:UniversityofDurham)|Badziahin, Dmitry(DmitryA.),editor.|Gorodnik,Alexander, 1975–editor.|Peyerimhoff,Norbert,1964–editor. Title:Dynamicsandanalyticnumbertheory:proceedingsoftheDurhamEaster School2014/editedbyDzmitryBadziahin,UniversityofDurham,Alexander Gorodnik,UniversityofBristol,NorbertPeyerimhoff,UniversityofDurham. Description:Cambridge:CambridgeUniversityPress,[2016]| Series:LondonMathematicalSocietylecturenoteseries;437| Includesbibliographicalreferencesandindex. Identifiers:LCCN2016044609|ISBN9781107552371(alk.paper) Subjects:LCSH:Numbertheory–Congresses.|Dynamics–Congresses. Classification:LCCQA241.D872014|DDC512.7/3–dc23 LCrecordavailableathttps://lccn.loc.gov/2016044609 ISBN978-1-107-55237-1Paperback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyInternetWebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchWebsitesis,orwillremain, accurateorappropriate. Contents Listofcontributors pagevii Preface ix 1 MetricDiophantineApproximation:AspectsofRecentWork 1 VictorBeresnevich,FelipeRamírezandSanjuVelani 1.1 Background:DirichletandBad 1 1.2 MetricDiophantineApproximation:TheClassicalLebesgue Theory 9 1.3 Metric Diophantine Approximation: The Classical HausdorffTheory 18 1.4 TheHigher-DimensionalTheory 27 1.5 UbiquitousSystemsofPoints 46 1.6 DiophantineApproximationonManifolds 53 1.7 TheBadlyApproximableTheory 74 2 ExponentsofDiophantineApproximation 96 YannBugeaud 2.1 IntroductionandGeneralities 96 2.2 FurtherDefinitionsandFirstResults 99 2.3 OverviewofKnownRelationsBetweenExponents 106 2.4 BoundsfortheExponentsofApproximation 109 2.5 Spectra 114 2.6 IntermediateExponents 121 2.7 ParametricGeometryofNumbers 124 2.8 Real Numbers Which Are Badly Approximable by AlgebraicNumbers 126 2.9 OpenProblems 127 v vi Contents 3 EffectiveEquidistributionofNilflowsandBoundsonWeylSums 136 GiovanniForni 3.1 Introduction 137 3.2 NilflowsandWeylSums 142 3.3 TheCohomologicalEquation 152 3.4 TheHeisenbergCase 159 3.5 Higher-StepFiliformNilflows 174 4 MultipleRecurrenceandFindingPatternsinDenseSets 189 TimAustin 4.1 Szemerédi’sTheoremandItsRelatives 189 4.2 MultipleRecurrence 192 4.3 BackgroundfromErgodicTheory 197 4.4 MultipleRecurrenceinTermsofSelf-Joinings 212 4.5 WeakMixing 222 4.6 Roth’sTheorem 230 4.7 TowardsConvergenceinGeneral 238 4.8 SatedSystemsandPleasantExtensions 242 4.9 FurtherReading 248 5 DiophantineProblemsandHomogeneousDynamics 258 ManfredEinsiedlerandTomWard 5.1 EquidistributionandtheGaussCircleProblem 258 5.2 CountingPointsinSL (Z)·i⊆H 267 2 5.3 Dirichlet’sTheoremandDani’sCorrespondence 278 6 ApplicationsofThinOrbits 289 AlexKontorovich 6.1 Lecture1:ClosedGeodesics,BinaryQuadraticForms,and Duke’sTheorem 289 6.2 Lecture2:ThreeProblemsinContinuedFractions:ELMV, McMullen,andZaremba 306 6.3 Lecture3:TheThinOrbitsPerspective 310 Index 318 Contributors TimAustin CourantInstitute,NYU NewYork,NY10012,USA Email:[email protected] VictorBeresnevich DepartmentofMathematics,UniversityofYork Heslington,York,Y0105DD,UnitedKingdom Email:[email protected] YannBugeaud Départementdemathématiques,UniversitédeStrasbourg F-67084Strasbourg,France Email:[email protected] ManfredEinsiedler DepartementMathematik,ETHZürich 8092Zürich,Switzerland Email:[email protected] GiovanniForni DepartmentofMathematics,UniversityofMaryland CollegePark,MD20742-4015,USA Email:[email protected] AlexKontorovich DepartmentofMathematics,RutgersUniversity Piscataway,NJ08854,USA Email:[email protected] vii viii Listofcontributors FelipeRamírez DepartmentofMathematicsandComputerScience WesleyanUniversity,Middletown,CT06459,USA Email:[email protected] SanjuVelani DepartmentofMathematics,UniversityofYork Heslington,York,Y0105DD,UnitedKingdom Email:[email protected] TomWard ExecutiveOffice,PalatineCentre,DurhamUniversity DurhamDH13LE,UnitedKingdom Email:[email protected]
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