Dynamics and Analytic Number Theory

341 Pages·2017·1.884 MB·English

by  Dzmitry Badziahin, Alexander Gorodnik, Norbert Peyerimhoff (eds.)

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
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Firstpublished2016
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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
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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]