Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics PDF
Preview Ergodic theory and dynamical systems in their interactions with arithmetics and combinatorics
Lecture Notes in Mathematics 2213 Sébastien Ferenczi Joanna Kułaga-Przymus Mariusz Lemańczyk Editors Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics CIRM Jean-Morlet Chair, Fall 2016 Lecture Notes in Mathematics 2213 Editors-in-Chief: Jean-MichelMorel,Cachan BernardTeissier,Paris AdvisoryBoard: MichelBrion,Grenoble CamilloDeLellis,Zurich AlessioFigalli,Zurich DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Athens GáborLugosi,Barcelona MarkPodolskij,Aarhus SylviaSerfaty,NewYork AnnaWienhard,Heidelberg Moreinformationaboutthisseriesathttp://www.springer.com/series/304 Sébastien Ferenczi (cid:129) Joanna Kułaga-Przymus (cid:129) Mariusz Leman´czyk Editors Ergodic Theory and Dynamical Systems in their Interactions with Arithmetics and Combinatorics CIRM Jean-Morlet Chair, Fall 2016 123 Editors SébastienFerenczi JoannaKułaga-Przymus CNRSUMR7373 FacultyofMathematicsandComputer InstitutdeMathématiquesdeMarseille Science Marseille,France NicolausCopernicusUniversity Torun´,Poland MariuszLeman´czyk FacultyofMathematicsandComputer Science NicolausCopernicusUniversity Torun´,Poland ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics ISBN978-3-319-74907-5 ISBN978-3-319-74908-2 (eBook) https://doi.org/10.1007/978-3-319-74908-2 LibraryofCongressControlNumber:2018940898 MathematicsSubjectClassification(2010):37-XX,11-XX,5-XX,51-XX ©SpringerInternationalPublishingAG,partofSpringerNature2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbytheregisteredcompanySpringerInternationalPublishingAGpart ofSpringerNature. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Foreword The interaction between number theory and ergodic theory can be traced to the birth of the latter with Birkhoff’s pointwise ergodic theorem. The early applications were naturally concerned with typical behavior, for example in the metrical theory of diophantine approximation. Most number theoretic questions which can be connected to ergodic theory are concerned with the dynamics of specific orbits or systems constructed from an arithmetic or combinatorial input, and it is with the classification or determination of the basic properties of the possiblesystemsthatcanarisethattheinteractionbecomespowerful.Startingwith Furstenberg’s introduction of such concepts as unique ergodicity, disjointness of dynamicalsystems,andnonconventionalergodicaverages,andthankstoadvances bymanyergodic/numbertheorists,thereisbynowabodyofstrikingapplications. Homogeneousdynamicstakesplaceonparameterspacesofarithmeticobjects,and as a consequence, rigidity theorems such as that of Ratner for unipotent orbits become powerful tools which underlie many of the most striking applications in homogeneous dynamics. There have also been major advances and arithmetic applicationsin variousnonhomogeneousdynamicalsettings. For example it turns out that Vinogradov’s bilinear method in the study of sums over primes for a sequence which is an observable in a dynamical system is intimately connected withtheBirkhoffsumsforjoiningsofthesystemwithitself.Anexampleexploiting this is the proof by Mauduit and Rivat of a conjecture of Gelfond about the distributionof the parity of the sum of the binarydigits of prime numbers.As far ascombinatorial/additivenumbertheory,thepathdevelopedbyFurstenberginhis proofofSzemeredi’stheoremonarithmeticprogressionsinsetsofpositivedensity isatthecenterofthiswell-developedmoderntoolfromergodictheory. Theabovearejustasmallsample(andbiasedtomytasteandknowledge)ofwhat istodayathrivinginteractionbetweenergodictheoryandnumbertheory.Thewell- timed 2016 fall semester activity at CIRM (Luminy) focused on this theme, with the aim of exposing these interactions and the theories that underlie the progress andthelatestdevelopments,aswelladvancingthem.Frommyownexperienceand accountsbyothers,the minicoursesandthe workshopsandseminarswerea great successandtherewereanumberofexcitingnewdevelopments. v vi Foreword Fortunately many of the experts who are responsible for this success prepared and expanded their presentations for this volume. The result is an instructive and insightfulaccountof the basic techniquesfromergodictheoryand numbertheory thathavefacilitatedtherecentdevelopments.Therearealsoexcellentsurveypapers thatbringthereaderuptoforefrontofthelatestdevelopmentsandopenproblems inthisfastmovingarea. Princeton,NJ,USA PeterSarnak October3,2017 Preface This volume consists of minicourses notes, survey, research/survey, and research articles that have arisen as an outcome of workshops, research in pairs, and other scientific work held under the auspices of the Jean Morlet Chair at CIRM betweenAugust1,2016andJanuary31,2017.Thesemesterhadasubstantialcore supportandfundingbyCIRM,Aix-MarseilleUniversity,andthecityofMarseille. Additionally, it was supported by the LABEX Archimède, and the ANR grants of Christian Mauduit (Aix-Marseille University) and Joël Rivat (Aix-Marseille University). The minicourses were those given in the framework of the doctoral school Applications of Ergodic Theory in Number Theory organized by Sébastien Fer- enczi (Aix-Marseille University), Joanna Kułaga-Przymus (Nicolaus Copernicus University Torun´ and Aix-Marseille University), Mariusz Leman´czyk (Nicolaus CopernicusUniversityTorun´),andSergeTroubetzkoy(Aix-MarseilleUniversity). Themainaimofthisschoolwas,ononehand,toprovideparticipantswithmodern methodsof ergodictheory and topologicaldynamicsoriented toward applications in numbertheoryandcombinatorics,and,ontheotherhand,to presentthemwith a broad spectrum of number theory problems that can be treated with the use of suchtools.ThesetaskswererealizedinfourminicoursesbyVitalyBergelson(Ohio State University), “Mutually enriching connections between ergodic theory and combinatorics,” Manfred Einsiedler (ETH Zürich), “Equidistribution on homoge- neousspaces,abridgebetweendynamicsandnumbertheory,”CarlosMatheusSilva Santos(CNRS-UniversitéParis13),“TheLagrangeandMarkovspectrafromthe dynamicalpointofview,”andJoëlRivat“Introductiontoanalyticnumbertheory." The main conference Ergodic Theory and its Connections with Arithmetic andCombinatoricswasorganizedbyJulienCassaigne(Aix-MarseilleUniversity), Sébastien Ferenczi, Pascal Hubert (Aix-Marseille University), Joanna Kułaga- Przymus, Mariusz Leman´czyk with the scientific committee consisting of Artur Avila (University Paris Diderot and IMPA, Rio de Janeiro), Vitaly Bergelson, Mandred Einsiedler, Hillel Furstenberg (The Hebrew University of Jerusalem), Anatole Katok (Penn State University), Christian Mauduit, Imre Ruzsa (Alfred Rényi Institute Budapest), and Peter Sarnak (IAS Princeton).The conferencewas vii viii Preface aimed at interactions between ergodic theory and dynamicalsystems and number theory. Its main subjects were disjointness in ergodic theory and randomness in numbertheory,ergodictheoryandcombinatorialnumbertheory,andhomogenous dynamicsanditsapplications. Importanteventsof the semester were two smaller specialized workshops.The firstoneErgodicTheoryandMöbiusDisjointnesswasorganizedbySébastienFer- enczi, Joanna Kułaga-Przymus,Mariusz Leman´czyk, Christian Mauduit, and Joël Rivat.ThemeetingfocusedontherecentprogressonSarnak’sconjectureonMöbius disjointness: methods, results, and the feedback in ergodic theory. The second one Spectral Theory of Dynamical Systems and Related Topics was organized by AlexanderBufetov(Aix-MarseilleUniversity),SébastienFerenczi,JoannaKułaga- Przymus,Mariusz Leman´czyk,and Arnaldo Nogueira(Aix-MarseilleUniversity). The meeting was aimed at the recent progress in the spectral theory and joinings of dynamical systems, especially, in the recent spectacular progress toward the solutionsof some openclassical problemsof ergodictheory:Rokhlin problemon mixing of all orders, stability of spectral propertiesunder smooth changes for the parabolicsystems,theBanachproblemontheexistenceofdynamicalsystemswith simpleLebesguespectrum,andtheproblemofspectralmultiplicity. The scientific part of the semester was completed by two research in pairs: Dynamical Properties of Systems Determined by Free Points in Lattices and On the Stability of Möbius Disjointnessin TopologicalModels and a special program of invitationswith participationof MichaelBaake (Universityof Bielefeld),Jean- Pierre Conze (University of Rennes 1), Alexandre Danilenko (Institute of Low Temperature,Kharkov),Christian Huck (Universityof Bielefeld), Joanna Kułaga- Przymus,ElHouceinElAbdalaoui(UniversityofRouen),MariuszLeman´czykand ThierrydelaRue(UniversityofRouen). Thecontentsofthisvolumeareasfollows.ItbeginswithPartIwhichisentirely thecourse. (cid:129) JoëlRivat, BasesofAnalytic NumberTheory. Amongotheraspects, the course contains a presentation of the main properties of the Riemann ζ function with a generous introduction to the theory of Dirichlet series. Large sieve method togetherwithabeautifulapplicationtoTwinPrimeconjectureanddeeprelations withthetheoryofmultiplicativefunctionsaredealtwith.Wefindalsoadetailed presentation of Vinogradov’smethod of major and minor arcs, together with a deepanalysisofsumsoftypeIandIIwhichareofgreatuseincurrentresearch. The final chapter is devoted to the van der Corput method of computing and estimatingtrigonometricsums. Part II of the volume consists of articles devoted to interactions between arithmeticanddynamics.Theyareallofresearch/survey/coursetype: (cid:129) M. Baake, A Brief Guide to Reversing andExtendedSymmetries ofDynamical Systemsisasurveywhichpresentsthebasicnotionsandreviewsfactsconcerning thereversingsymmetryofdynamicalsystems,focusingonsystems(subshifts)of algebraicandnumber-theoreticorigin. Preface ix (cid:129) M.Einsiedler,M.Luethi,KloostermanSums,Disjointness,andEquidistribution summarizes the aforementioned minicourse of M. Einsiedler. Various appli- cations of Kloosterman sums are shown: equidistribution properties of sparse subsetsofhorocycleorbitsinthemodularcase,disjointnessresultsonthetorus, mixingproperties. (cid:129) S. Ferenczi, J. Kułaga-Przymus, M. Leman´czyk, Sarnak’s Conjecture: What’s New?isasurveypresentinganexhaustivelistofmethodsandresultsconcerning theproblemofMöbiusdisjointness.Somenewresultsarealsoincluded. (cid:129) A. Gomilko, D. Kwietniak, M. Leman´czyk, Sarnak’s Conjecture Implies the ChowlaConjectureAlongaSubsequenceprovesthiselementarybutnewresult. (cid:129) C.Huck,OntheLogarithmicProbabilityThataRandomIntegralIdealIsA-free isanarticlewhichextendsatheoremofDavenportandErdösonsetsofmultiples withintegerstotheexistenceoflogarithmicdensityforunionsofintegralideals innumberfields. (cid:129) C. Matheus, The Lagrange and Markov Spectra from the Dynamical Point of View summarizes the aforementioned minicourse of C. Matheus. The notes introducetheworldofLagrangeandMarkovspectrawithaspecialfocusonthe proofofMoreira’stheoremontheintricatestructureofsuchspectra. (cid:129) O. Ramaré,On theMissing Log Factoris a “journey”aroundthe Axer-Landau Equivalence Theorem of the Prime Number Theorem and properties of the MöbiusandvonMangoldtfunctions. (cid:129) O. Ramaré, Chowla’s Conjecture: From the Liouville Function to the Möbius Functionis a notefocusingonproofsofimplicationsbetweenvariousversions oftheChowlaconjectureinwhichweuseeitherLiouvilleorMöbiusfunction. PartIIIofthevolumeconsistsofthreearticlesofsurveyorresearch/surveytype fromselectedtopicsindynamics: (cid:129) T.Adams,C.Silva,WeakMixingforInfiniteMeasureInvertibleTransformations surveys and studies mixing properties of transformations preserving infinite measure. (cid:129) E. Glasner, M. Megrelishvili, More on Tame Dynamical Systems surveys and amplifiesoldresultsin(topological)tamedynamicalsystems,provessomenew results,andprovidesnewexamplesoftamesystems. (cid:129) K. Inoue, H. Nakada, A Piecewise Rotation of the Circle, IPR Maps and TheirConnectionwithTranslationSurfacesreviewsaconstructionoftranslation surfacesintermsofacontinuousversionofthecutting-and-stackingsystemsand provesanewresultofrealizationofRauzyclasses. Marseille,France SébastienFerenczi Torun´,Poland JoannaKułaga-Przymus Torun´,Poland MariuszLeman´czyk
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