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Arithmetic Structures in Small Subsets of Euclidean Space Arithmetic Structures in Small Subsets of Euclidean Space Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Marc Carnovale, M.S. Graduate Program in Mathematics The Ohio State University 2019 Dissertation Committee: Prof. Vitaly Bergelson, Advisor Prof. Alexander Leibman Prof. Krystal Taylor c Copyright by Marc Carnovale 2019 Abstract In this thesis we extend techniques from additive combinatorics to the setting of harmonic analysis and geometric measure theory. We focus on studying the distribution of three- term arithmetic progressions (3APs) within the supports of singular measures in Euclidean space. In Chapter 2 we prove a relativized version of Roth's theorem on existence of 3APs for positive measure subsets of pseudorandom measures on R and show that a positive measure of points are the basepoints for three-term arithmetic progressions within these measures' supports. In Chapter 3 we combine Mattila's approach to the Falconer distance d conjecture with Green and Tao's arithmetic regularity lemma to show that measures on R with sufficiently small Fourier transform as measured by an Lp-norm have supports with an abundance of three-term arithmetic progressions of various step-sizes. In Chapter 4 d we develop a novel regularity lemma to show that measures on R with sufficiently large dimension, as measured by a gauge function, must either contain non-trivial three-term arithmetic progressions in their supports or else be structured in a specific quantitative manner, which can be qualitatively described as, at infinitely many scales, placing a large amount of mass on at least two distinct cosets of a long arithmetic progression. ii Aknowlegements First and foremost, I would like to thank my Ph.D. advisor Vitaly Bergelson, who set me on the road many years ago. His constant support and generosity have had a greater impact on me than I could ever express. In addition to Vitaly Bergelson, who taught me so much of what I know about math- ematics, I would especially like to thank my M.Sc. advisors IzabellaLaba and Malabika Pramanik who taught me much of the rest. Many thanks also to my thesis defense committee members Alexander Leibman and Krystal Taylor for their time and encouragement. Finally I would like to thank my parents, family, and friends, who have always been there for me and provided endless support. iii Vita 2010 . .B.S. Mathematics Ohio State University. 2013 . .M.S. Mathematics University of British Columbia. 2013-present . .Graduate Teaching Associate, Ohio State University. Publications Research Publications M. Carnovale \A relative Roth theorem in dense subsets of pseudorandom fractals." Online Journal of Analytic Combinatorics, 27{55, June. 2015. Fields of Study Major Field: Mathematics iv Table of Contents Page Abstract . ii Vita............................................. iv List of Tables . vii 1. Introduction . .1 1.1 Notation . .2 1.2 A selected history of additive combinatorics . .4 1.3 A selected history of geometric measure theory . .6 1.4 A selected history of harmonic analysis . .9 1.4.1 The Restriction Conjecture . 10 1.4.2 The multilinear Hilbert transform . 12 1.5 Content of this thesis. 14 2. A relative Roth theorem in dense subsets of sparse pseudorandom fractals . 16 2.1 Notation and approach . 19 2.2 Existence of the restricted measure on 3APs . 21 2.3 The Main Estimate : Proof of Lemma 2.3.2 . 25 2.4 3APs in dense subsets : Proof of Theorem 2.0.1 . 32 2.5 Existence of the measure on 3APs . 33 2.6 Further Remarks . 40 3. On the number of 3APs in fractal subsets of Euclidean Space . 42 3.0.1 Integer sets . 43 3.0.2 Fractal sets . 44 3.0.3 Finite fields . 50 3.1 Integer results . 51 3.1.1 3AP counts of L2 functions . 58 v 3.2 Fractal results . 62 3.2.1 Proof of Theorem 3.0.7 . 63 3.2.2 Proof of Theorem 3.0.8 . 73 3.3 Finite field results . 77 3.3.1 Proof of Corollary 3.0.11 . 77 3.3.2 Gauss sums . 78 3.3.3 Circular bounds and the proof of Theorem 3.0.14 . 82 3.3.4 Decay and the proof of Theorem 3.0.12 . 83 4. 3APs in dense subsets of Euclidean Space . 86 4.1 Introduction . 86 4.2 Large gauges without 3APs . 88 4.3 Almost flat fractals . 89 4.4 Counting 3APs in almost-flat fractals . 90 4.4.1 Counting 3APs between three progressions . 90 4.4.2 The counting lemma . 95 4.5 An (L1, U 2) regularity lemma . 98 4.5.1 The regularity lemma . 98 4.5.2 Bohr sets . 104 4.6 Decomposing a large fractal into an almost-flat portion and a small pseu- dorandom error . 106 4.7 Counting 3APs in large fractals . 108 Bibliography . 109 vi List of Tables Table Page 1.1 Comparison of discrete versus fractal results on 3APs. 15 vii Chapter 1: Introduction This thesis explores applications of additive combinatorics to Euclidean harmonic analysis and geometric measure theory through the lens of one very specific family of structures: arithmetic progressions. We are particularly interested in the study of their presence within subsets of Hausdorff dimension less than d of d-dimensional Euclidean space (we will refer to such subsets as `fractals'). Such a study was initiated by the seminal 2009 paper ofLaba and Pramanik [35] (see Theorem 1.4.1 in Section 1.4 below). There are several reasons such a study may be of interest. First, a number of recent advances in Harmonic Analysis have seen connections to questions about the distribution of arithmetic progressions. Second, it is natural to count the number of k-term arithmetic progressions (kAPs for short) in a set A with characteristic function 1A by introducing the multilinear functional ZZ Λk(f0; : : : ; fk−1) := f0(x)f1(x − r) : : : fk−1(x + (k − 1)r) dx dr and applying it to the tuple (f0; : : : ; fk−1) = (1A;:::; 1A); it is the consequence of monu- mental advances in combinatorial number theory, additive combinatorics, and ergodic theory that in many contexts these multilinear funtionals are fairly well-understood compared to two to three decades ago. Finally, many of the tools being developed in additive combina- torics to understand discrete problems are of a finitistic Fourier-analytic nature but largely novel to Euclidean harmonic analysis, and extending them to this setting is often non-trivial. 1 It is our belief that important future advances in Euclidean harmonic analysis will continue to benefit from tools developed in additive combinatorics. 1.1 Notation Given quantities A and B which may depend on auxiliary parameters ~s = hs1; : : : ; ski we write A . B to indicate the existence of a finite positive constant C (independent of these parameters unless otherwise noted) such that A ≤ CB: uniformly in these parameters. N Similarly, when A = A , B = B , and lim (1+As) = 0 for some power N > 1, we s s s ! 1 Bs will write A B. We use also the notation A & B when B . A and A B when B A, and we write A ≈ B when both A . B and B . A hold. Given an integer N 2 N, write [N] to denote the set [N] := fn 2 N : 0 ≤ n < Ng ; (note that this differs from standard usage where [N] denotes the integers from 1 to N). d Define the Fourier transform of a function f or Borel measure µ on R by Z f^(ξ) = f(x)e−2πiξ·x dx; d R Z µ(ξ) = e−2πiξ·x dµ(x) b d R d for ξ 2 R . 2 For a function f on ZN := Z =N Z define for ξ 2 ZN 1 X f^(ξ) = f(x)e−2πinξ=N : N x2ZN ^ For a function f :[N] ! C, we define f(ξ) via first identifying f with a function on Zp for p a prime between 3N and 6N via the embedding [N] 7! Zp, n 7! (n mod p) 2 Zp and then taking the Fourier transform on this group. That such a prime, p, always exists is guaranteed by Bertrand's postulate. Given functions f; g : G 7! C on an abelian group, G with Haar measure m, we denote convolution with an asterisk: Z f ∗ g(x) = f(x − y)g(y) dm(y); G and extend this to finite Borel measures in the usual way, R f dµ∗ν := R f(x+y) dµ(x) dν(y). We will denote the Lebesgue measure of a set, E, by jEj. We will at times write L^q to refer to the space of functions whose Fourier transforms possess finite Lq norms. d Denote the ball of radius r centered at a point x in R by B(x; r), and let σ denote the surface measure on the unit sphere. In this thesis, we will very often be interested in positive Borel measures µ on [0; 1]d satisfying the following Hausdorff dimension and Fourier decay conditions α (a) µ(B(x; r)) ≤ CH r and −β=2 (b) jµb(ξ)j ≤ CF (1 + jξj) for some positive constants CH and CF . We will refer back to conditions (a) and (b) frequently. 3 1.2 A selected history of additive combinatorics The study of the distribution of kAPs within various subsets of abelian groups is one of the central questions in combinatorial number theory and additive combinatorics, dating back to a 1927 theorem of van der Waerden, [53], stating that any parition of the natu- ral numbers into finitely many subsets must contain a cell which contains arbitrarily long arithmetic progressions, and a later conjecture of Erd}osand Tur´an,[14], asking whether any large enough subset, A, of the natural numbers must contain arbitrarily long arithmetic progressions.
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