THE UNIVERSITY OF CHICAGO BESICOVITCH SETS, RECTIFIABILITY, AND PROJECTIONS A DISSERTATION SUBMITTED TO THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS BY ALAN CHANG CHICAGO, ILLINOIS JUNE 2020 Copyright c 2020 by Alan Chang All Rights Reserved This thesis is dedicated to my family, and also to you, the reader! \We will use this process to generate our monster, which will have a tiny heart and many arms." { Elias Stein, Harmonic Analysis [51, xX.1: The Besicovitch set] TABLE OF CONTENTS LIST OF FIGURES . viii ACKNOWLEDGMENTS . ix ABSTRACT . x 1 INTRODUCTION . 1 1.1 Besicovitch sets and the Kakeya conjecture . .1 1.2 Rectifiability and projections . .2 1.3 Main results of each chapter . .2 1.3.1 Chapter 2: Kakeya needle problem and Besicovitch sets . .2 1.3.2 Chapter 3: Small unions of affine subspaces and skeletons . .3 1.3.3 Chapter 4: Analytic capacity and projections . .5 2 THE KAKEYA NEEDLE PROBLEM AND THE EXISTENCE OF BESICOVITCH AND NIKODYM SETS FOR RECTIFIABLE SETS . 7 2.1 Introduction . .7 2.1.1 History . .8 2.1.2 Translations . .9 2.1.3 Rotations . 10 2.1.4 Besicovitch and Nikodym sets . 13 2.1.5 The sharpness of our results, and dilations . 15 2.2 Main ideas of the proof of Theorems 2.1.2 and 2.5.1 . 16 2.2.1 The first key idea . 16 2.2.2 The second key idea . 17 2.2.3 Combining the key ideas . 17 2.3 Preliminaries . 19 2.3.1 Tangents of rectifiable sets . 19 2.3.2 Rotations . 20 2.3.3 The \second key idea" for translations . 21 2.3.4 The \second key idea" for rotations . 22 2.4 Kakeya needle problem for translations . 24 2.4.1 Notation . 24 2.4.2 Basic zigzag . 25 2.4.3 Venetian blind . 25 2.4.4 Main construction . 27 2.4.5 Choosing the parameters βi; γi ...................... 27 2.4.6 Choosing the signs σi ........................... 28 2.4.7 Stopping time . 29 2.4.8 Fineness of the zigzags, and the small neighborhood lemma . 30 2.4.9 Area estimate . 31 2.5 Kakeya needle problem for rotations . 33 2.5.1 Basic zigzags, deconstructed . 33 v 2.5.2 The structure of intrinsic compositions . 34 2.5.3 Basic zigzag construction for rotations . 36 2.5.4 Iterating the basic zigzag . 37 2.5.5 Turning the translations into rotations . 40 2.5.6 Ignoring small parts of E and of P~ ................... 41 2.5.7 Area estimates . 42 2.5.8 Further remarks . 44 2.6 Besicovitch and Nikodym sets . 45 2.6.1 Construction of a Besicovitch set for translations . 45 2.6.2 Construction of a Besicovitch set for rotations . 48 2.6.3 The main theorem . 49 2.6.4 Construction of a Nikodym set . 51 2.7 Dilations and similarity transformations . 53 2.7.1 Circles . 57 3 SMALL UNIONS OF AFFINE SUBSPACES AND SKELETONS VIA BAIRE CAT- EGORY . 60 3.1 Introduction . 60 3.2 Scaled copies . 64 3.3 Scaled and rotated copies . 70 3.4 Rotated copies: dimension . 73 3.5 Rotated copies: measure . 77 3.5.1 Translating cones . 78 3.5.2 Proof of Theorem 3.5.1 . 82 4 ANALYTIC CAPACITY AND PROJECTIONS . 87 4.1 Introduction . 87 4.2 Notation and preliminaries . 93 4.2.1 Generalities . 93 4.2.2 Measures and rectifiability . 94 4.2.3 Cauchy and Riesz transform, and capacities . 95 4.3 The Fourier calculation . 97 4.3.1 The planar case . 97 4.3.2 The higher dimensional case . 100 4.4 The dyadic lattice of David and Mattila . 105 4.5 The corona decomposition . 110 4.6 The construction of an approximate Lipschitz graph . 112 4.6.1 The stopping cubes . 112 4.6.2 Preliminary estimates . 113 4.6.3 A key estimate . 114 4.6.4 An algorithm to construct a Lipschitz graph close to the stopping cubes118 4.6.5 The small measure of the low density set . 121 4.6.6 The approximate Lipschitz graph . 125 4.7 The family of Top cubes . 126 4.7.1 The family Top .............................. 126 vi 4.7.2 The family of cubes ID ......................... 127 4.8 The packing condition . 128 4.9 Proof of Lemma 4.5.1 . 132 4.10 Application to curvature, Riesz transforms, and capacities . 133 4.10.1 Curvature of measures and Riesz transforms . 133 4.10.2 Analytic capacity . 135 4.10.3 The capacities Γd;n ............................ 137 4.A The reverse inequality . 137 REFERENCES . 142 vii LIST OF FIGURES 1 Words of encouragement. (Produced by Kevin Chang and used with permission.) ix 2.1 Two steps in the Venetian blind construction, starting with (a) as the initial segment. 27 viii ACKNOWLEDGMENTS I would like to thank Marianna Cs¨ornyei for her guidance and mentorship over the last six years. When we first met in my first year of graduate school, I knew very little analysis. Now, I know a little more. I would like to thank Luis Silvestre for being the second reader of my thesis and for teaching me many interesting things about PDEs. I would like to thank Xavier Tolsa for hosting me at Universitat Aut`onomade Barcelona in the fall of 2017. One chapter of this dissertation is work we did together during my visit. I am also extremely grateful to Elias Stein, who introduced me to the beauty of analysis during my first year as an undergraduate, and who continued to provide invaluable advice during and after my time at Princeton. My family, friends, and mentors have supported me in so many ways during my time in graduate school. Your words of encouragement have been very helpful, so let me express my gratitude to you by providing some encouragement of my own (see Figure 1). There is another picture I would like to include but cannot, due to copyright laws. Figure 1: Words of encouragement. (Produced by Kevin Chang and used with permission.) ix ABSTRACT In this thesis, we use the connections between projections and rectifiability to study problems in geometric measure theory, harmonic analysis, and complex analysis. In Chapter 2 (joint work with Marianna Cs¨ornyei), we study \curved" versions of the Kakeya needle problem and Besicovitch sets; i.e., we consider what happens if we replace 1 2 the line segment with a C curve, or more generally, a rectifiable set E ⊂ R . Roughly speaking, our main result states that we can move a rectifiable set E between any two prescribed positions in a set of measure zero, provided that at each time moment t of the movement we are allowed to \hide" a set Et ⊂ E of linear measure zero. In Chapter 3 (joint work with Marianna Cs¨ornyei, Korn´eliaH´eraand Tam´asKeleti), our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the k-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we show that a typical arrangement in the sense of Baire category gives minimal Hausdorff dimension. In particular, this proves a conjecture of R. Thornton. Our results n also show that Nikodym sets are typical among all sets which contain, for every x 2 R , a punctured hyperplane H n fxg through x. With similar methods we also construct a Borel n subset of R of Lebesgue measure zero containing a hyperplane at every positive distance from every point. In Chapter 4 (joint work with Xavier Tolsa), we study the connection between the analytic capacity of a set and the size of its orthogonal projections. More precisely, we prove that if E ⊂ C is compact and µ is a Borel measure supported on E, then the analytic capacity of E satisfies µ(E)2 γ(E) ≥ c R 2 ; I kPθµk2 dθ where c is some positive constant, I ⊂ [0; π) is an arbitrary interval, and Pθµ is the image iθ measure of µ by Pθ, the orthogonal projection onto the line fre : r 2 Rg. This result is related to an old conjecture of Vitushkin about the relationship between the Favard length x and analytic capacity. xi CHAPTER 1 INTRODUCTION 1.1 Besicovitch sets and the Kakeya conjecture In the early 20th century, Besicovitch proved the following results. Theorem 1.1.1 ([3, 4]). 1. (Solution to the Kakeya needle problem) For all " > 0, we can rotate a unit line segment in the plane 180 degrees within a set of area less than ". 2 2. (Existence of Besicovitch sets) There exists a set in R of Lebesgue measure zero which contains a unit line segment in every direction. n A set in R of measure zero which contains a unit line segment in every direction is called a Besicovitch set (or Kakeya set). Theorem 1.1.1 implies the existence of Besicovitch sets n n in R for any n ≥ 2. The Kakeya conjecture asserts that Besicovitch sets in R must have Hausdorff dimension n. Charles Fefferman showed that the existence of Besicovitch sets implies the unbounded- p n ness of the ball multiplier operator on L (R ) for p =6 2 and n ≥ 2 [25]. Since then, a wide variety of problems have been related to the Kakeya conjecture, for example: local smoothing conj. ) Bochner{Riesz conj. ) restriction conj. ) Kakeya conj. The conjectures mentioned above have applications ranging from regularity of solutions to the wave equation to estimates on Fourier operators.