Lecture Notes in Mathematics 2213

Transcription

1 Lecture Notes in Mathematics 2213 Editors-in-Chief: Jean-Michel Morel, Cachan Bernard Teissier, Paris Advisory Board: Michel Brion, Grenoble Camillo De Lellis, Zurich Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gábor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg More information about this series at

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3 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

4 Editors Sébastien Ferenczi CNRS UMR 7373 Institut de Mathématiques de Marseille Marseille, France Joanna Kułaga-Przymus Faculty of Mathematics and Computer Science Nicolaus Copernicus University Toruń, Poland Mariusz Lemańczyk Faculty of Mathematics and Computer Science Nicolaus Copernicus University Toruń, Poland ISSN ISSN (electronic) Lecture Notes in Mathematics ISBN ISBN (ebook) Library of Congress Control Number: Mathematics Subject Classification (2010): 37-XX, 11-XX, 5-XX, 51-XX Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

5 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 possible systems that can arise that the interaction becomes powerful. Starting with Furstenberg s introduction of such concepts as unique ergodicity, disjointness of dynamical systems, and nonconventional ergodic averages, and thanks to advances by many ergodic/number theorists, there is by now a body of striking applications. Homogeneous dynamics takes place on parameter spaces of arithmetic objects, 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 applications in various nonhomogeneous dynamical settings. 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 with the Birkhoff sums for joinings of the system with itself. An example exploiting this is the proof by Mauduit and Rivat of a conjecture of Gelfond about the distribution of the parity of the sum of the binary digits of prime numbers. As far as combinatorial/additive number theory, the path developed by Furstenberg in his proof of Szemeredi s theorem on arithmetic progressions in sets of positive density is 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 what is today a thriving interaction between ergodic theory and number theory. The welltimed 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 and the latest developments, as well advancing them. From my own experience and accounts by others, the minicourses and the workshops and seminars were a great success and there were a number of exciting new developments. v

6 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 insightful account of the basic techniques from ergodic theory and number theory that have facilitated the recent developments. There are also excellent survey papers that bring the reader up to forefront of the latest developments and open problems in this fast moving area. Princeton, NJ, USA October 3, 2017 Peter Sarnak

7 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 between August 1, 2016 and January 31, The semester had a substantial core support and funding by CIRM, Aix-Marseille University, and the city of Marseille. Additionally, it was supported by the LABEX Archimède, and the ANR grants of Christian Mauduit (Aix-Marseille University) and (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 Ferenczi (Aix-Marseille University), Joanna Kułaga-Przymus (Nicolaus Copernicus University Toruń and Aix-Marseille University), Mariusz Lemańczyk (Nicolaus Copernicus University Toruń), and Serge Troubetzkoy (Aix-Marseille University). The main aim of this school was, on one hand, to provide participants with modern methods of ergodic theory and topological dynamics oriented toward applications in number theory and combinatorics, and, on the other hand, to present them with a broad spectrum of number theory problems that can be treated with the use of such tools. These tasks were realized in four minicourses by Vitaly Bergelson (Ohio State University), Mutually enriching connections between ergodic theory and combinatorics, Manfred Einsiedler (ETH Zürich), Equidistribution on homogeneous spaces, a bridge between dynamics and number theory, Carlos Matheus Silva Santos (CNRS - Université Paris 13), The Lagrange and Markov spectra from the dynamical point of view, and Introduction to analytic number theory." The main conference Ergodic Theory and its Connections with Arithmetic and Combinatorics was organized by Julien Cassaigne (Aix-Marseille University), Sébastien Ferenczi, Pascal Hubert (Aix-Marseille University), Joanna Kułaga- Przymus, Mariusz Lemańczyk 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 conference was vii

8 viii Preface aimed at interactions between ergodic theory and dynamical systems and number theory. Its main subjects were disjointness in ergodic theory and randomness in number theory, ergodic theory and combinatorial number theory, and homogenous dynamics and its applications. Important events of the semester were two smaller specialized workshops. The first one Ergodic Theory and Möbius Disjointness was organized by Sébastien Ferenczi, Joanna Kułaga-Przymus, Mariusz Lemańczyk, Christian Mauduit, and Joël Rivat. The meeting focused on the recent progress on Sarnak s conjecture on Möbius disjointness: methods, results, and the feedback in ergodic theory. The second one Spectral Theory of Dynamical Systems and Related Topics was organized by Alexander Bufetov (Aix-Marseille University), Sébastien Ferenczi, Joanna Kułaga- Przymus, Mariusz Lemańczyk, and Arnaldo Nogueira (Aix-Marseille University). 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 solutions of some open classical problems of ergodic theory: Rokhlin problem on mixing of all orders, stability of spectral properties under smooth changes for the parabolic systems, the Banach problem on the existence of dynamical systems with simple 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 On the Stability of Möbius Disjointness in Topological Models and a special program of invitations with participation of Michael Baake (University of Bielefeld), Jean- Pierre Conze (University of Rennes 1), Alexandre Danilenko (Institute of Low Temperature, Kharkov), Christian Huck (University of Bielefeld), Joanna Kułaga- Przymus, El Houcein El Abdalaoui (University of Rouen), Mariusz Lemańczyk and Thierry de la Rue (University of Rouen). The contents of this volume are as follows. It begins with Part I which is entirely the course., Bases of Analytic Number Theory. Among other aspects, 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 together with a beautiful application to Twin Prime conjecture and deep relations with the theory of multiplicative functions are dealt with. We find also a detailed presentation of Vinogradov s method of major and minor arcs, together with a deep 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 and estimating trigonometric sums. Part II of the volume consists of articles devoted to interactions between arithmetic and dynamics. They are all of research/survey/course type: M. Baake, A Brief Guide to Reversing and Extended Symmetries of Dynamical Systems is a survey which presents the basic notions and reviews facts concerning the reversing symmetry of dynamical systems, focusing on systems (subshifts) of algebraic and number-theoretic origin.

9 Preface ix M. Einsiedler, M. Luethi, Kloosterman Sums, Disjointness, and Equidistribution summarizes the aforementioned minicourse of M. Einsiedler. Various applications of Kloosterman sums are shown: equidistribution properties of sparse subsets of horocycle orbits in the modular case, disjointness results on the torus, mixing properties. S. Ferenczi, J. Kułaga-Przymus, M. Lemańczyk, Sarnak s Conjecture: What s New? is a survey presenting an exhaustive list of methods and results concerning the problem of Möbius disjointness. Some new results are also included. A. Gomilko, D. Kwietniak, M. Lemańczyk, Sarnak s Conjecture Implies the Chowla Conjecture Along a Subsequence proves this elementary but new result. C. Huck, On the Logarithmic Probability That a Random Integral Ideal Is A-free is an article which extends a theorem of Davenport and Erdös on sets of multiples with integers to the existence of logarithmic density for unions of integral ideals in number fields. C. Matheus, The Lagrange and Markov Spectra from the Dynamical Point of View summarizes the aforementioned minicourse of C. Matheus. The notes introduce the world of Lagrange and Markov spectra with a special focus on the proof of Moreira s theorem on the intricate structure of such spectra. O. Ramaré, On the Missing Log Factor is a journey around the Axer-Landau Equivalence Theorem of the Prime Number Theorem and properties of the Möbius and von Mangoldt functions. O. Ramaré, Chowla s Conjecture: From the Liouville Function to the Möbius Function is a note focusing on proofs of implications between various versions of 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 type from selected topics in dynamics: T.Adams,C.Silva,Weak Mixing for Infinite Measure Invertible Transformations surveys and studies mixing properties of transformations preserving infinite measure. E. Glasner, M. Megrelishvili, More on Tame Dynamical Systems surveys and amplifies old results in (topological) tame dynamical systems, proves some new results, and provides new examples of tame systems. K. Inoue, H. Nakada, A Piecewise Rotation of the Circle, IPR Maps and Their Connection with Translation Surfaces reviews a construction of translation surfaces in terms of a continuous version of the cutting-and-stacking systems and proves a new result of realization of Rauzy classes. Marseille, France Toruń, Poland Toruń, Poland Sébastien Ferenczi Joanna Kułaga-Przymus Mariusz Lemańczyk

10 Contents Part I Bases of Analytic Number Theory 1 Prime Numbers Arithmetic Functions Dirichlet Series Euler s Gamma Function Riemann s Zeta Function The LargeSieve The Theorem of Vinogradov The van der Corput Method Part II Interactions Between Arithmetic and Dynamics 9 A Brief Guide to Reversing and Extended Symmetries of Dynamical Systems M. Baake 10 Kloosterman Sums, Disjointness, and Equidistribution M. Einsiedler and M. Luethi xi

11 xii Contents 11 Sarnak s Conjecture: What s New Sébastien Ferenczi, Joanna Kułaga-Przymus, and Mariusz Lemańczyk 12 Sarnak s Conjecture Implies the Chowla Conjecture Along a Subsequence Alexander Gomilko, Dominik Kwietniak, and Mariusz Lemańczyk 13 On the Logarithmic Probability That a Random Integral Ideal Is A-Free Christian Huck 14 The Lagrange and Markov Spectra from the Dynamical Point of View Carlos Matheus 15 On the Missing Log Factor Olivier Ramaré 16 Chowla s Conjecture: From the Liouville Function to the Moebius Function Olivier Ramaré Part III Selected Topics in Dynamics 17 Weak Mixing for Infinite Measure Invertible Transformations Terrence Adams and Cesar E. Silva 18 More on Tame Dynamical Systems Eli Glasner and Michael Megrelishvili 19 A Piecewise Rotation of the Circle, IPR Maps and Their Connection with Translation Surfaces Kae Inoue and Hitoshi Nakada

12 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, Institut de 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, Nicolaus Copernicus University, Toruń, 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, Nicolaus Copernicus University, Toruń, Poland Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, I2M UMR 7373, Marseille, France Dominik Kwietniak Faculty of Mathematics and Computer Science, Jagiellonian University in Kraków, Kraków, Poland Institute of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil Mariusz Lemańczyk Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland Manuel Luethi Departement Mathematik, ETH Zürich, Zürich, Switzerland xiii

13 xiv Contributors Carlos Matheus Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 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 de Mathématiques de Marseille, I2M UMR 7373, Marseille, France Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, I2M UMR 7373, Marseille, France Cesar E. Silva Department of Mathematics, Williams College, Williamstown, MA, USA