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The Dichotomy Between Structure and Randomness and Applications to Combinatorial Number Theory

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The Dichotomy Between Structure and Randomness and Applications to Combinatorial Number Theory THE DICHOTOMY BETWEEN STRUCTURE AND RANDOMNESS AND APPLICATIONS TO COMBINATORIAL NUMBER THEORY DISSERTATION Presented in partial fulfillment of the requirements for the degree Doctor of Philosophy in the Graduate School of The Ohio State University By Florian K. Richter, MASt, Bsc Graduate Program in Mathematics The Ohio State University 2018 Dissertation Committee: Vitaly Bergelson, Advisor Alexander Leibman David Penneys Copyright by Florian K. Richter 2018 ABSTRACT The study of the long-term behavior of dynamical systems has far-reaching applications to other areas of mathematics. The employment of analytic tools coming from measurable, topological, and symbolic dynamics offers novel possibilities for analyzing seemingly static number-theoretic and combinatorial situations and has proven to be a powerful method in solving numerous open problems in Ramsey theory and combinatorial number theory that previously appeared to be intractable. In this thesis we develop new techniques that are inspired by dynamical heuristics and lead to a variety of applications in discrete mathematics. One theme featured prominently in this work is the idea of dichotomy between structure and randomness. This dichotomy manifests itself via decomposition theorems that deal with splittings of arithmetic functions into two components, one of which is structured and the other is pseudo-random. From these decomposition theorems we then derive results in ergodic theory and density Ramsey theory. Among other things, we obtain generalizations and refinements of Szemerédi’s theorem and Sárközy’s theorem, and present a solution to a long-standing open sumset conjecture of Erdős. ii ACKNOWLEDGEMENTS My accomplishments as a graduate student would not have been possible without the sup- port of the kind people around me. First and foremost, I would like to express my sincere gratitude to Vitaly Bergelson, my advisor, whose mentorship has formed me as a mathematician. His never-ending patience and care towards me and all his other students has set a great example of how to become an effective and successful advisor. I could not have imagined having a better mentor for my Ph.D. studies. I also want to thank the members of the Ohio State University mathematics department, in particular Alexander Leibman, Nimish Shah and Daniel Thompson, for their help and guidance throughout the years. My thanks also go to Mariusz Lemańczyk. I benefited greatly from my collaboration with him. I am grateful to the graduate school at the Ohio State University for awarding me the Presidential Fellowship, and to the Phi Kappa Phi Honor Society for awarding me the Louise B. Vetter Award, which provided me with the time and resources to finish this dissertation. To my colleagues, co-authors and friends, including Daniel Glasscock, John H. John- son, Andreas Koutsogiannis, Joanna Kułaga-Przymus, Joel Moreira and Donald Robertson, thank you for sharing with me your skills and enthusiasm for mathematics. Finally, I would like to thank my loving family for their help and support and for all the sacrifices that they have made to help me achieve my goals. Thank you! iii VITA June 2010 . .Bachelor of Science Technische Universität Wien June 2011 . .Master of Advanced Studies in Mathematics Cambridge University September 2012 to Present . Graduate Teaching Associate Department of Mathematics Ohio State University PUBLICATIONS • V. Bergelson, F. K. Richter. “On the density of coprime tuples of the form (n, bf1(n)c, ..., bfk(n)c), where f1, . , fk are functions from a Hardy field”. In: Number Theory – Diophantine Problems, Uniform Distribution and Applications, Festschrift in Hon- our of Robert F. Tichy’s 60th Birthday (C. Elsholtz and P. Grabner, eds.), Springer International Publishing, Cham, (2017), pp. 109 – 135. • J. H. Johnson, F. K. Richter. “Revisiting the Nilpotent Polynomial Hales-Jewett Theorem”. In: Advances in Mathematics 321 (2017), pp. 269 – 286. d • J. Moreira, F. K. Richter. “Large subsets of discrete hypersurfaces in Z contain arbitrarily many collinear points”. In: European Journal of Combinatorics 54 (2016), pp. 163 – 176. iv FIELDS OF STUDY Major Field: Mathematics Specialization: Ergodic Theory, Additive and Combinatorial Number Theory v TABLE OF CONTENTS Abstract ........................................ ii Acknowledgements.................................. iii Vita........................................... iv List of Figures..................................... ix 1 Introduction.................................... 1 1.1 The dichotomy between structure and randomness.............. 1 1.2 Notions of structure ............................... 3 1.3 Notions of randomness.............................. 9 1.4 Decomposition theorems............................. 12 1.4.1 Decomposition theorems for arithmetic functions........... 12 1.4.2 Decomposition theorems for multicorrelation sequences . 14 1.4.3 A dichotomy theorem for multiplicative functions and a structure theorem for level sets of multiplicative functions ........... 17 1.5 Main Results ................................... 19 1.5.1 Generalizations of Furstenberg’s multiple recurrence theorem and re- finements of Szemerédi’s theorem.................... 19 1.5.2 The Erdős sumset conjecture...................... 24 2 Proofs of decomposition theorems....................... 26 2 2.1 Proofs of decomposition theorems for L (N, Φ) . 26 2 2.1.1 A completeness lemma for L (N, Φ) ................... 26 vi 2 2.1.2 Decomposing functions from L (N, Φ) into almost periodic and cope- riodic components ............................ 28 2 2.1.3 The Jacobs–de Leeuw–Glicksberg splitting for L (N, Φ) . 31 2.2 Proofs of decomposition theorems for multicorrelation sequences . 35 2.2.1 Preliminaries on nilmanifolds, nilsystems and nilsequences . 35 2.2.2 Preliminaries on almost periodic functions............... 40 2.2.3 Proving Theorem 18 for the special case of nilsystems . 41 2.2.4 Host-Kra-Ziegler factors......................... 47 2.2.5 Spectrum of the orbit of the diagonal ................. 49 2.2.6 A useful reduction............................ 51 2.2.7 The subgroup H ............................. 53 2.2.8 A Theorem for eliminating the rational spectrum........... 56 2.3 Multiplicative functions and their level sets .................. 62 2.3.1 Preliminaries............................... 62 2.3.2 Dichotomy theorem for M0 ....................... 74 2.3.3 Structure theorem for D ......................... 76 3 Applications of decomposition theorems to the theory of multiple re- currence and to combinatorial number theory................ 90 3.1 Multiple ergodic averages along Beatty sequences and a proof of Theorem 28 90 3.2 Multiple ergodic averages along rational sets and applications . 93 3.2.1 Rational sequences are good weights for polynomial multiple conver- gence ................................... 94 3.2.2 Divisible rational sets are good for polynomial multiple recurrence . 96 3.2.3 Applications to additive combinatorics................. 97 3.3 Multiple ergodic averages along level sets of multiplicative functions and applications to ergodic theory and combinatorics............... 99 3.3.1 The class Drat ............................... 99 3.3.2 Proofs of Theorem 35 and Proposition 36 . 103 vii 3.4 Completing the proof of the Erdős sumset conjecture . 105 3.4.1 An ultrafilter reformulation of the Erdős sumset conjecture . 105 3.4.2 Proving the ultrafilter reformulation.................. 110 3.4.3 Establishing properties U1 - U4 . 112 3.4.4 An application of the pointwise ergodic theorem . 115 3.4.5 A variant of an argument of Beiglböck . 117 Bibliography......................................119 viii LIST OF FIGURES 1.1 Notions of structure ................................. 9 1.2 Notions of randomness................................ 12 ix CHAPTER 1 INTRODUCTION 1.1 The dichotomy between structure and randomness Density Ramsey theory is a rich and active area of research in mathematics at the interface of combinatorics and measure theory. Broadly speaking, it deals with finding arithmetic, geometric or combinatorial patterns in large subsets of spaces that admit a natural notion of density. The most classical and most studied space of this kind – and also the space that we focus on in this dissertation – is the set of positive integers N := {1, 2, 3,...}. On N, a notion of density that is natural to consider is the so-called upper density. Given a subset A ⊂ N the upper density of A is defined as |A ∩ {1,...,N}| d(A) := lim sup . N→∞ N One of the basic principles of density Ramsey theory is that any set A with d(A) > 0 is combinatorially and arithmetically rich. Two celebrated theorems that showcase this principle are Szemerédi’s theorem on arithmetic progressions and Sárközy’s theorem. Szemerédi’s theorem ([Sze75]). Any set A ⊂ N with d(A) > 0 contains arbitrarily long arithmetic progressions. Sárközy’s theorem ([Sár78], see also [Fur77]). Any set A ⊂ N with d(A) > 0 contains two elements whose difference is a perfect square. How does one prove results such as Szemerédi’s theorem or Sárközy’s theorem? The class of all subsets of N with positive upper density is so large that, if no further restrictions are made, the nature of an arbitrary set A with d(A) > 0 can be rather intricate and 1 difficult to describe, especially since A might exhibit a blend of different qualities. An effective approach for proving such theorems is to decompose A into manageable pieces. At the center of any such approach lies a decomposition theorem which guarantees that an
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