Lecture Notes in Mathematics 2213

Editors-in-Chief:Jean-Michel Morel, CachanBernard Teissier, Paris

Advisory Board:Michel Brion, GrenobleCamillo De Lellis, ZurichAlessio Figalli, ZurichDavar Khoshnevisan, Salt Lake CityIoannis Kontoyiannis, AthensGábor Lugosi, BarcelonaMark Podolskij, AarhusSylvia Serfaty, New YorkAnna Wienhard, Heidelberg

More information about this series at http://www.springer.com/series/304

Sébastien Ferenczi • Joanna Kułaga-Przymus •Mariusz LemanczykEditors

Ergodic Theory andDynamical Systemsin their Interactionswith Arithmeticsand CombinatoricsCIRM Jean-Morlet Chair, Fall 2016

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EditorsSébastien FerencziCNRS UMR 7373Institut de Mathématiques de MarseilleMarseille, France

Joanna Kułaga-PrzymusFaculty of Mathematics and ComputerScienceNicolaus Copernicus UniversityTorun, Poland

Mariusz LemanczykFaculty of Mathematics and ComputerScienceNicolaus Copernicus UniversityTorun, Poland

ISSN 0075-8434 ISSN 1617-9692 (electronic)Lecture Notes in MathematicsISBN 978-3-319-74907-5 ISBN 978-3-319-74908-2 (eBook)https://doi.org/10.1007/978-3-319-74908-2

Library of Congress Control Number: 2018940898

Mathematics Subject Classification (2010): 37-XX, 11-XX, 5-XX, 51-XX

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Foreword

The interaction between number theory and ergodic theory can be traced tothe birth of the latter with Birkhoff’s pointwise ergodic theorem. The earlyapplications were naturally concerned with typical behavior, for example in themetrical theory of diophantine approximation. Most number theoretic questionswhich can be connected to ergodic theory are concerned with the dynamics ofspecific orbits or systems constructed from an arithmetic or combinatorial input,and it is with the classification or determination of the basic properties of thepossible systems that can arise that the interaction becomes powerful. Starting withFurstenberg’s introduction of such concepts as unique ergodicity, disjointness ofdynamical systems, and nonconventional ergodic averages, and thanks to advancesby many ergodic/number theorists, there is by now a body of striking applications.Homogeneous dynamics takes place on parameter spaces of arithmetic objects, andas a consequence, rigidity theorems such as that of Ratner for unipotent orbitsbecome powerful tools which underlie many of the most striking applicationsin homogeneous dynamics. There have also been major advances and arithmeticapplications in various nonhomogeneous dynamical settings. For example it turnsout that Vinogradov’s bilinear method in the study of sums over primes for asequence which is an observable in a dynamical system is intimately connectedwith the Birkhoff sums for joinings of the system with itself. An example exploitingthis is the proof by Mauduit and Rivat of a conjecture of Gelfond about thedistribution of the parity of the sum of the binary digits of prime numbers. As faras combinatorial/additive number theory, the path developed by Furstenberg in hisproof of Szemeredi’s theorem on arithmetic progressions in sets of positive densityis at the center of this well-developed modern tool from ergodic theory.

The above are just a small sample (and biased to my taste and knowledge) of whatis today a thriving interaction between ergodic theory and number theory. The well-timed 2016 fall semester activity at CIRM (Luminy) focused on this theme, withthe aim of exposing these interactions and the theories that underlie the progressand the latest developments, as well advancing them. From my own experience andaccounts by others, the minicourses and the workshops and seminars were a greatsuccess and there were a number of exciting new developments.

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Fortunately many of the experts who are responsible for this success preparedand expanded their presentations for this volume. The result is an instructive andinsightful account of the basic techniques from ergodic theory and number theorythat have facilitated the recent developments. There are also excellent survey papersthat bring the reader up to forefront of the latest developments and open problemsin this fast moving area.

Princeton, NJ, USA Peter SarnakOctober 3, 2017

Preface

This volume consists of minicourses notes, survey, research/survey, and researcharticles that have arisen as an outcome of workshops, research in pairs, andother scientific work held under the auspices of the Jean Morlet Chair at CIRMbetween August 1, 2016 and January 31, 2017. The semester had a substantial coresupport and funding by CIRM, Aix-Marseille University, and the city of Marseille.Additionally, it was supported by the LABEX Archimède, and the ANR grantsof Christian Mauduit (Aix-Marseille University) and Joël Rivat (Aix-MarseilleUniversity).

The minicourses were those given in the framework of the doctoral schoolApplications of Ergodic Theory in Number Theory organized by Sébastien Fer-enczi (Aix-Marseille University), Joanna Kułaga-Przymus (Nicolaus CopernicusUniversity Torun and Aix-Marseille University), Mariusz Lemanczyk (NicolausCopernicus University Torun), and Serge Troubetzkoy (Aix-Marseille University).The main aim of this school was, on one hand, to provide participants with modernmethods of ergodic theory and topological dynamics oriented toward applicationsin number theory and combinatorics, and, on the other hand, to present them witha broad spectrum of number theory problems that can be treated with the use ofsuch tools. These tasks were realized in four minicourses by Vitaly Bergelson (OhioState University), “Mutually enriching connections between ergodic theory andcombinatorics,” Manfred Einsiedler (ETH Zürich), “Equidistribution on homoge-neous spaces, a bridge between dynamics and number theory,” Carlos Matheus SilvaSantos (CNRS - Université Paris 13), “The Lagrange and Markov spectra from thedynamical point of view,” and Joël Rivat “Introduction to analytic number theory."

The main conference Ergodic Theory and its Connections with Arithmeticand Combinatorics was organized by Julien Cassaigne (Aix-Marseille University),Sébastien Ferenczi, Pascal Hubert (Aix-Marseille University), Joanna Kułaga-Przymus, Mariusz Lemanczyk with the scientific committee consisting of ArturAvila (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 (AlfredRényi Institute Budapest), and Peter Sarnak (IAS Princeton). The conference was

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aimed at interactions between ergodic theory and dynamical systems and numbertheory. Its main subjects were disjointness in ergodic theory and randomness innumber theory, ergodic theory and combinatorial number theory, and homogenousdynamics and its applications.

Important events of the semester were two smaller specialized workshops. Thefirst one Ergodic Theory and Möbius Disjointness was organized by Sébastien Fer-enczi, Joanna Kułaga-Przymus, Mariusz Lemanczyk, Christian Mauduit, and JoëlRivat. The meeting focused on the recent progress on Sarnak’s conjecture on Möbiusdisjointness: methods, results, and the feedback in ergodic theory. The secondone Spectral Theory of Dynamical Systems and Related Topics was organized byAlexander Bufetov (Aix-Marseille University), Sébastien Ferenczi, Joanna Kułaga-Przymus, Mariusz Lemanczyk, and Arnaldo Nogueira (Aix-Marseille University).The meeting was aimed at the recent progress in the spectral theory and joiningsof dynamical systems, especially, in the recent spectacular progress toward thesolutions of some open classical problems of ergodic theory: Rokhlin problem onmixing of all orders, stability of spectral properties under smooth changes for theparabolic systems, the Banach problem on the existence of dynamical systems withsimple Lebesgue spectrum, and the problem of spectral multiplicity.

The scientific part of the semester was completed by two research in pairs:Dynamical Properties of Systems Determined by Free Points in Lattices and Onthe Stability of Möbius Disjointness in Topological Models and a special programof invitations with participation of Michael Baake (University of Bielefeld), Jean-Pierre Conze (University of Rennes 1), Alexandre Danilenko (Institute of LowTemperature, Kharkov), Christian Huck (University of Bielefeld), Joanna Kułaga-Przymus, El Houcein El Abdalaoui (University of Rouen), Mariusz Lemanczyk andThierry de la Rue (University of Rouen).

The contents of this volume are as follows. It begins with Part I which is entirelythe course.

• Joël Rivat, Bases of Analytic Number Theory. Among other aspects, the coursecontains a presentation of the main properties of the Riemann ζ function witha generous introduction to the theory of Dirichlet series. Large sieve methodtogether with a beautiful application to Twin Prime conjecture and deep relationswith the theory of multiplicative functions are dealt with. We find also a detailedpresentation of Vinogradov’s method of major and minor arcs, together with adeep analysis of sums of type I and II which are of great use in current research.The final chapter is devoted to the van der Corput method of computing andestimating trigonometric sums.

Part II of the volume consists of articles devoted to interactions betweenarithmetic and dynamics. They are all of research/survey/course type:

• M. Baake, A Brief Guide to Reversing and Extended Symmetries of DynamicalSystems is a survey which presents the basic notions and reviews facts concerningthe reversing symmetry of dynamical systems, focusing on systems (subshifts) ofalgebraic and number-theoretic origin.

Preface ix

• M. Einsiedler, M. Luethi, Kloosterman Sums, Disjointness, and Equidistributionsummarizes the aforementioned minicourse of M. Einsiedler. Various appli-cations of Kloosterman sums are shown: equidistribution properties of sparsesubsets of horocycle orbits in the modular case, disjointness results on the torus,mixing properties.

• S. Ferenczi, J. Kułaga-Przymus, M. Lemanczyk, Sarnak’s Conjecture: What’sNew? is a survey presenting an exhaustive list of methods and results concerningthe problem of Möbius disjointness. Some new results are also included.

• A. Gomilko, D. Kwietniak, M. Lemanczyk, Sarnak’s Conjecture Implies theChowla Conjecture Along a Subsequence proves this elementary but new result.

• C. Huck, On the Logarithmic Probability That a Random Integral Ideal Is A-freeis an article which extends a theorem of Davenport and Erdös on sets of multipleswith integers to the existence of logarithmic density for unions of integral idealsin number fields.

• C. Matheus, The Lagrange and Markov Spectra from the Dynamical Pointof View summarizes the aforementioned minicourse of C. Matheus. The notesintroduce the world of Lagrange and Markov spectra with a special focus on theproof of Moreira’s theorem on the intricate structure of such spectra.

• O. Ramaré, On the Missing Log Factor is a “journey” around the Axer-LandauEquivalence Theorem of the Prime Number Theorem and properties of theMöbius and von Mangoldt functions.

• O. Ramaré, Chowla’s Conjecture: From the Liouville Function to the MöbiusFunction is a note focusing on proofs of implications between various versionsof the Chowla conjecture in which we use either Liouville or Möbius function.

Part III of the volume consists of three articles of survey or research/survey typefrom selected topics in dynamics:

• T. Adams, C. Silva, Weak Mixing for Infinite Measure Invertible Transformationssurveys and studies mixing properties of transformations preserving infinitemeasure.

• E. Glasner, M. Megrelishvili, More on Tame Dynamical Systems surveys andamplifies old results in (topological) tame dynamical systems, proves some newresults, and provides new examples of tame systems.

• K. Inoue, H. Nakada, A Piecewise Rotation of the Circle, IPR Maps andTheir Connection with Translation Surfaces reviews a construction of translationsurfaces in terms of a continuous version of the cutting-and-stacking systems andproves a new result of realization of Rauzy classes.

Marseille, France Sébastien FerencziTorun, Poland Joanna Kułaga-PrzymusTorun, Poland Mariusz Lemanczyk

Contents

Part I Bases of Analytic Number TheoryJoël Rivat

1 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Joël Rivat

2 Arithmetic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Joël Rivat

3 Dirichlet Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Joël Rivat

4 Euler’s Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Joël Rivat

5 Riemann’s Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Joël Rivat

6 The Large Sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Joël Rivat

7 The Theorem of Vinogradov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Joël Rivat

8 The van der Corput Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Joël Rivat

Part II Interactions Between Arithmetic and Dynamics

9 A Brief Guide to Reversing and Extended Symmetriesof Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117M. Baake

10 Kloosterman Sums, Disjointness, and Equidistribution. . . . . . . . . . . . . . . . 137M. Einsiedler and M. Luethi

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11 Sarnak’s Conjecture: What’s New . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Sébastien Ferenczi, Joanna Kułaga-Przymus,and Mariusz Lemanczyk

12 Sarnak’s Conjecture Implies the Chowla Conjecture Alonga Subsequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Alexander Gomilko, Dominik Kwietniak, and Mariusz Lemanczyk

13 On the Logarithmic Probability That a Random Integral IdealIs AAA-Free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249Christian Huck

14 The Lagrange and Markov Spectra from the Dynamical Pointof View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259Carlos Matheus

15 On the Missing Log Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293Olivier Ramaré

16 Chowla’s Conjecture: From the Liouville Function to theMoebius Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317Olivier Ramaré

Part III Selected Topics in Dynamics

17 Weak Mixing for Infinite Measure Invertible Transformations . . . . . . . 327Terrence Adams and Cesar E. Silva

18 More on Tame Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351Eli Glasner and Michael Megrelishvili

19 A Piecewise Rotation of the Circle, IPR Maps and TheirConnection with Translation Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393Kae Inoue and Hitoshi Nakada

Contributors

Terrence Adams U.S. Government, Ft. Meade, MD, USA

Michael Baake Faculty of Mathematics, Universität Bielefeld, Bielefeld, Germany

Manfred Einsiedler Departement Mathematik, ETH Zürich, Rämistrasse, Zürich,Switzerland

Sébastien Ferenczi Aix Marseille Université, CNRS, Centrale Marseille, Institutde Mathématiques de Marseille, I2M – UMR 7373, Marseille, France

Eli Glasner Department of Mathematics, Tel-Aviv University, Ramat Aviv, Israel

Alexander Gomilko Faculty of Mathematics and Computer Science, NicolausCopernicus University, Torun, Poland

Christian Huck Fakultät für Mathematik, Universität Bielefeld, Bielefeld,Germany

Kae Inoue Faculty of Pharmacy, Keio University, Tokyo, Japan

Joanna Kułaga-Przymus Faculty of Mathematics and Computer Science, Nico-laus Copernicus University, Torun, Poland

Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques deMarseille, I2M – UMR 7373, Marseille, France

Dominik Kwietniak Faculty of Mathematics and Computer Science, JagiellonianUniversity in Kraków, Kraków, Poland

Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro,Brazil

Mariusz Lemanczyk Faculty of Mathematics and Computer Science, NicolausCopernicus University, Torun, Poland

Manuel Luethi Departement Mathematik, ETH Zürich, Zürich, Switzerland

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Carlos Matheus Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR7539), Villetaneuse, France

Michael Megrelishvili Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

Hitoshi Nakada Department of Mathematics, Keio University, Yokohama, Japan

Olivier Ramaré Aix Marseille Université, CNRS, Centrale Marseille, Institut deMathématiques de Marseille, I2M – UMR 7373, Marseille, France

Joël Rivat Aix Marseille Université, CNRS, Centrale Marseille, Institut de Math-ématiques de Marseille, I2M – UMR 7373, Marseille, France

Cesar E. Silva Department of Mathematics, Williams College, Williamstown,MA, USA