Kakeya-type Sets, Lacunarity, and Directional Maximal Operators in Euclidean Space by Edward Kroc A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in The Faculty of Graduate and Postdoctoral Studies (Mathematics) The University of British Columbia (Vancouver) April 2015 ⃝c Edward Kroc, 2015 Abstract Given a Cantor-type subset Ω of a smooth curve in Rd+1, we construct random examples of Euclidean sets that contain unit line segments with directions from Ω and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. We also develop a notion of finite order lacunarity for direction sets in Rd+1, and use it to extend our construction to direction sets Ω that are sublacunary according to this definition. This generalizes to higher dimensions a pair of planar results due to Bateman and Katz [4], [3]. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set Ω is unbounded on Lp(Rd+1) for all 1 ≤ p < 1. ii Preface Much of the proceeding document is adapted from two research papers authored by myself and Malabika Pramanik, currently unpublished. These materials are used with permission. Chapters 6 through 11 form the main content of [31], Kakeya-type sets over Cantor sets of directions in Rd+1, while Chapters 2, 3.7, and 12 through 19 are adapted from [32], Lacunarity, Kakeya-type sets and directional maximal op- erators. The first of these two manuscripts has recently been conditionally accepted for publication in the Journal of Fourier Analysis and Applications. iii Table of Contents Abstract ...................................... ii Preface ....................................... iii Table of Contents ................................ iv List of Figures .................................. viii Acknowledgements ............................... x 1 Introduction .................................. 1 1.1 Summary of results ............................ 2 1.2 Notations, conventions, and structure of the document . 6 1.3 Early history of Kakeya sets ....................... 10 1.4 Background: maximal averages over lines with prescribed directions . 17 1.5 Kakeya-type sets and the property of stickiness . 21 2 Finite order lacunarity ........................... 24 2.1 Lacunarity on the real line ........................ 24 2.1.1 Examples ............................. 26 2.1.2 Non-closure of finite order lacunarity under algebraic sums . 28 2.2 Finite order lacunarity in general dimensions . 29 2.2.1 Examples of admissible lacunary and sublacunary sets in Rd . 32 2.3 Finite order lacunarity for direction sets . 35 2.3.1 Examples of admissible lacunary and sublacunary direction sets 36 iv 3 Rooted, labelled trees ............................ 40 3.1 The terminology of trees ......................... 40 3.2 Encoding bounded subsets of the unit interval by trees . 43 3.3 Encoding higher dimensional bounded subsets of Euclidean space by trees .................................... 45 3.4 Stickiness as a kind of mapping between trees . 46 3.5 The tree structure of Cantor-type and sublacunary sets . 48 3.6 The splitting number of a tree ...................... 50 3.6.1 Preliminary facts about splitting numbers . 52 3.6.2 A reformulation of Theorem 1.3 . 53 3.7 Lacunarity on trees ............................ 55 4 Electrical circuits and percolation on trees .............. 60 4.1 The percolation process associated to a tree . 60 4.2 Trees as electrical networks ........................ 62 4.3 Estimating the survival probability after percolation . 65 5 Kakeya-type sets over a Cantor set of directions in the plane ... 68 5.1 The result of Bateman and Katz ..................... 68 5.1.1 Proof of inequality (5.4) ..................... 70 5.1.2 Proof of inequality (5.5) ..................... 74 5.2 Points of distinction between the construction of Kakeya-type sets over Cantor directions in the plane and over arbitrary sublacunary sets in any dimension ............................... 77 6 Setup of construction of Kakeya-type sets in Rd+1 over a Cantor set of directions: a reformulation of Theorem 1.2 .......... 80 7 Families of intersecting tubes ....................... 85 8 The random mechanism and sticky collections of tubes in Rd+1 over a Cantor set of directions ...................... 91 v 8.1 Slope assignment algorithm ....................... 93 8.2 Construction of Kakeya-type sets revisited . 96 9 Slope probabilities and root configurations, Cantor case ...... 97 9.1 Four point root configurations . 100 9.2 Three point root configurations . 103 10 Proposition 8.4: proof of the lower bound (6.6) ............ 105 10.1 Proof of Proposition 10.1 . 107 10.2 Proof of Proposition 10.2 . 109 10.3 Expected intersection counts . 113 11 Proposition 8.4: proof of the upper bound (6.7) ........... 122 12 Construction of Kakeya-type sets in Rd+1 over an arbitrary sub- lacunary set of directions .......................... 128 12.1 Pruning of the slope tree . 129 12.2 Splitting and basic slope cubes . 135 12.3 Binary representation of ΩN . 138 13 Families of intersecting tubes, revisited ................. 140 13.1 Tubes and a point .............................142 13.2 Weakly sticky maps ............................148 14 Random construction of Kakeya-type sets ............... 152 14.1 Features of the construction . 155 14.2 Theorem 1.3 revisited . 157 15 Proof of the upper bound (14.10) ..................... 159 16 Probability estimates for slope assignments .............. 163 16.1 A general rule ...............................163 16.2 Root configurations ............................166 16.3 Notation ..................................167 vi 16.4 The case of two roots . 168 16.5 The case of three roots . 169 16.6 The case of four roots . 173 17 Tube counts .................................. 183 17.1 Collections of two intersecting tubes . 183 17.2 Counting slope tuples . 188 17.3 Collections of four tubes with at least two pairwise intersections . 190 17.3.1 Four roots of type 1 . 190 17.3.2 Four roots of type 2 . 191 17.3.3 Four roots of type 3 . 194 17.4 Collections of three tubes with at least two pairwise intersections . 198 18 Sums over root and slope vertices .................... 200 19 Proof of the lower bound (14.9) ...................... 205 19.1 Proof of Proposition 19.1 . 206 19.2 Proof of Proposition 19.2 . 208 19.2.1 Expected value of S41 . 210 19.2.2 Expected value of S42 . 213 19.2.3 Expected value of S43 . 219 20 Future work .................................. 224 20.1 Maximal functions over other collections of sticky objects . 224 20.2 A characterization of the Lp(Rd+1)-boundedness of directional maxi- mal operators over an arbitrary set of directions . 227 20.2.1 Boundedness of directional maximal operators, sketch . 228 20.2.2 Boundedness of directional maximal operators, a detailed ex- ample ...............................233 Bibliography ................................... 237 vii List of Figures 1.1 A direction set in the plane ....................... 6 1.2 Flowchart of thesis chapters ....................... 7 1.3 Flowchart of thesis chapters, proof of the main theorems . 8 1.4 Besicovitch construction of a zero measure Kakeya set . 11 1.5 Besicovitch's solution to the Kakeya needle problem . 13 1.6 A sticky collection of tubes ........................ 22 3.1 Cantor set tree .............................. 49 3.2 Example trees encoding simple Euclidean sets . 51 3.3 A figure explaining inequality (3.15) when M = 2 and nk = k. 59 4.1 Diagram of proof of Proposition 4.1 ................... 63 6.1 Two tubes in the Kakeya-type set .................... 82 7.1 Triangle defined by the intersection of two tubes . 86 7.2 The cone generated by a set of Cantor directions . 88 9.1 A sticky assignment between trees .................... 98 9.2 Intersections of tubes, four point configurations of type 1 . 101 9.3 Intersections of tubes, four point configurations of type 2 . 103 9.4 Intersections of tubes, three point configurations . 104 10.1 Diagram of all root cubes restricted to lie within a prescribed distance from the boundary of a child of some tree vertex . 114 viii 12.1 Euclidean separation in the pruning mechanism of the slope tree . 133 12.2 Illustration of a pruned tree at the second step of pruning . 135 12.3 Two basic slope cubes . 137 13.1 The pull-back mechanism used to define the compressed root tree . 145 14.1 Diagram of a typical weakly sticky slope assignment . 154 16.1 Identification of basic slope cubes . 168 16.2 Intersections of tubes, three point configurations . 170 16.3 Intersections of tubes, four point configurations . 175 16.4 Root configurations not preserved under sticky mappings . 182 17.1 Illustration of the proof of Lemma 17.1 . 185 17.2 Illustration of the proof of Lemma 17.2 . 187 17.3 Illustration of the spatial restriction of a root cube paired with a cer- tain structured partner root cube . 194 19.1 Slope configurations for four roots of type 3 . 223 ix Acknowledgements Above all others, I would like to thank my supervisor Malabika Pramanik. This work would not have been remotely possible without her unwavering support and tireless commitment. Her advice, her guidance, and her enthusiasm have been invaluable. I am grateful for the financial support she has granted me throughout my graduate studies at UBC, support that has allowed me to fulfill a personal dream of studying mathematics professionally, while simultaneously pursuing my myriad other interests and ambitions. Malabika has generously supported me in all my scientific and profes- sional endeavours, regardless of their bearing on my status as a mathematician. She has guided me through the sometimes existential perils of life in academia, providing truly life changing direction and support. She has been there for me at each and every step of my academic career over the past six years, always happy to help in any way possible. I will be forever grateful for her kindness, her assistance, and her wisdom. I consider myself immeasurably lucky to have had the chance to work with and learn from her. She is my mentor, my colleague, and my friend.