Character Sum Estimates in Finite Fields and Applications by Brandon Hanson A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto c Copyright 2015 by Brandon Hanson Abstract Character Sum Estimates in Finite Fields and Applications Brandon Hanson Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2015 In this thesis we present a number of character sum estimates for sums of various types occurring in finite fields. The sums in question generally have an arithmetic combinatorial flavour and we give applications of such estimates to problems in arithmetic combinatorics and analytic number theory. Conversely, we demonstrate ways in which the theory of arithmetic combinatorics can be used to obtain certain character sum estimates. ii Dedication To my friends, for all the laughs. To my family, for their support. To my teachers, for inspiring me. To John, for his patience and commitment. To Michelle, for everything. iii Acknowledgements First and foremost I must thank John Friedlander, my thesis advisor, for giving me so much of his time and patience. This thesis would not have been possible without all of the helpful discussions we had. I also want to thank Leo Goldmakher for getting me interested in the field and providing much encouragement along the way. Thanks to Kumar Murty and Antal Balog for being a part of my thesis committee and providing fruitful discussion. Finally, like every graduate student in math at the University of Toronto, I am indebted to Ida Bulat and Jemima Merisca. They made my life in the graduate program so much easier. I want to thank them for ensuring that my headaches were purely mathematical ones. iv Contents 1 Introduction and Motivation 1 1.1 Primes in arithmetic progressions: A fundamental example of equidistribution in number theory ............................................... 1 1.2 Random oscillatory sums and the square-root law: The nature of random sequences . 2 1.3 Weyl's Equidistribution Criterion: Fourier analysis enters the scene ............ 3 1.4 The Sum-Product Phenomenon: A source of equidistribution ................ 5 1.5 Character sums: The star of the show ............................. 6 1.6 An outline of this thesis . 8 2 Notation and relevant background 10 2.1 Asymptotic Notation . 10 2.2 Fourier analysis on finite abelian groups . 10 2.3 Finite fields . 13 2.4 Additive combinatorics and the Sum-Product Phenomenon . 14 2.5 Bohr sets and their structure . 18 2.6 Character sums . 21 3 Capturing forms in dense subsets of finite fields 25 3.1 Introduction . 25 3.2 Statement of results . 26 3.3 Upper Bound . 27 3.4 Lower Bound . 29 3.5 Remarks for Composite Modulus . 31 4 Character sum estimates for Bohr sets and applications 33 4.1 Introduction . 33 4.2 Statement of Results and Applications . 33 4.2.1 Main Results . 33 4.2.2 Applications . 34 4.3 The P´olya-Vinogradov Argument . 35 4.4 The Burgess Argument . 36 4.5 Application to Polynomial Recurrence . 38 v 5 Character sum estimates for various convolutions 41 5.1 Introduction . 41 5.2 Statement of Results . 43 5.3 Trivariate sums . 44 5.4 Mixed multivariate sums . 47 Bibliography 49 vi Chapter 1 Introduction and Motivation Much of this thesis is concerned with equidistribution as it pertains to arithmetic. This notion is a fundamental one in number theory which measures the extent to which an object behaves randomly. In the next five sections we illustrate some results in number theory, old and new, which will motivate the thesis. The hope is that through this exposition, our train of thought will be made clear, so that the reader has context for the results of the following chapters. In the first section we present the quintessential example of equidistribution in analytic number theory - the distribution of primes in arithmetic progressions. Though not explicitly related to the results of this thesis, the problem of understanding the distribution of primes seems like the most natural starting point for any discussion about equidistribution in number theory. In the second section we digress a bit in order to recall some of the properties of uniform random sequences. We hope this diversion will suggest which qualities a deterministic object should have in order to deem it random-like. The third section of this introduction is devoted to Weyl's Equidistribution Criterion. This is a basic result which relates the problem of measuring the uniformity of a sequence with its Fourier analytic behaviour. As the criterion suggests, Fourier analysis plays a large r^ole,in analytic number theory and it will be used at length in this book. In the fourth section of the introduction we discuss the Sum-Product Problem of combinatorial number theory. This problem seeks to quantify the extent to which additive structure and multiplicative structure are uncorrelated. The spirit of the Sum-Product Problem was the motivation for work on the character sum estimates proved in Chapters 4 and 5. In the fifth section, we hope to capture the reader's interest in the question of character sum estimates. Such questions began with Dirichlet's work on the distribution of primes in arithmetic progressions, but we hope that throughout this chapter we can convince the reader that these estimates are interesting in their own right. In the final section of this introduction we give an outline for the rest of this thesis and a statement of the results to come. 1.1 Primes in arithmetic progressions: A fundamental example of equidistribution in number theory The first, and perhaps most famous instance of equidistribution of arithmetic objects is the equidistri- bution of the primes into arithmetic progressions. While we do not investigate the distribution of primes in this thesis, the question provides a good starting point for our exposition. The primes are mysterious numbers, mostly because they are defined by what they are not rather than by what they are. As such, 1 Chapter 1. Introduction and Motivation 2 stating facts about primes is rarely easy. We begin by examining some basic properties. Certainly, each prime other than 2 is odd. And no primes other than 3 and 5 should have a common factor with 15. In general, when we divide p by q, which is to say we write p = nq + a with 0 ≤ a ≤ q − 1, the remainder a is necessarily relatively prime with q. Indeed, if q and a had a factor in common, that factor would also divide p. In short, if p = a mod q then (a; q) = 1. Beyond this obvious pattern, it is hard to deduce anything structural about the number a. Arguments going back to Euclid tell us that if we divide the odd primes by 4 then the remainders 1 and 3 occur infinitely often (0 and 2 are forbidden). This fact was famously generalized by Dirichlet, who proved that each of the eligible remainders that come from dividing a prime p by a number q also occur infinitely often as we run over the primes. His work and subsequent work in analytic number theory lead to the Prime Number Theorem in Arithmetic Progressions, which says that each eligible remainder occurs with roughly the same frequency. In other words, primes fall uniformly into the φ(q) eligible residue classes modulo q. Theorem (Prime Number Theorem in Arithmetic Progressions). Let π(x; q; a) denote the number of primes up to x which have remainder a when divided by q, and let π(x) denote to total number of primes up to x. Then as x ! 1 we have π(x; q; a) 1 ! : π(x) φ(q) Suppose we were given a large prime and asked which residue class p lies in modulo q. Without any further information this task seems hopeless - the prime is really, really big. We should not be to hard on ourselves however, because the above theorem is telling us we might do just as well to choose one class at random. So, it is not the case that we do not understand patterns in the distribution of primes beyond the obvious ones, but rather that (at least at this scope) there aren't any. The main tool for the study of primes in arithmetic progressions is the Dirichlet character. These characters will be of central interest in Chapter 4 and Chapter 5. We will give further exposition to Dirichlet characters in Section 1.5. 1.2 Random oscillatory sums and the square-root law: The na- ture of random sequences Most of the equidistribution problems investigated in this thesis are concerned with points on the unit circle in the complex plane, 1 S = fz 2 C : jzj = 1g: We identify the circle S1 with the group R=Z = [0; 1], the group operation being addition modulo 1. This identification is via the map e : R=Z ! S1 defined by e(θ) = e2πiθ. Given a real number α, the expression α mod 1 means the fractional part fαg of α up to translation by integers. We now divert briefly from the topic of equidistribution to discuss what we might expect on random grounds when summing complex unit vectors. Suppose we choose N numbers θ1; : : : ; θN uniformly at random from [0; 1] and send them to the circle by the map e defined above. These new points have uniformly distributed angles and so are likely to point in all sorts of directions. In particular, given an arc C of length l(C) on the circle, we would expect that a proportion l(C)=2π - the proportion of the circle occupied by C - of the points lie in the arc C.