A Kakeya Estimate for Sticky Sets Using a Planebrush Citation Kulkarni, Neeraja Raghavendra (2024) A Kakeya Estimate for Sticky Sets Using a Planebrush. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/japt-b214. https://resolver.caltech.edu/CaltechTHESIS:06102024-225449252 Abstract A Besicovitch set is defined as a compact subset of ℝⁿ which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Besicovitch set has Minkowski and Hausdorff dimensions equal to n. This thesis gives an improved Hausdorff dimension estimate, d ⩾ 0.60376707287 n + O(1), for Besicovitch sets displaying a special structural property called "stickiness." The improved estimate comes from using an incidence geometry argument called a "k-planebrush," which is a higher dimensional analogue of Wolff's "hairbrush" argument from 1995. In addition, an x-ray transform estimate is obtained as a corollary of Zahl's k-linear estimate in 2019. The x-ray estimate, together with the estimate for sticky sets, implies that all Besicovitch sets in ℝⁿ must have Minkowski dimension greater than (2 - √2 + ε)n. Though this Minkowski dimension estimate is not as good as one previously known from Katz-Tao(2000), it provides a new proof of the same result. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Kakeya conjecture, Besicovitch set, sticky Kakeya sets Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Apostol Award for Excellence in Teaching in Mathematics, 2021, 2023. Thesis Availability: Public (worldwide access) Research Advisor(s): Conlon, David Thesis Committee: Graber, Thomas B. (chair) Conlon, David Isett, Philip Katz, Nets H. Defense Date: 7 June 2024 Record Number: CaltechTHESIS:06102024-225449252 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06102024-225449252 DOI: 10.7907/japt-b214 ORCID: Author ORCID Kulkarni, Neeraja Raghavendra 0000-0001-7747-9177 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16515 Collection: CaltechTHESIS Deposited By: Neeraja Kulkarni Deposited On: 11 Jun 2024 22:02 Last Modified: 18 Jun 2024 19:09 Thesis Files PDF - Final Version See Usage Policy. 521kB Repository Staff Only: item control page

A Kakeya estimate for sticky sets using a planebrush Thesis by Neeraja Raghavendra Kulkarni In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematics CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2024 Defended June 7, 2024 ii © 2024 Neeraja Raghavendra Kulkarni ORCID: 0000-0001-7747-9177 All rights reserved except where otherwise noted iii ACKNOWLEDGEMENTS I am very grateful to my advisor, Professor Nets Katz, for suggesting this thesis topic and supervising the project. Thanks also to Professor David Conlon, for acting as my advisor during my last quarter at Caltech, and to Professors Tom Graber and Phil Isett, for being on my dissertation committee. I am grateful to Professor Joshua Zahl for many helpful discussions and for telling me that the x-ray estimate in Chapter 4 of this thesis could be derived as a corollary of his 𝑘-linear Kakeya theorem. I would also like to thank Polona Durcik and Shukun Wu for teaching me about many topics in harmonic analysis. Finally, I am glad to have a chance to thank my father, my mother, my sister, and my grandmother, for their loving support through all the years of my education. iv ABSTRACT A Besicovitch set is defined as a compact subset of R𝑛 which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Besicovitch set has Minkowski and Hausdorff dimensions equal to 𝑛. This thesis gives an improved Hausdorff dimension estimate, 𝑑 ě 0.60376707287 𝑛 + O (1), for Besicovitch sets displaying a special structural property called “stickiness.” The improved estimate comes from using an incidence geometry argument called a “𝑘-planebrush,” which is a higher dimensional analogue of Wolff’s “hairbrush” argument from 1995. In addition, an x-ray transform estimate is obtained as a corollary of Zahl’s 𝑘-linear estimate in 2019. The x-ray estimate, together with the estimate for sticky sets, implies that all Besicovitch sets in R𝑛 must have Minkowski dimension greater than ? (2 ´ 2 + 𝜀)𝑛. Though this Minkowski dimension estimate is not as good as one previously known from Katz-Tao (2000), it provides a new proof of the same result. v TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Outline of thesis and summary of notation . . . . . . . . . . . . . . 1.2 From Kakeya needle sets to the Kakeya maximal conjecture . . . . . 1.3 Formulation of the Kakeya maximal conjecture . . . . . . . . . . . . 1.4 Importance of the Kakeya maximal conjecture . . . . . . . . . . . . 1.5 Equivalent versions of the Kakeya maximal conjecture . . . . . . . . 1.6 Proof of the Kakeya maximal conjecture in R2 . . . . . . . . . . . . 1.7 Proof that the Kakeya maximal conjecture implies the Kakeya set conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Introduction to the x-ray transform estimate and sticky sets . . . . . . Chapter II: Recent progress in the 𝑘-linear Kakeya conjecture . . . . . . . . . 2.1 The multilinear Kakeya theorem . . . . . . . . . . . . . . . . . . . . 2.2 Polynomial partitioning method . . . . . . . . . . . . . . . . . . . . 2.3 Polynomial Wolff axioms . . . . . . . . . . . . . . . . . . . . . . . Chapter III: New Kakeya estimate for 𝑘-plany sets . . . . . . . . . . . . . . . 3.1 Two-ends reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Broad-narrow reduction . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Volume of a planebrush . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Proof of k-planebrush estimate . . . . . . . . . . . . . . . . . . . . Chapter IV: New Kakeya estimate for sticky sets . . . . . . . . . . . . . . . . 4.1 Shading and refinement of sticky sets . . . . . . . . . . . . . . . . . 4.2 Extremal sticky sets . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Kakeya estimate for sticky sets . . . . . . . . . . . . . . . . . . . . . 4.4 Relation to sticky sets in Wang-Zahl . . . . . . . . . . . . . . . . . . 4.5 An x-ray estimate derived as a corollary of Zahl’s k-linear estimate . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv v 1 1 2 3 5 6 10 11 13 15 15 17 18 20 21 24 28 34 37 38 40 45 49 52 59 1 Chapter 1 INTRODUCTION 1.1 Outline of thesis and summary of notation Chapter 1 of this document contains a brief exposition of the Kakeya problem. It introduces the Kakeya set conjecture and the Kakeya maximal conjecture along with the concept of a “sticky set,” providing context for the results proved in later chapters. Chapter 2 introduces some variants of the Kakeya problem, including the multilinear Kakeya problem and the 𝑘-linear Kakeya problem, and outlines some recent results in the 𝑘-linear Kakeya problem (Hickman-Rogers-Zhang [12] and Zahl [27]) which are used in Chapters 3 and 4. Chapter 3 gives a Kakeya maximal estimate, at a dimension depending on a parameter 𝑘, for sets of 𝛿-tubes obeying a special structural property called 𝑘-planiness. Chapter 4 combines the results of Chapter 3 with recent results in the 𝑘-linear Kakeya conjecture to prove an improved Hausdorff dimension estimate for sticky sets. In the final section of Chapter 4, an x-ray transform estimate is obtained as a corollary of Zahl ([27]). The x-ray transform estimate is used to give a Minkowski dimension ? estimate, 𝑑 ě (2 ´ 2) + 10´10 )𝑛 + O (1), for all Besicovitch sets. Though this Minkowski dimension estimate is not as good as one previously known (from Katz and Tao [18]), it provides a different proof of the result. Table 1.1: Summary of notation 𝐴≲𝐵 𝐴„𝐵 𝐴⪅𝐵 𝐴«𝐵 dim𝐻 (𝐸) dim 𝑀 (𝐸) dim 𝑀 (𝐸) dimP (𝐸) |𝐸 | 𝜈 #𝐸 𝑁 𝛿 (𝐸) D𝐶 ą 0 : 𝐴 ď 𝐶𝐵 𝐴 ≲ 𝐵 and 𝐵 ≲ 𝐴 D𝐶𝜀 ą 0 : 𝐴 ď 𝐶𝜀 𝛿´𝜀 𝐵 𝐴 ⪅ 𝐵 and 𝐵 ⪅ 𝐴 Hausdorff dimension of 𝐸 Minkowski dimension of 𝐸 upper Minkowski dimension of 𝐸 Packing dimension of 𝐸 Lebesgue measure of 𝐸 normalized surface measure on S𝑛´1 Cardinality of 𝐸 𝛿-neighborhood of 𝐸 2 1.2 From Kakeya needle sets to the Kakeya maximal conjecture In 1920, the Russian mathematician A.S. Besicovitch studied the following problem in a paper ([3]) published in a Russian monthly periodical: Given a function of two variables, Riemann integrable on a plane domain, does there always exist a pair of mutually perpendicular directions such that the repeated simple integration along the two directions exists and gives the value of the integral over the domain? This problem reduced to proving the existence a set of Lebesgue plane measure 0 which contained a line segment of length ě 1 in every direction. Besicovitch solved the problem, answering the above question in the negative, by demonstrating the construction of such a set. Later, in 1928, Besicovitch wrote that he had become aware of a “twin problem” posed by the Japanese mathematician S. Kakeya in 1917 ([2]). The problem had never before come to his attention because of the isolation of Russia following the civil war. Kakeya’s problem was this: In the class of figures in which a line segment of length 1 can be rotated by 360𝑜 without ever leaving the figure, which one has the smallest area? Each of these two problems was concerned with sets containing a line segment of length 1 in every direction, with Kakeya’s problem imposing the additional constraint that there should be a continuous transition. Definition 1. A Kakeya needle set is a set in R𝑛 in which a line segment of length 1 can be rotated continuously by 360𝑜 without leaving the set. Definition 2. A Besicovitch set is a compact set in R𝑛 which contains a line segment of length 1 in every direction. In his paper [2], Besicovitch gave a modified version of his construction of a measure zero Besicovitch set in [3] to solve Kakeya’s problem, thereby disproving Kakeya’s original conjecture that the smallest figure in which one could continuously rotate a unit line segment by 360𝑜 was the hypocycloid of area 𝜋/8 ([13]). 3 Decades later, in 1971, Besicovitch sets resurfaced in the work of Charles Fefferman, who was studying the following question. If 𝑇 is an operator on 𝐿 𝑝 (R𝑛 ) defined by 𝑇c𝑓 = 𝜒𝐵 𝑓ˆ, where 𝜒𝐵 is the characteristic function of the unit ball in R𝑛 , for which values of 𝑝 is it the case that, for all 𝑓 P 𝐿 𝑝 (R𝑛 ), ||𝑇 𝑓 || 𝐿 𝑝 (R𝑛 ) ď 𝐶 || 𝑓 || 𝐿 𝑝 (R𝑛 ) ? The “disc conjecture” stated the the above estimate holds for all 2𝑛 2𝑛 ă𝑝ă ; 𝑛+1 𝑛´1 these necessary conditions were demonstrated by Hertz using the example where 𝑓 is a Schwartz function whose Fourier transform is identically 1 on the unit ball. Prior to Fefferman’s work, the disc conjecture was known to be true in R1 . It thus came as a surprise when Fefferman showed that the disc conjecture is false whenever 𝑝 ≠ 2 (𝑛 ą 1) using, as a lemma, Besicovitch’s construction of measure zero Besicovitch sets ([9]). The work of Fefferman and Hertz led to the following further conjecture, which has come to be called the Kakeya set conjecture, about Besicovitch sets: Conjecture 3 (Kakeya set conjecture). Every Besicovitch set in R𝑛 has Minkowski and Hausdorff dimensions equal to 𝑛. The Kakeya set conjecture in R2 was solved by Davies ([8]) but remains open for 𝑛 ě 3. The best known results are due to Katz-Zahl ([19]) in R3 and due to KatzZahl ([14]) in R4 . The best known bounds in higher dimensions are due to Katz-Tao ([18]). Corollary 47 of this thesis gives an new proof of a Minkowski dimension estimate for Besicovitch sets which was previously known from [18]. The current state-of-the-art for the Kakeya set conjecture (Hausdorff dimension version) is given in Table 1.2. The motivation for the Kakeya set conjecture is discussed in Section 1.3 below. 1.3 Formulation of the Kakeya maximal conjecture The Kakeya maximal conjecture is a more quantitative formulation of the Kakeya set conjecture in terms of a maximal operator. The formulation of the conjecture given below was given by Bourgain [5], though a similar formulation had been given by Córdoba [7], who also proved the conjecture in R2 . 4 Table 1.2: State of the art for the Kakeya conjecture 𝑛 3 4 5 6 7 .. . dim𝐻 (𝐾) ě 2.5 + 𝜀 3.058. . . 3.6? 7´2 2 34/7 .. . Proved by Katz-Zahl [15] Katz-Zahl [14] Hickman-Rogers-Zhang [12] Katz-Tao [18] Hickman-Rogers-Zhang [12] For any small number 𝛿 ą 0, 𝑒 P S𝑛´1 and 𝑎 P R𝑛 , define 𝑇𝑒𝛿 (𝑎) to be the 𝛿neighborhood of a line segment pointing in direction 𝑒 and centered at point 𝑎, so 1 (R𝑛 ), its Kakeya 𝑇𝑒𝛿 (𝑎) is a tube (or cylinder) of radius 𝛿 and length 1. If 𝑓 P 𝐿 loc maximal function 𝑓𝛿˚ : S𝑛´1 Ñ R is defined by ∫ ˚ 1´𝑛 𝑓𝛿 (𝑒) = 𝛿 sup | 𝑓 |. 𝑎 PR 𝑛 𝑇𝑒𝛿 (𝑎) If 𝑓 is the indicator function of the set 𝑁 𝛿 (𝐵), where 𝐵 is any Besicovch set, any bound of the type || 𝑓𝛿˚ || 𝑞 ď 𝐶 || 𝑓 || 𝑝 with 1 ă 𝑞 ă 8, 𝑝 ă 8, and 𝐶 independent of 𝛿, fails to hold. In addition, if 𝑓 is the indicator function of the set 𝑁 𝛿 (𝐵), where 𝐵 is any Besicovch set of zero Lebesgue measure, a bound of the form || 𝑓𝛿˚ || 𝑝 ď 𝐶𝜀 𝛿´𝜀 || 𝑓 || 𝑝 cannot hold if 𝑝 ą 𝑛. Thus the strongest estimate which may be expected is of the following form. Conjecture 4 (Kakeya maximal conjecture, version 1). For every 𝜀 ą 0, there is a constant 𝐶𝜀 ą 0 such that for all 𝛿 ą 0 and 𝑓 P 𝐿 𝑛 (R𝑛 ), || 𝑓𝛿˚ || 𝐿 𝑛 (S𝑛´1 ) ď 𝐶𝜖 𝛿´𝜀 || 𝑓 || 𝐿 𝑛 (R𝑛 ) . Interpolating this (by the Riesz-Thorin interpolation theorem) with the trivial estimate || 𝑓𝛿˚ || 𝐿 8 (S𝑛´1 ) ≲ 𝛿1´𝑛 || 𝑓 || 𝐿 1 (R𝑛 ) , one obtains a conjecture || 𝑓𝛿˚ || 𝐿 𝑞 (S𝑛´1 ) ⪅ 𝛿1´𝑛/𝑝 || 𝑓 || 𝐿 𝑝 (R𝑛 ) where 1 ď 𝑝 ď 𝑛 and 𝑞 = (𝑛 ´ 1) 𝑝 1 . 5 Partial progress in the Kakeya conjecture is given by estimates of this form. The full Kakeya maximal conjecture is solved only in R2 ([7]) and remains open in higher dimensions. The Kakeya maximal conjecture implies the Kakeya set conjecture, as is demonstrated in Section 1.5 below. Table 1.3 gives the best known results for the Kakeya maximal conjecture in higher dimensions. Table 1.3: State of the art for the Kakeya maximal conjecture 𝑛 2 3 4 5 6 7 8 9 10 .. . 1.4 dim𝐻 (𝐾) ě 2 5/2 + 𝜀 3.057. . . 18/5 4 34/7 21/4 6 13/2 Result proved by Córdoba Katz-Zahl Katz-Zahl Hickman-Rogers-Zhang Wolff Hickman-Rogers-Zhang Hickman-Rogers-Zhang Hickman-Rogers-Zhang Hickman-Rogers-Zhang .. . Importance of the Kakeya maximal conjecture There are many prominent open conjectures in diverse areas of math which have been shown, if they are true, to each imply that the Kakeya set conjecture and the Kakeya maximal conjecture are true. Some of these conjectures are mentioned below. Restriction conjecture. If 𝜎 is the surface measure on the unit sphere S𝑛´1 , is there an estimate || š 𝑓 𝑑𝜎|| 𝐿 𝑝 (R𝑛 ) ≲ || 𝑓 || 𝐿 𝑝 (𝑑𝜎) for all 𝑝 ě 𝑛2𝑛 ´1 . (There are also versions of the restriction conjecture for other hypersurfaces with boundary, rather than the sphere.) It was proved by Fefferman in [9] that the restriction conjecture implies the Kakeya maximal conjecture. Bochner-Riesz conjecture. 𝛿 𝑓 (𝜉) = (1´|𝜉 | 2 ) 𝛿 𝜒 (𝜉) 𝑓ˆ(𝜉), If 𝛿 ą 0, and if 𝑆 𝛿 is an operator on 𝐿 𝑝 (R𝑛 ) defined by 𝑆d 𝐵 𝑛 where 𝜒𝐵 is the characteristic function of the unit ball in R , then ||𝑆 𝛿 𝑓 || 𝐿 𝑝 (R𝑛 ) ď 𝐶 || 𝑓 || 𝐿 𝑝 (R𝑛 ) 6 for all 𝑝 satisfying 1𝑝 ´ 12 ď 2𝛿+1 2𝑛 . Tao showed that the Bochner Riesz conjecture implies the restriction conjecture ([22]). The greater importance of the Kakeya maximal conjecture is that there are methods, first developed by Bourgain in 1991, whereby progress on the Kakeya maximal conjecture would imply progress on the Bochner-Riesz conjecture and restriction conjecture ([6]). However, even if the Kakeya maximal conjecture were proved in full, the Bochner Riesz and restriction conjectures would not be completely solved. Local smoothing conjecture. Let 𝑢 be the solution of the initial value problem for the wave equation in 𝑛 space dimensions: B𝑢 (¨, 0) = 0. □𝑢 = 0, 𝑢(¨, 0) = 𝑓 , B𝑡 Then @𝜀 ą 0 D𝐶𝜀 ą 0 : ||𝑢|| 𝐿 𝑝 (R𝑛 ˆ [1,2]) ď 𝐶𝜀 || 𝑓 || 𝑝,𝜀 , 𝑝 for all 𝑝 P [2, 𝑛2𝑛 ´1 ]. Here || ¨ || 𝑝,𝜀 is the inhomogeneous 𝐿 Sobolev norm with 𝜀 derivatives. This conjecture was made by Sogge and is known to imply the Bochner Riesz conjecture ([20]). Montgomery’s conjecture. Assume 𝑇 ď 𝑁 2 . Let 𝐷 (𝑠) = 𝑁 ∑︁ 𝑎 𝑛 𝑛𝑖𝑠 𝑛=1 where ||{𝑎 𝑛 }|| ℓ8 ď 1. Let T be a 1-separated subset of [0, 𝑇]. Then is there an estimate ∑︁ @𝜀 ą 0 D𝐶𝜀 ą 0 : |𝐷 (𝑡)| 2 ď 𝐶𝜀 𝑁 1+𝜀 (𝑁 + |T |)? 𝑡 PT Bourgain showed in [4] that Montgomery’s conjecture implies the Kakeya set conjecture. 1.5 Equivalent versions of the Kakeya maximal conjecture For the following sections, it may be more convenient to work with some equivalent formulations of the Kakeya maximal conjecture, which are written below. 7 Conjecture 5 (Kakeya maximal conjecture, version 2). For every 𝜀 ą 0, if T is a family of 𝛿-tubes pointing in 𝛿-separated directions, there is a number 𝐶𝜀 ą 0 such that ∑︁ ∑︁
(𝑛´1)/𝑛 || 𝜒𝑇 || 𝐿 𝑛 (R𝑛´1 ) ď 𝐶𝜀 𝛿´𝜀 |𝑇 | . 𝑇 PT 𝑇 PT Conjecture 6 (Kakeya maximal conjecture, version 3). For every 𝜀 ą 0, if (i) T is a set of 𝛿-tubes in R𝑛 that point in 𝛿-separated directions, and (ii) 𝑌 (𝑇) is a subset of 𝑇 for every 𝑇 P T, and 0 ď 𝜆 ď 1 is a number such that |𝑌 (𝑇)| ě 𝜆|𝑇 | for every 𝑇 P T, then there exists a constant 𝐶𝜀 ą 0 such that Ø 𝑌 (𝑇) ě 𝐶𝜖 𝛿 𝜀 𝜆𝑛 𝑇 PT ∑︁ |𝑇 |. (1.1) 𝑇 PT Theorem 7. The three versions of the Kakeya conjecture are equivalent. It will be shown that version 3 ñ version 2 ñ version 1 ñ version 3. Proof that version 3 implies version 2. Assume that version 3 of the Kakeya conjecture is true. Let T be a set of 𝛿-tubes pointing in 𝛿-separated directions. For every nonnegative integer 𝑘, define 𝐸 𝑘 = {𝑥 P R𝑛 : 2𝑘 ď #T(𝑥) ă 2 𝑘 ´1 }. The statement of version 2 of the Kakeya conjecture is equivalent to ∑︁ ∑︁ |𝐸 𝑘 |2 𝑘𝑛/(𝑛´1) ď 𝐶𝜖 𝛿´𝜖/(𝑛´1) |𝑇 | 𝑇 PT 𝑘 By the pigeonhole principle, there exists a value of 𝑘 for which |𝐸 𝑘 |2 𝑘𝑛/(𝑛´1) ≳ ∑︁ 1 |𝐸 𝑘 |2 𝑘𝑛/(𝑛´1) . log(1/𝛿) 𝑘 Since log(1/𝛿) ≲ 𝛿 𝜀 for any 𝜀 ą 0, it is sufficient to show that for this value of 𝑘, ! 𝑛´1 |𝐸 𝑘 | 𝑛´1 2 𝑘𝑛 ď 𝐶𝜖 𝛿´𝜀 ∑︁ 𝑇 PT |𝑇 | 8 For each 𝑇 P T, define a shading 𝑌𝑘 (𝑇) by 𝑌𝑘 (𝑇) = 𝑇 X 𝐸 𝑘 . Note that for a certain value of ℓ, we have ∑︁ 1 ∑︁ ´ℓ 1 Ø |𝐸 𝑘 | ¨ 2 𝑘 „ |𝑌𝑘 (𝑇)| „ | 2 |𝑇 | „ 𝑌𝑘 (𝑇)| ¨ 2 𝑘 , log 𝛿 𝑇 PT log 𝛿 𝑇 PT 𝑇 PT ℓ ℓ where Tℓ = {𝑇 P T : 2´ℓ´1 ă |𝑌𝑘 (𝑇)| ď 2´ℓ }. This leads to two conclusions, first |𝐸 𝑘 | « | Ø 𝑌𝑘 (𝑇)| 𝑇 PTℓ and second 2´ℓ 𝛿 𝑛´1 #Tℓ . |𝐸 𝑘 | From the second conclusion, the statement to show becomes ! 𝑛 ´1 Í ∑︁ 2´ℓ𝑑 ( 𝑇 PTℓ |𝑇 |) 𝑛 |𝑇 | . ď 𝐶𝜖 𝛿 ´𝜖 |𝐸 𝑘 | 𝑇 PT 2𝑘 « Rearranging and using the first conclusion above, this is Ø ∑︁ | 𝑌𝑘 (𝑇)| ≳ 𝛿 𝜖 2´ℓ𝑛 |𝑇 |, 𝑇 PTℓ 𝑇 PTℓ □ which is true from version 3 of the Kakeya conjecture. 1 (R𝑛 ). Proof that version 2 implies version 1. Let 𝜀 ą 0 be given, and let 𝑓 P 𝐿 loc Let Ω Ď S𝑛´1 be a maximal 𝛿-separated set of directions. For every 𝑒 P Ω define 𝑇𝑒 as the 𝛿-tube centered at the position that achieves the supremum ∫ sup | 𝑓 (𝑥)| 𝑑𝑥, 𝑎 PR𝑛 𝑇𝑒𝛿 (𝑎) and then set T := {𝑇𝑒 : 𝑒 P Ω}. From version 2 of the conjecture, ∑︁ || 𝜒𝑇 || 𝐿 𝑛 (R𝑛 ) ≲ 𝛿´𝜀 . 𝑇 PT The crucial observation used is that, provided |𝑒 ´ 𝑒 1 | ă 𝛿, there is some 𝐶 ą 0 such that | 𝑓𝛿˚ (𝑒)| „ | 𝑓𝐶˚𝛿 (𝑒 1 )|. Thus ∑︁ ∫ ∑︁ ˚ 𝑛 ˚ || 𝑓𝛿 || 𝐿 𝑛 (S𝑛´1 ) ≲ | 𝑓𝛿˚ (𝜔)| 𝑛 𝑑𝜔 ≲ 𝛿 𝑛´1 | 𝑓𝐶𝛿 (𝑒)| 𝑛 . 𝑒 PΩ 𝑁 𝛿 ({𝑒}) 𝑒 PΩ 9 Using duality between ℓ𝑛 and ℓ𝑛1 , there exist numbers {𝑦 𝑒 } 𝑒PΩ such that ! 1/𝑛1 ! 1/𝑛 𝛿 𝑛 ´1 ∑︁ 𝑦 𝑒 𝑓𝛿˚ (𝑒) = 𝛿 𝑛´1 𝑒 PΩ ∑︁ | 𝑓𝛿˚ (𝑒)| 𝑛 𝛿 𝑛 ´1 𝑒 PΩ ∑︁ |𝑦 𝑒 | 𝑛 1 𝑒 PΩ and furthermore 𝛿 𝑛 ´1 ∑︁ 1 |𝑦 𝑒 | 𝑛 = 1. 𝑒 PΩ In the following estimates we will abuse notation by using 𝑦 𝑒 interchangably with 𝑦𝑇 , where 𝑇 P T is understood to be 𝑇𝑒 as defined above in the proof. ! 1/𝑛 || 𝑓𝛿˚ || 𝐿 𝑛 (S𝑛´1 ) ≲ 𝛿 𝑛´1 ≲𝛿 𝑛 ´1 ∑︁ | 𝑓𝛿˚ (𝑒)| 𝑛 𝑒 PΩ ∑︁ 𝑦 𝑒 ¨ | 𝑓𝛿˚ (𝑒)| 𝑒 PΩ ∫ ∑︁ ≲ 𝜒𝑇 (𝑥) ¨ 𝑦𝑇 ¨ 𝑓 (𝑥) 𝑑𝑥 𝑇 PT ∑︁ ď || 𝑦𝑇 ¨ 𝜒𝑇 || 𝑛/(𝑛´1) ¨ || 𝑓 || 𝐿 𝑛 (R𝑛 ) . 𝑇 PT It now remains to show that ∑︁ || 𝑦𝑇 ¨ 𝜒𝑇 || 𝑛/(𝑛´1) ≲ 𝛿´𝜀 . 𝑇 PT By dyadic pigeonholing, it is possible to choose a subset T1 Ď T for which ∑︁ ∑︁ 𝑦𝑇 ¨ 𝜒𝑇 || 𝑛/(𝑛´1) ⪆ || 𝑦𝑇 ¨ 𝜒𝑇 || 𝑛/(𝑛´1) || 𝑇 PT1 𝑇 PT and for which there is number 𝑁 P N such that 𝑦𝑇 „ 2𝑁 for all 𝑇 P T1 . It now follows that ∑︁ ∑︁ || 𝑦𝑇 ¨ 𝜒𝑇 || 𝑛/(𝑛´1) ≲ 2𝑁 ¨ || 𝜒𝑇 || 𝑛/(𝑛´1) 𝑇 PT ≲ 2𝑁 𝛿 𝑇 PT1 ´ 𝜀 𝑛 ´1 (𝛿 #T1 ) 1/𝑛 1 ! 1/𝑛1 „ 𝛿 ´ 𝜀 𝛿 𝑛 ´1 ∑︁ |𝑦𝑇 | 𝑛 1 𝑇 PT1 ď𝛿 and this completes the proof. ´𝜀 , □ 10 Proof that version 1 implies version 3. Let 𝜀 ą 0 be given, and let T be a set of 𝛿-tubes which point in 𝛿-separated directions. Let 𝑌 (𝑇) Ď 𝑇 be given for every 𝑇 P T and let 0 ă 𝜆 ă 1 be chosen so that |𝑌 (𝑇)| ě 𝜆|𝑇 | for every 𝑇 P T. By dyadic pigeonholing, it is possible to find a number 𝜆1 and a subset T1 Ă T for which 𝜆1 |𝑇 | ď |𝑌 (𝑇)| ă 2𝜆1 |𝑇 | for all 𝑇 P T1 and ∑︁ |𝑌 (𝑇)| ≳ 𝑇 PT1 ∑︁ 1 |𝑌 (𝑇)|. log(1/𝛿) 𝑇 PT The above statement implies that 𝜆1 ≳ 𝜆/log(1/𝛿). (1.2) Ð Let 𝑓 be the indicator function of 𝑇 PT1 𝑌 (𝑇). Let Ω = {dir(𝑇) : 𝑇 P T1 } and let 𝑇𝑒 P T1 be the 𝛿-tube pointing in direction 𝑒 for every 𝑒 P Ω. Now ∑︁ ∑︁ ∫ 1 𝑛 ´1 1 𝑓 (𝑥) 𝑑𝑥 𝜆 𝛿 #T „ |𝑌 (𝑇)| ≲ 𝑒 PΩ 𝑇𝑒 𝑇 PT1 ≲ ∑︁ 𝛿 𝑛´1 𝑓𝛿˚ (𝑒) 𝑒 PΩ ≲ ∑︁ ∫ 𝑓𝛿˚ (𝜔) 𝑑𝜎(𝜔) 𝑒 PΩ 𝑁 𝛿 ({𝑒}) ≲ || 𝑓𝛿˚ || 𝐿 𝑛 (S𝑛´1 ) ¨ (𝛿 𝑛´1 #Ω) (𝑛´1)/𝑛 . Thus, from version 1 of the conjecture, || 𝑓 || 𝐿 𝑛 (R𝑛 ) ⪆ 𝜆1 (𝛿 𝑛´1 #T1 ) 1/𝑛 . From (1.2), ! | Ø 𝜀 𝑛 𝑌 (𝑇)| ≳ 𝛿 𝜆 𝑇 PT ∑︁ |𝑇 | , 𝑇 PT which is the conclusion of version 3 of the Kakeya maximal conjecture. 1.6 □ Proof of the Kakeya maximal conjecture in R2 In this section, it will be shown that version 2 of the Kakeya conjecture is true for R2 . The statement to prove is the following. For every 𝜀 ą 0, there is a constant 𝐶𝜀 ą 0 such that whenever T is a set of 𝛿-tubes pointing in 𝛿-separated directions in R2 , ∫ ∑︁ | 𝜒𝑇 | 2 ď 𝐶𝜀 𝛿´𝜀 𝛿#T. R 𝑛 𝑇 PT 11 The above statement may be re-written as ∫ ∑︁ ∑︁ 𝜒𝑇1 𝜒𝑇2 = |𝑇1 X 𝑇2 | ď 𝐶𝜀 𝛿´𝜀 (𝛿#T) 2 . R𝑛 𝑇 ,𝑇 PT 1 (1.3) 𝑇1 ,𝑇2 PT 2 Let 𝜀 ą 0 be given. Observe that for any two 𝛿-tubes 𝑇1 and 𝑇2 , the area of the intersection 𝑇1 X 𝑇2 depends on the angle 𝜃 = ∠(dir(𝑇1 ), dir(𝑇2 )): 𝛿2 𝛿2 ≲ . sin 𝜃 𝜃 |𝑇1 X 𝑇2 | ď For each dyadic number 𝜃, let T𝜃 [𝑇1 ] = {𝑇2 P T : 𝜃 ď ∠(dir(𝑇1 ), dir(𝑇2 )) ă 2𝜃}. Since the tubes in T point in 𝛿-separated directions, #T𝜃 [𝑇1 ] ≲ 𝜃𝛿´1 for every 𝑇1 P T. Thus, ∑︁ ∑︁ ∑︁ ∑︁ 𝛿2 |𝑇1 X 𝑇2 | ≲ 𝜃 𝑇1 ,𝑇2 PT 𝑇1 PT 𝜃 𝑇2 PT 𝜃 [𝑇1 ] ∑︁ ∑︁ ≲ 𝛿. 𝑇1 PT 𝜃 Since there are „ log(1/𝛿) possible values of 𝜃, and since there always exists a constant 𝐶𝜀 ą 0 such that log(1/𝛿) ď 𝐶𝜀 𝛿´𝜀 , this proves (1.3). 1.7 Proof that the Kakeya maximal conjecture implies the Kakeya set conjecture Slightly different versions of the following proof have been written down by Wolff ([24]) and by Tao ([21]) in collections of lecture notes. The proof will show that version 3 of the Kakeya maximal conjecture implies that every Besicovitch set in R𝑛 has Hausdorff dimension equal to 𝑛. Since the Minkowski dimension of a set is never less than its Hausdorff dimension, this will prove that that every Besicovitch set in R𝑛 also has Minkowski dimension 𝑛. The reader will observe that the full strength of the Kakeya maximal conjecture (version 3) will not be needed in the proof; rather, the subcase of the Kakeya maximal conjecture where 𝜆 ≳ (1/log(1/𝛿)) 2 will be sufficient. Let 𝐵 be a Besicovitch set in R𝑛 and let 𝜀 ą 0 be arbitrary. The goal is to prove that the (𝑛 ´ 𝜀)-dimensional Hausdorff measure H 𝑛´𝜀 (𝐵) is infinite. For each direction 𝜔 P S𝑛´1 , let ℓ𝜔 be a unit line segment which is contained in 𝐵 and which points in the direction 𝜔. For any set 𝐸 Ď R𝑛 , define a measure 𝜇 by ∫ 𝜇(𝐸) = |𝐸 X ℓ𝜔 | 𝑑𝜔, S𝑛´1 12 where |𝐸 X ℓ𝜔 | is the 1-dimensional Lebesgue measure of 𝐸 X ℓ𝜔 . Thus 𝜇(𝐵) = 1. This measure is sometimes called the canonical measure associated to a Besicovitch set. Let 𝛿0 ą 0 be given and let U be a covering of 𝐵 by balls of any radius smaller than 𝛿0 . For any 𝑘 P Z, let U𝑘 be the subcollection of balls given by U𝑘 = {𝑈 P U : 2´ 𝑘 ´1 ă diam(𝑈) ď 2´ 𝑘 } and define 𝐵 𝑘 be the union of all 𝑈 P U𝑘 . Since ∑︁ 1 ≲ 1, 2 𝑘 ´𝑘 𝑘 PZ : 2 ă𝛿 0 there must exist a value of 𝑘 for which 𝜇(𝐵 𝑘 ) ≳ 1 . 𝑘2 Fix 𝛿 := 2´ 𝑘 . Let T0 be the collection of 𝛿-tubes 𝑇𝜔 , each defined as the 𝛿neighborhood of the line segment ℓ𝜔 , for every 𝜔 P S𝑛´1 . Observe that ∫ ∫ 1´ 𝑛 𝜇(𝐵 𝑘 ) = 𝛿 1𝑇𝜔 (𝑥) 𝑑𝑥 𝑑𝜔. S 𝑛 ´1 𝐵𝑘 Consider the following tiling of S𝑛´1 : let R𝑛´1 be tiled by 𝛿-cubes given by translates (0, 𝛿] 𝑛´1 placed end-to-end and let a “tile” on the sphere be given by the image of a one of these 𝛿-cubes via the map 𝜉 ÞÑ (𝜉, 1 ´ |𝜉 | 2 ) sending R𝑛´1 to the northern hemisphere. Thus the sphere S𝑛´1 may be covered by tiles 𝜃 so that ∑︁ ∫ ∫ 1´ 𝑛 𝜇(𝐵 𝑘 ) = 𝛿 1𝑇𝜔 (𝑥) 𝑑𝑥 𝑑𝜔. 𝜃 𝜃 𝐵𝑘 Further, it may arranged that half of all the tiles 𝜃 are removed from the sum above without reducing the right hand side above by a factor of more than one half, so that any two tiles are separated by a distance of at least 𝛿. For every remaining tile 𝜃, ∫ ∫ ∫ there must exist 𝜔 P 𝜃 such that 𝐵 1𝑇𝜔 (𝑥) 𝑑𝑥 ě 𝜃 𝐵 1𝑇𝜔 (𝑥) 𝑑𝑥 𝑑𝜔. Let T be 𝑘 𝑘 the subcollection of T0 consisting of this choice of 𝜔 for every tile 𝜃; then there are „ 𝛿1´𝑛 tubes in T and they point in 𝛿-separated directions. Define a shading for each tube 𝑇 P T by 𝑌 (𝑇) := 𝑇 X 𝐵 𝑘 , where 𝑘 = log(1/𝛿) is the chosen value of 𝑘 above, for which 𝜇(𝐵 𝑘 ) ≳ 𝑘 ´2 . Thus  2 ∑︁ 1 |𝑌 (𝑇) X 𝐵 𝑘 | ≳ 𝜇(𝐵 𝑘 ) ě . log(1/𝛿) 𝑇 PT 13 Using dyadic pigeonholing, it is possible to find a refinement T1 Ď T and a number Í 0 ă 𝜆 ă 1 such that |𝑌 (𝑇)| „ 𝜆|𝑇 | for every 𝑇 P T1 , and further 𝑇 PT1 |𝑌 (𝑇)| ≳ Í 1 1 log(1/𝛿) 𝑇 PT 𝑌 (𝑇). From this, and the fact that T continues to be direction separated, it follows that 𝜆 ≳ (1/log(1/𝛿)) 3 . With an appropriate implicit constant, it may then be arranged that 𝜆 ≳ 𝛿 𝜀/4𝑛 , and if the Kakeya maximal conjecture is true, Ø | 𝑌 (𝑇)| ≳ 𝛿 𝜀/2 , 𝑇 PT then it follows that |𝐵 𝑘 | ≳ 𝛿 𝜀 . Since each ball in U𝑘 has volume „ 𝛿 𝑛 , it must be that #U𝑘 ≳ 𝛿´𝑛+𝜀/2 . Thus, ∑︁ 𝛿 𝑛´𝜀 ě 𝛿´𝜀/2 , 𝑈 PU𝑘 and H 𝑛´𝜀 (𝐵) = 8. This completes the proof. 1.8 Introduction to the x-ray transform estimate and sticky sets In 1998, Wolff observed a certain estimate involving the x-ray transform, defined below, had a close connection to the Kakeya problem ([25]). The question considered by Wolff about the x-ray transform is written below: Let G be the space of all lines ℓ = ℓ𝑥,𝑣 Ă R𝑛 , where 𝑥, 𝑣 are points within the unit ball centered at the origin in R𝑛´1 , and ℓ𝑥,𝑣 is parametrized by ℓ𝑥,𝑣 = {(𝑥 + 𝑣𝑡, 𝑡) : 𝑡 P [0, 1]}. For any function 𝑓 P 𝐿 1 (R𝑛 ), the x-ray transform 𝑋 𝑓 : G Ñ R is defined by ∫ 𝑋 𝑓 (ℓ) = 𝑓. ℓ For which 1 ď 𝑝, 𝑞, 𝑟 ď 8 and 𝛼 ě 0 does the estimate ||𝑋 𝑓 || 𝐿 𝑞𝜈 𝐿 𝑟𝑥 ď || 𝑓 || 𝐿 𝛼𝑝 , 𝑝 hold? Here 𝐿 𝛼 is the Sobolev space (1 + ? ´Δ) ´𝛼 𝐿 𝑝 . Various estimates of this kind have been proved (e.g. [25]) and more are conjectured to hold. The relation of this question to the Kakeya problem becomes more evident in the following equivalent formulation of a certain mixed norm x-ray estimate: 14 Definition 8. We say there is an x-ray estimate at dimension 𝑑 if there exist 0 ď 𝛼, 𝛾 ď 1 for which the following statement holds: for any 𝛿-separated set Ω of directions and any collection T of 𝛿-tubes pointing in directions in Ω, of which no tube is contained in the tenfold dilate of another tube, the following estimate holds: ∑︁ 𝑛 || 𝜒𝑇 || 𝑑 1 ⪅ 𝛿1´ 𝑑 𝑚 1´𝛼 (𝛿 𝑛´1 #Ω) 𝛾 , 𝑇 PT where 𝑚 is the directional multiplicity of T. It was first observed by Katz, Łaba and Tao that an x-ray transform estimate could be used to deduce the prevalence of special structure in “extremal” Besicovitch sets. More precisely, the following result is proved in [17]: Proposition 9. Suppose that an x-ray transform estimate holds at dimension 𝑑 in R𝑛 . If there exists a Besicovitch set 𝐵 satisfying dim 𝑀 (𝐵) ď 𝑑 + 𝜀, then there exists a set of 𝛿-tubes T with the following properties: (i) The tubes in T point in 𝛿-separated directions, (ii) There exists a set of 𝛿1/2 -tubes T𝛿1/2 and a partition of T into sets T[𝑇𝛿1/2 ] given by T[𝑇𝛿1/2 ] = {𝑇 P T : 𝑇 Ď 𝑇𝛿1/2 }; satisfying the following properties: #T « 𝛿1´𝑛 , #T𝛿1/2 « 𝛿 (1/2) (1´𝑛) and #T[𝑇𝛿1/2 ] « 𝛿 (1/2) (1´𝑛) (iii) | Ð (iv) | Ð 𝑇 PT 𝑇 | ⪅ 𝛿 𝑛 ´ 𝑑 ´𝜀 . 𝑇 𝛿 1/2 PT 𝛿 𝑇 | ⪅ (𝛿 1/2 ) 𝑛´𝑑 ´𝜀 . The proof of a slightly different version of this proposition is given in Section 4.5. Sets of 𝛿-tubes satisfying properties (i) and (ii) in the list above are called sticky sets. Katz-Łaba-Tao ([17]) demonstrated that sticky sets have a self-similar structure and exploited the structural properties of sticky sets to give an improved Minkowski dimension estimate for Besicovitch sets in R3 . Very recently, the Kakeya conjecture was proved by Wang and Zahl ([23]) in the special case of sticky sets in R3 . Chapter 4 in this thesis gives an improved Hausdorff dimension estimate for sticky sets in higher dimensions. 15 Chapter 2 RECENT PROGRESS IN THE 𝑘-LINEAR KAKEYA CONJECTURE 2.1 The multilinear Kakeya theorem The most recent results in the higher-dimensional Kakeya problem have been proved by means of progress on a variant of the Kakeya problem called the multilinear Kakeya problem. The multilinear variant of the Kakeya problem was first introduced by Benett, Carbery and Tao in [1]. The following theorem was proved up to endpoint in [1] and the endpoint was proved by Guth in [10]. Theorem 10 (Multilinear Kakeya theorem). Let 2 ď 𝑘 ď 𝑛. There exists a constant 𝐶 so that, if T1 , . . . , T 𝑘 are collections of 𝛿-tubes in R𝑛 , then ∑︁ ∑︁
1/𝑘 || ¨¨¨ 𝜒𝑇1 ¨ ¨ ¨ 𝜒𝑇𝑘 |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| || 𝑘/(𝑘 ´1) 𝑇1 PT1 𝑇𝑘 PT 𝑘 𝑛 ď 𝐶𝛿1´ 𝑘 𝑘 Ö ∑︁ 𝑖=1 |𝑇𝑖 |
1/𝑘 𝑇𝑖 PT𝑖 As was observed by Bourgain and Guth in 2010, the multlinear Kakeya estimates could lead to estimates for the Kakeya maximal function, but the resulting estimates were weaker than the maximal estimates proved previously by Wolff ([26]) using incidence geometry techniques. Though the multilinear Kakeya theorem is sharp, the multilinear variant of the Kakeya problem does not impose the requirement that the 𝛿-tubes point in 𝛿separated directions. The direction-separated multilinear Kakeya conjecture is still open. The best known estimates for this problem are due to Zahl ([28]). A very similar though slightly weaker estimate was proved by Hickman-Rogers-Zhang in [12]. Theorem 11 (Zahl, 𝑘-linear estimate). Let 2 ď 𝑘 ď 𝑛. For every 𝜖 ą 0, if T is a set of 𝛿-tubes pointing in 𝛿-separated directions, then there is a constant 𝐶𝜖 ą 0 such . 16 ´𝑘 that, for all 𝑑 ď 𝑛 +𝑛+𝑘 , 2𝑛 ∑︁ ||( 1𝑇1 ¨ ¨ ¨ 1𝑇𝑘 |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| 𝑘/𝑑 ) 1/𝑘 || 𝑑/(𝑑 ´1) 2 2 𝑇1 ,...,𝑇𝑘 PT 𝑛 ď 𝐶 𝜖 𝛿 1´ 𝑑 ´ 𝜖 ( ∑︁ |𝑇 |) 𝑛(𝑑 ´1)/(𝑛´1)𝑑 . 𝑇 PT The next two sections (on polynomial partitioning and the polynomial Wolff axioms) mention some of the prominent insights in the proof of the above theorem. Directly below is a corollary of Zahl’s 𝑘-linear estimate that will be applied in Chapter 4. Corollary 12. Let 2 ď 𝑘 ď 𝑛. For every 𝜖 ą 0, if T is a set of 𝛿-tubes satisfying the following hypotheses: i. The tubes in T point in 𝛿-separated directions, ii. 𝑌 is a shading of T satisfying |𝑌 (𝑇)| ě 𝜆|𝑇 | for every 𝑇 P T, iii. At every point 𝑥 P Ð 𝑇 PT 𝑌 (𝑇), #{(𝑇1 , . . . , 𝑇𝑘 ) P T 𝑘 (𝑥)) : |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| ą 𝛽} ≳ 1 #T 𝑘 (𝑥), log(1/𝛿) where 𝛽 ą 0 is a fixed constant that does not depend on 𝛿, ´𝑘 , then there is a constant 𝐶𝜖 ą 0 such that for all 𝑑 ď 𝑛 +𝑛+𝑘 2𝑛 2 Ø 𝑌 (𝑇) ě 𝐶𝜖 𝛽𝜆 𝑑 𝛿 𝑛´𝑑+𝜖 ( 𝑇 PT ∑︁ 2 |𝑇 |) (𝑛´𝑑)/(𝑛´1) . 𝑇 PT Proof of Corollary 12 from Zahl’s 𝑘-linear theorem. Let 𝜖 ą 0 be given, and let T be a set of 𝛿-tubes that obeys hypotheses i, ii and iii. Then, from condition ii, ∑︁ ∑︁ ||( 1𝑌 (𝑇1 ) ¨ ¨ ¨ 1𝑌 (𝑇𝑘 ) |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| 𝑘/𝑑 ) 1/𝑘 || 𝐿 1 ě 𝛽1/𝑑 𝜆|𝑇 |. 𝑇1 ,...,𝑇𝑘 PT 𝑇 PT By Hölder’s inequality, ∑︁ ||( 1𝑌 (𝑇1 ) ¨ ¨ ¨ 1𝑌 (𝑇𝑘 ) |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| 𝑘/𝑑 ) 1/𝑘 || 1 𝑇1 ,...,𝑇𝑘 PT ď ||( ∑︁ 𝑇1 ,...,𝑇𝑘 PT 1𝑇1 ¨ ¨ ¨ 1𝑇𝑘 |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| 𝑘/𝑑 ) 1/𝑘 || 𝑑/(𝑑 ´1) ¨ | Ø 𝑇 PT 𝑌 (𝑇)| 1/𝑑 . 17 Applying Theorem 11 to the statement above, one obtains ∑︁ Ø |𝑇 |) 𝑑 ´𝑛(𝑑 ´1)/(𝑛´1) ď 𝐶𝜖 𝛿 𝑑 ´𝑛´𝜖 | 𝑌 (𝑇)|. 𝛽𝜆 𝑑 ( 𝑇 PT 𝑇 PT Rearranging this gives the statement Ø ∑︁ | 𝑌 (𝑇)| ě 𝐶𝜖 𝛽𝛿 𝑛´𝑑+𝜖 𝜆 𝑑 ( |𝑇 |) (𝑛´𝑑)/(𝑛´1) . 𝑇 PT 𝑇 PT □ 2.2 Polynomial partitioning method The techniques used for proving Zahl’s 𝑘-linear estimate involve polynomial partitioning, which was introduced by Guth and Katz ([11]) in 2014. For any polynomial 𝑃 : R𝑛 Ñ R, the zero set of 𝑃, 𝑍 (𝑃) := {𝑥 P R𝑛 : 𝑃(𝑥) = 0}, partitions R𝑛 into “cells”, which are the disjoint connected components of R𝑛 z𝑍 (𝑃). Further, a 𝛿-shrunken cell is a connected component of R𝑛 z𝑁 𝛿 (𝑍 (𝑃)). The “idea” of the polynomial partitioning technique is to consider the contributions to the norm of the function to be bounded (in this case, Í ||( 𝑇1 ,...,𝑇𝑘 PT 1𝑇1 ¨ ¨ ¨ 1𝑇𝑘 |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| 𝑘/𝑑 ) 1/𝑘 || 𝑑/(𝑑 ´1) ) from the different cells, and from 𝑁 𝛿 (𝑍 (𝑃)), which is the 𝛿-neighborhood of an algebraic variety. If the polynomial 𝑃 is correctly chosen, the contribution to the function norm in every cell is roughly equal. If the contribution from the cells dominates, the “cellular case” is said to hold; alternatively, if the dominant contribution comes from the 𝛿-neighborhood of the zero set of the polynomial, then the “algebraic case” is said to hold. The following polynomial partitioning theorem is due to Guth. Theorem 13. Fix 0 ă 𝛿 ă 𝑟, 𝑥 0 P R𝑛 and suppose 𝐹 P 𝐿 1 (R𝑛 ) is nonnegative and supported on 𝐵(𝑥 0 , 𝑟) X 𝑁4𝛿 Z where Z is an 𝑚-dimensional transverse complete intersection with deg Z ď 𝑑. At least one of the following cases holds: 1. Cellular case. There exists a polynomial 𝑃 : R𝑛 Ñ R of degree 𝑂 (𝑑) with the following properties: a) #cell(𝑃) „ 𝑑 𝑚 and each 𝑂 P cell(𝑃) has diameter at most 𝑟/2. b) One may pass to a refinement of cell(𝑃) such that if H is defined as a family of 𝛿-shrunken cells as above, then ∫ ∫ ´𝑚 𝐹„𝑑 𝐹 for all 𝑂 P H. 𝑂 R𝑛 18 2. Algebraic case. There exists an (𝑚 ´ 1)-dimensional algebraic variety Y of degree at most 𝑂 (𝑑) such that ∫ ∫ 𝐹 ≲ log 𝑑 𝐹. 𝐵(𝑥 0 ,𝑟) X 𝑁4 𝛿 Z 𝐵(𝑥 0 ,𝑟) X 𝑁 𝛿 𝑌 In the cellular case, Bezout’s theorem is used to deduce that the number of 𝛿-tubes entering each cell can be controlled since by Bezout’s theorem, a 𝛿-tube 𝑇 can cross 𝑍 (𝑃) at most deg 𝑃 times. Thus, each 𝛿-tube 𝑇 can enter at most deg 𝑃 + 1 shrunken cells. As long as the cellular case holds, each cell can be re-partitioned into smaller cells, until the cells have radius „ 𝛿 and the contribution from each cell is bounded. In the algebraic case, one can obtain a chain of nested algebraic varieties, of decreasing dimensions, through which passing 𝛿-tubes makes the dominant contribution to the norm of the function to be estimated. In this case, a “multi-scale” polynomial Wolff axiom, discussed briefly in the next section, is used to estimate the maximum number of 𝛿-tubes that lie at a tangent to several nested algebraic varieties. 2.3 Polynomial Wolff axioms The following estimate for the cardinality of a family of direction-separated 𝛿-tubes partially contained in a semialgebraic subset of R𝑛 is a key result used in proving Theorem 11. The following theorem was proved for 𝑛 = 3 by Larry Guth, for 𝑛 = 4 by Zahl ([27]), and for 𝑛 ě 5 by Katz and Rogers ([16]). Theorem 14 (Polynomial Wolff axioms). Let 𝑛 and 𝐸 be integers, with 𝑛 ě 2, and let 𝜖 ą 0.Then there is a constant 𝐶 (𝑛, 𝐸, 𝜖) so that for every semialgebraic set 𝑆 Ă R𝑛 of complexity at most 𝐸 and for every set T of direction-separated 𝛿-tubes, we have #{𝑇 P T : 𝑇 X 𝑆| ě 𝜆|𝑇 |} ď 𝐶 (𝑛, 𝐸, 𝜖)|𝑆|𝛿1´𝑛´𝜖 𝜆´𝑛 The new feature used in Zahl’s 𝑘-linear estimate, and in Hickman-Rogers-Zhang ([12]), is the “multiscale polynomial Wolff axiom”, which provides a sharp bound for the cardinality of a set of 𝛿-separated 𝛿-tubes that intersect several algebraic varieties of different dimensions at once. Theorem 15 (Multiscale polynomial Wolff axioms). For all 𝑛 ě 𝑚 ě 𝑘 ď 1 and 𝜖 ą 0, there is a constant 𝐶𝑛,𝑑,𝜖 such that   𝑛 ´𝑚 𝑚 ´1 ©Ö 𝜌 ª 𝜌 𝛿´ (𝑛´1) ´𝜖 # {𝑇 P T : |𝑇 X 𝐵𝜆 𝑗 X 𝑁 𝜌 Z 𝑗 | ě 𝜆 𝑗 |𝑇 |} ď 𝐶𝑛,𝑑,𝜖 ­ ® 𝜆 𝑗 𝜆𝑚 𝑗=𝑘 « 𝑗=𝑘 ¬ 𝑚 Ù 19 whenever 0 ă 𝛿 ď 𝑝 ď 𝜆 𝑘 ď ¨ ¨ ¨ ď 𝜆 𝑚 ď 1, T is a direction-separated family of 𝛿-tubes, Z 𝑗 Ă R𝑛 are 𝑗-dimensional algebraic varieties of degree ď 𝑑 and the balls 𝐵𝜆 𝑗 are nested: 𝐵𝜆 𝑘 Ď . . . Ď 𝐵𝜆 𝑚 Ă R𝑛 . This theorem is proved using tools from real algebraic geometry. There are some open questions related to the polynomial Wolff axioms which, if solved, could lead to progress in proving x-ray transform estimates at higher dimensions. Namely, is there a cardinality bound for sets of 𝛿-tubes partially contained in the neighborhood of an algebraic variety, which do not necessarily satisfy the hypothesis that the tubes point in 𝛿-separated directions, but instead, satisfy the hypothesis that the directional multiplicity of tubes is at most 𝛿´ 𝛽𝑛 for some number 𝛽 between 0 and 1? Such a set of 𝛿-tubes would also have to satisfy the condition that no tube in the set contains the 5-fold dilate of another one. 20 Chapter 3 NEW KAKEYA ESTIMATE FOR 𝑘-PLANY SETS The Kakeya estimate obtained in this chapter is proved using a geometric structure called the “𝑘-planebrush”. A planebrush structure has been used before to prove Kakeya maximal estimates in R4 by Katz and Zahl in [14]. The planebrush is a higher-dimensional analogue of the “hairbrush” appearing in Wolff ([26]). Wolff’s hairbrush consisted of all 𝛿-tubes intersecting a single, central 𝛿-tube 𝑇 at large angles in R3 . Each 𝛿-tube passing through 𝑇 at a large angle lies in the 𝛿-neighborhood of a plane that contains the axis of 𝑇. The 𝛿-thickened planes containing the axis of 𝑇 are disjoint from each other away from 𝑇, and the each tube in the hairbrush lies in exactly one thickened 𝑘-plane. The Kakeya maximal estimate for the plane is applied separately to each thickened plane to get a Kakeya estimate for all the tubes in the hairbrush. In the argument used in this chapter, the 𝑘-planebrush consist of all tubes 𝑇 1 which intersect at least one of many tubes 𝑇 lying tangent to a (𝑘 ´ 1)-plane 𝑉. The tubes in the planebrush each lie in exactly one of up to 𝛿´𝑛+𝑘 𝛿-thickened 𝑘-planes that all contain 𝑉. Away from a neighborhood of 𝑉, the different thickened 𝑘-planes are disjoint, and the best known result for the Kakeya maximal conjecture in R 𝑘 is applied to separately to each thickened 𝑘-plane. In order to ensure that each 𝑘-planebrush contains enough 𝛿-tubes to give a good estimate, the 𝑘-planebrush technique requires the hypothesis that the collection of T is (𝑘 ´ 1)-plany. The meaning of this term is explained below: Definition 16 (m-Planiness). A family of 𝛿-tubes T with shading 𝑌 is called 𝑚plany if for every 𝛿-cube 𝑄, there exists an 𝑚-plane 𝑉 (𝑄) such that for every 𝑇 P T satisfying 𝑌 (𝑇) X 𝑄 ≠ H, ∠(𝑇, 𝑉 (𝑄)) ≲ 𝛿. The main result in this chapter is the following: Proposition 17. Let 3 ď 𝑘 ď 𝑛. For any 𝜀 ą 0, if T is a set of (𝑘 ´ 1)-plany 𝛿-tubes pointing in 𝛿-separated directions, and if each 𝑇 P T has a shading 𝑌 (𝑇) Ď 𝑇 21 satisfying |𝑌 (𝑇)| ě 𝜆|𝑇 |, then there exists a constant 𝐶𝜀 ą 0 such that Ø ∑︁ | 𝑌 (𝑇)| ≳ 𝜆 𝑑 𝛿 𝑛´𝑑+𝜀 |𝑇 |, 𝑇 PT 𝑇 PT ? where 𝑑 ď 2𝑛+2´ (3 2´1)𝑘 . The above result is obtained by combining the next proposition with the Kakeya maximal estimate obtained from Hickman-Rogers-Zhang. The proof of Proposition 17 is given at the end of this chapter. Proposition 18. Let 𝜀 ą 0. If T is a set of 𝛿-tubes satisfying the following hypotheses: i. The tubes in T point in 𝛿-separated directions, ii. Each tube 𝑇 P 𝑇 has a subset 𝑌 (𝑇) and |𝑌 (𝑇)| ě 𝜆|𝑇 | for all 𝑇 P T, iii. The family T with shading 𝑌 is (𝑘-1)-plany, then there exists a constant 𝐶𝜖 ą 0 such that Ø 𝑌 (𝑇) ě 𝐶𝜖 𝛿 (𝑛´2+𝑘 ´𝑑 (𝑘))/3+𝜖 𝜆 𝑑 (𝑘)/3+1 ( 𝑇 PT ∑︁ |𝑇 |), (3.1) 𝑇 PT where 𝑑 (𝑘) is the largest exponent for which a Kakeya maximal estimate holds in R𝑘 . This proposition is proved in section 3.4 of this chapter. 3.1 Two-ends reduction The “two-ends reduction” was first introduced by Wolff ([25]) and has been reused by many others in proving Kakeya estimates. The proof given below imitates one given by Tao’s lecture notes [21]. Definition 19 (Two-ends condition). Let 𝜖 ą 0. A set 𝐹 Ď R𝑛 obeys the “two-ends” condition with exponent 𝜖 if |𝐹 X 𝐵(𝑥, 𝑟)| ≲ 𝑟 𝜖 |𝐹 | for all balls 𝐵(𝑥, 𝑟). 22 Proposition 20. Let 𝜖 ą 0. If T is a set of 𝛿-tubes satisfying the following hypotheses: i. The tubes in T point in 𝛿-separated directions, ii. Each tube 𝑇 P 𝑇 has a subset 𝑌 (𝑇) and |𝑌 (𝑇)| ě 𝜆|𝑇 | for all 𝑇 P T, iii. The family T with shading 𝑌 is (𝑘-1)-plany, iv. For every 𝑇 P T, 𝑌 (𝑇) obeys the two-ends conjecture with exponent 𝜖, then there exists a constant 𝐶𝜖 ą 0 such that Ø 𝑌 (𝑇) ě 𝐶𝜖 𝛿 (𝑛´2+𝑘 ´𝑑 (𝑘))/3+𝜖 𝑑 (𝑘)/3+1 𝜆 𝑇 PT ( ∑︁ |𝑇 |), 𝑇 PT where 𝑑 (𝑘) is the largest exponent for which a Kakeya maximal estimate holds in R𝑘 . Lemma 21 (Two-ends reduction). Proposition 20 implies Proposition 18. Proof. Let 𝜖 ą 0 be given. Suppose that Proposition 20 is true, and let T be a set of 𝛿-tubes obeying hypotheses i, ii and iii in the statement of Proposition 18. For any tube 𝑇 P T, consider the ratio |𝑌 (𝑇) X 𝐵(𝑥, 𝑟)| 𝑟𝜖 (‹) where 𝐵(𝑥, 𝑟) is a ball of radius 𝑟 centered at point 𝑥. If 𝛿 ď 𝑟 ď 1, then |𝑌 (𝑇) X 𝐵(𝑥, 𝑟)| ď 𝛿 𝑛´1𝑟 ď 𝑟 𝜖 , so (‹) is bounded above by 1. If 𝑟 ă 𝛿, then |𝑌 (𝑇) X 𝐵(𝑥, 𝑟)| ≲ 𝑟 𝑛 ď 𝑟 𝜖 . Thus, the set   |𝑌 (𝑇) X 𝐵(𝑥, 𝑟)| : 𝐵(𝑥, 𝑟) is a ball with 𝑟 ď 1 . 𝑟𝜖 is bounded above and has a supremum. If 𝑟 = 1, and the point 𝑥 is chosen so that 𝐵(𝑥, 𝑟) contains 𝑌 (𝑇), one can see that the supremum must exceed 𝜆|𝑇 |. For each 𝑇 P T, it is possible to choose a ball 𝐵(𝑥𝑇 , 𝑟𝑇 ) that comes within a factor of 2 of the supremum, so that 𝑟 𝜖 𝜆|𝑇 | |𝑌 (𝑇) X 𝐵(𝑥𝑇 , 𝑟𝑇 )| ě 𝑇 . (3.2) 2 Since |𝑌 (𝑇) X 𝐵(𝑥𝑇 , 𝑟𝑇 )| ≲ 𝑟𝑇𝑛 , and since 𝜆 ě 𝛿1´𝜖 , the above statement implies that 𝑟𝑇 ≳ 𝛿. For each 𝑇, define a new shading 𝑌1 (𝑇) by 𝑌1 (𝑇) = 𝑌 (𝑇) X 𝐵(𝑥𝑇 , 𝑟𝑇 ). 23 By performing dyadic pigeonholing on 𝑟𝑇 , we may choose a subcollection T1 Ď T of tubes such that 𝜌 ď 𝑟𝑇 ă 2𝜌 for every 𝑇 P T1 , where 𝜌 is a number between 𝛿1´𝜖 and 1, and #T1 ≳ 1 #T. log 1/𝛿 Then for every 𝑇 P T1 , (3.2) implies that |𝑌1 (𝑇)| ≳ 𝜌 𝜖 𝜆|𝑇 | ě 𝛿 𝜖 𝜆|𝑇 | ≳ 𝛿 𝜖 𝜆 |𝑇 X 𝐵(𝑥𝑇 , 𝑟𝑇 )| . 𝜌 (3.3) The intersection 𝑇 X 𝐵(𝑥𝑇 , 𝑟𝑇 ) is contained in a tube of radius 𝛿 and length 2𝜌. By means of an isotropic linear rescaling 𝜎 which takes 𝑥 ÞÑ 𝑥/2𝜌, each set ˜ Then 𝜎(𝑌1 (𝑇)) Ď 𝑇˜ and from 𝑇 X 𝐵(𝑥𝑇 , 𝑟𝑇 ) may be scaled to a (𝛿/𝜌)-tube 𝑇. (3.3), 𝜆 𝜎(𝑌1 (𝑇))| ⪆ |𝑇˜ |. 𝜌 We will now show that each 𝛿/𝜌-tube 𝑇˜ with shading 𝜎(𝑌1 (𝑇)) satisfies the two-ends condition. For any ball 𝐵(𝑥 1 , 𝑟 1 ), we have |𝑌1 (𝑇) X 𝐵(𝑥 1 , 𝑟 1 )| |𝑌 (𝑇) X 𝐵(𝑥 1 , 𝑟 1 )| 2 ¨ |𝑌 (𝑇) X 𝐵(𝑥𝑇 , 𝑟𝑇 )| ď ď ď 2𝜌 ´𝜖 |𝑌1 (𝑇)|, 1 𝜖 1 𝜖 (𝑟 ) (𝑟 ) (𝑟𝑇 ) 𝜖 because of the way 𝐵(𝑥𝑇 , 𝑟𝑇 ) were previously chosen. Re-arranging the previous statement gives  1 𝜖 𝑟 1 1 |𝑌1 (𝑇) X 𝐵(𝑥 , 𝑟 )| ≲ |𝑌1 (𝑇)|. 𝜌 This implies that  1 𝜖 𝑟 1 1
|𝜎 (𝑌1 (𝑇))| , 𝜎 𝑌1 (𝑇) X 𝐵(𝑥 , 𝑟 ) ≲ 𝜌 which means that 𝜎(𝑌1 (𝑇)) obeys the two-ends condition because 𝜎(𝐵(𝑥 1 , 𝑟 1 )) is a ball of radius 𝑟 1 /𝜌. Since the number of 𝛿/𝜌 tubes 𝑇˜ is #T1 , which is „ 1/log(1/𝛿) ¨ #T, we will need to choose a subcollection T2 Ď T1 so that the rescaled tubes 𝑇˜ = 𝜎(𝑇) for 𝑇 P T2 will point in 𝛿/𝜌-separated directions. Observe that #T2 ≳ 𝜌 𝑛´1 #T1 ⪆ 𝜌 𝑛´1 #T. We may now apply the conclusion of Proposition 20 to the rescaled tubes from T2 to get   𝑑   𝑛´𝑑+𝜖 ∑︁ Ø 𝜆 𝛿 𝜎 (𝑌1 (𝑇)) ě 𝐶𝜖 ( |𝑇˜ |). 𝜌 𝜌 𝑇 PT 𝑇 PT 2 2 24 Reversing the rescaling 𝜎, we now find Ø 𝑌 (𝑇) ě 𝐶𝜖 𝜆 𝑑 𝛿 𝑛´𝑑+𝜖 (#T2 )(𝛿/𝜌) 𝑛´1 ⪆ 𝐶𝜖 𝜆 𝑑 𝛿 𝑛´𝑑+𝜖 ( 𝑇 PT2 ∑︁ |𝑇 |), 𝑇 PT □ which is the desired conclusion for Proposition 18. 3.2 Broad-narrow reduction If T is a family of 𝛿-tubes with shading 𝑌 , we will say that a point 𝑥 P 𝑠-narrow if there exists a unit vector 𝑣 𝑥 P S𝑛´1 such that Ð 𝑇 PT 𝑌 (𝑇) is 1 {𝑇 P T(𝑥) : ∠(dir(𝑇), 𝑣 𝑥 ) ď 𝑠} ě #T(𝑥). 2 Ð If this condition fails at a point 𝑥 P 𝑇 PT 𝑌 (𝑇), then we say that the point 𝑥 is 𝑠-broad. Proposition 22. Let 𝜖 ą 0. If T is a set of 𝛿-tubes satisfying the following hypotheses: i. The tubes in T point in 𝛿-separated directions, ii. Each tube 𝑇 P 𝑇 has a subset 𝑌 (𝑇) and |𝑌 (𝑇)| ě 𝜆|𝑇 | for all 𝑇 P T, iii. The family T with shading 𝑌 is (𝑘-1)-plany, iv. For every 𝑇 P T, 𝑌 (𝑇) obeys the two-ends conjecture with exponent 𝜖, v. There is a set 𝐸 Ď Ð 𝑇 PT 𝑌 (𝑇) satisfying the condition ∑︁ |𝑌 (𝑇) X 𝐸 | ≳ 𝑇 PT ∑︁ 1 |𝑌 (𝑇)|, log(1/𝛿) 𝑇 PT for which any point 𝑥 P 𝐸 satisfies: #{(𝑇1 , 𝑇2 ) P T(𝑥) ˆ T(𝑥) : ∠(dir(𝑇1 ), dir(𝑇2 )) ≳ 𝛿 𝜖 } ≳ 1 # (T(𝑥)) 2 ; log(1/𝛿) then there exists a constant 𝐶𝜖 ą 0 such that Ø 𝑇 PT 𝑌 (𝑇) ě 𝐶𝜖 𝛿 (𝑛´2+𝑘 ´𝑑 (𝑘))/3+𝜖 𝜆 𝑑 (𝑘)/3+1 ( ∑︁ |𝑇 |) 𝑇 PT where 𝑑 (𝑘) is the largest exponent for which a Kakeya maximal estimate holds. Lemma 23 (Broad-narrow reduction). Proposition 22 implies Proposition 20. 25 To prove the broad-narrow reduction, we will first make some observations on broad and narrow points in the following simple lemmas. Ð Lemma 24. If a point 𝑥 P 𝑇 PT 𝑌 (𝑇) is 𝑠-broad, then   3 # (𝑇1 , 𝑇2 ) P T(𝑥) ˆ T(𝑥) : ∠(dir(𝑇1 ), dir(𝑇2 ) ą 𝑠 ě # (T(𝑥)) 2 . 4 (3.4) Proof. If a point 𝑥 is 𝑠-broad with respect to (T, 𝑌 ), then one may find a set of 𝐶𝑠1´𝑛 , where 𝐶 „ 1 is a fixed constant, such that 4𝑠-separated directions {𝑣 𝑖 P S𝑛´1 }𝑖=1 for every 𝑖, 1 #{𝑇 P T(𝑥) : ∠(dir(𝑇), 𝑣 𝑖 ) ď 𝑠} ď #T(𝑥). 2 1´ 𝑛 𝐶𝑠 is 4𝑠-separated, we observe that if Since the set of directions {𝑣 𝑖 P S𝑛´1 }𝑖=1 𝑇𝑖 P {𝑇 P T : ∠(dir(𝑇), 𝑣𝑖 ) ď 𝑠}, 𝑇 𝑗 P {𝑇 P T : ∠(dir(𝑇), 𝑣 𝑗 ) ď 𝑠}, and 𝑖 ≠ 𝑗, then ∠(dir(𝑇𝑖 ), dir(𝑇 𝑗 )) ě 𝑠. The total number of ordered pairs (𝑇1 , 𝑇2 ) P T(𝑥) ˆ T(𝑥) in which both 𝑇1 and 𝑇2 come are in the 𝑠-neighborhood of the same direction is less than 1/4 (#T(𝑥)) 2 , and the conclusion follows. □ Lemma 25. If T is a family of 𝛿-tubes with shading 𝑌 , there is a number 𝑠 and a set Ð 𝐸 Ď 𝑇 PT 𝑌 (𝑇) such that ∑︁ 𝑇 PT |𝑌 (𝑇) X 𝐸 | ≳ ∑︁ 1 |𝑌 (𝑇)| log(1/𝛿) 𝑇 PT and every point 𝑝 P 𝐸 satisfies two conditions: (i) 𝑝 is 𝑠-narrow, and (ii) we have   𝑠 3 # (𝑇1 , 𝑇2 ) P T(𝑥) ˆ T(𝑥) : ∠(dir(𝑇1 ), dir(𝑇2 )) ě ě (#T(𝑥)) 2 . 4 4 Proof. For every point 𝑝, define 𝑘 𝑝 by   1 𝑛 ´1 ´𝑘 𝑘 𝑝 = max 𝑘 P Z : D𝑣 𝑝 P S such that #{𝑇 P T( 𝑝) : ∠(𝑣 𝑝 , dir(𝑇)) ≲ 2 } ě #T( 𝑝). 2 (3.5) 26 Note that this maximum must always exist since 𝑘 must lie between 0 and „ log(𝛿´1 ). By dyadic pigeonholing, we may choose a value of 𝑠 such that 𝑠 ď 2´ 𝑘 𝑝 ă 𝑠 2 (3.6) Ð for every point 𝑝 in a set 𝐸 Ď 𝑇 PT 𝑌 (𝑇), such that the Lebesgue measure of 𝐸 satisfies ∑︁ ∑︁ 1 |𝑌 (𝑇) X 𝐸 | ≳ |𝑌 (𝑇)|. log(1/𝛿) 𝑇 PT 𝑇 PT We claim that every point 𝑝 P 𝐸 satisfies the two required conditions. To see this, note from (3.5) and (3.6) that 𝑝 satisfies the first condition. For the second condition, we observe from the definition of 𝑘 𝑝 that at every 𝑝 P 𝐸,   1 𝑠 𝑛 ´1 ě #T( 𝑝), @𝑣 P S : # 𝑇 P T( 𝑝) : ∠(dir(𝑇), 𝑣) ě 4 2 which means 𝑝 is 𝑠/4-broad. By applying the result of Lemma 24, we have   𝑠 3 # (𝑇1 , 𝑇2 ) P T(𝑥) ˆ T(𝑥) : dir(𝑇1 ), dir(𝑇2 ) ą ě (#T( 𝑝)) 2 . 4 4 □ Proof of Lemma 23. Assume that Proposition 22 is true. Let 𝜀 ą 0 be given and let T be a family of 𝛿-tubes that satisfies hypotheses (i)–(iv) of Proposition 20. Apply Lemma 25 and let 𝐸 and 𝑠 be as in the statement of that lemma. For every point 𝑝 P 𝐸, let 𝑣 𝑝 P S𝑛´1 be as in (3.5). Define a new shading by 𝑌 1 (𝑇) = {𝑝 P 𝑌 (𝑇) X 𝐸 : ∠(𝑣 𝑝 , dir(𝑇) ď 𝑠}, and correspondingly define T1 ( 𝑝) = {𝑇 P T : 𝑝 P 𝑌 1 (𝑇)}. From Lemma 25 we have, for every 𝑝 P 𝐸, 1 #{𝑇 P T : 𝑝 P 𝑌 1 (𝑇)} ě #T( 𝑝), 2 and from this we can see that ∫ ∫ ∑︁ ∑︁ 1 1 |𝑌 (𝑇)| „ #T ( 𝑝) 𝑑𝑝 ≳ #T( 𝑝) 𝑑𝑝 „ |𝑌 (𝑇) X 𝐸 |. 𝑇 PT 𝐸 𝐸 𝑇 PT 27 Also from Lemma 25, ∑︁ |𝑌 (𝑇) X 𝐸 | ≳ 𝑇 PT ∑︁ 1 |𝑌 (𝑇)|. log(1/𝛿) 𝑇 PT Combining the above two statements, we obtain ∑︁ ∑︁ 1 |𝑌 1 (𝑇)| ≳ |𝑌 (𝑇)|. log(1/𝛿) 𝑇 PT 𝑇 PT (3.7) Next we may choose a shading 𝑌 2 of T such that 𝑌 2 (𝑇) Ď 𝑌 1 (𝑇) for every 𝑇 P T, there is a number 0 ă 𝜆1 ă 1 such that |𝑌 2 (𝑇)| „ 𝜆1 |𝑇 | for all 𝑇 P T and in addition ∑︁ |𝑌 2 (𝑇)| ≳ 𝑇 PT ∑︁ 1 |𝑌 1 (𝑇)|. log(1/𝛿) 𝑇 PT (3.8) From (3.7), (3.8) and hypothesis (ii) of Proposition 20, we can conclude that 𝜆1 ≳ 𝜆 . (log(1/𝛿)) 2 (3.9) It is possible to cover the upper hemisphere of S𝑛´1 with „ 𝑠1´𝑛 finitely overlapping tiles of side length 𝑠, where a tile of side length 𝑠 means the image via the map a 𝜉 ÞÑ (𝜉, 1 ´ |𝜉 | 2 ) of translates of tiles [0, 𝑠) 𝑛´1 tiling R𝑛´1 . For each tile 𝜏 of radius 𝑠 covering the sphere, define T(𝜏) := {𝑇 P T : dir(𝑇) P 𝜏}. Let 𝜏1 and 𝜏2 be two non-adjacent tiles so dist(𝜏1 , 𝜏2 ) ą 2𝑠. If 𝑇1 P T(𝜏1 ) and 𝑇2 P T(𝜏2 ), the angle between the tubes would be greater than 2𝑠: ∠(𝑇1 , 𝑇2 ) ą 2𝑠. By the definition of the shading 𝑌 2 , this means that 𝑌 2 (𝑇1 ) X 𝑌 2 (𝑇2 ) = H. Under the shading 𝑌 2 , the collections T(𝜏1 ) and T(𝜏2 ) are disjoint. Within each 𝑠-tile 𝜏 covering S𝑛´1 , there are up to (𝑠/𝛿) 𝑛´1 𝛿-tubes. Without loss of generality, we may assume that the center of the tile 𝜏 is the north pole of the sphere, i.e. the point (0, . . . , 0, 1). Perform a non-isotropic rescaling 𝜙 which takes (𝑥 1 , . . . , 𝑥 𝑛 ) ÞÑ (𝑥 1 /𝑠, . . . , 𝑥 𝑛´1 /𝑠, 𝑥 𝑛 ). After this rescaling, 𝜙(𝑇) is contained in the 𝑛-fold dilate of a 𝛿/𝑠-tube 𝑇˜ that points in the same direction as 𝑇. Define a shading 𝑌˜ for each 𝑇˜ by ˜ := 𝜙(𝑌 2 (𝑇)). 𝑌˜ (𝑇) 28 From (3.9), we have ˜ „ 𝜆1 |𝑇˜ |. |𝑌˜ (𝑇)| ˜ 𝑌˜ ) satisfies hypotheses (i) — (v) in the statement of PropoWe now claim that (𝑇, sition 22. Hypothesis (ii) has already been shown, and hypothesis (i) holds because of the non-isotropic way in which the re-scaling 𝜙 was done. The family 𝑇˜ is (𝑘 ´ 1)-plany because the family T is assumed to be (𝑘 ´ 1)-plany. For hypothesis (iv), we observe that since 𝑌 (𝑇) obeys the two-ends condition with exponent 𝜀 for ˜ obeys the two-ends condition every 𝑇 P T, so does 𝑌 2 (𝑇), and it follows that 𝑌˜ (𝑇) ˜ 𝑌˜ satisfy hypothesis (v), consider that at with exponent 𝜀 as well. To show that 𝑇, Ð every point 𝑝 P 𝑇 PT[𝜏] 𝑌 1 (𝑇), we know from Lemma 25 that  
2 𝑠 3 1 1 # (𝑇1 , 𝑇2 ) P T ( 𝑝) ˆ T ( 𝑝) : ∠(dir(𝑇1 ), dir(𝑇2 )) ≳ ě # T1 ( 𝑝) . 4 4 Finally, we observe that after the rescaling, ∠(dir(𝑇1 ), dir(𝑇2 )) ě 𝑠/4 implies that ∠(dir(𝑇˜1 , 𝑇˜2 ) ě 𝑐 𝜖 𝛿 𝜖 for some 𝑐 𝜖 ą 0. Thus the family ( T̃, 𝑌˜ ) satisfies hypothesis (v), and by Proposition 22, there exists 𝐶𝜀 ą 0 such that Ø ˜ ě 𝐶𝜀 (𝛿/𝑠) 𝑛´𝑑+𝜀 𝜆 𝑑 𝑌˜ (𝑇) 𝑇˜ PT̃ ∑︁ |𝑇˜ |. 𝑇˜ PT̃ Undoing the rescaling 𝜙 (which has the effect of multiplying both sides of the above equation by 𝑠𝑛´1 ), we obtain the estimate Ø 𝑌 2 (𝑇) ě 𝐶𝜀 𝛿 𝑛´𝑑+𝜀 𝑠´𝑛+𝑑 ´𝜀 𝜆 𝑑 𝑇 PT(𝜏) ∑︁ |𝑇 |. 𝑇 PT[𝜏] Sum over a collection of pairwise non-adjacent tiles covering the sphere to get Ø 𝑌 2 (𝑇) ě 𝐶𝜀 𝛿 𝑛´𝑑+𝜀 𝑠´𝑛+𝑑 ´𝜖 𝜆 𝑑 𝑇 PT ∑︁ |𝑇 |. 𝑇 PT Since 𝑠 ă 1, this estimate is stronger than the desired estimate by a factor of 𝑠´𝑛+𝑑 ´𝜀 . This completes the proof. □ 3.3 Volume of a planebrush In this section, an estimate will be given for the volume of 𝛿-tubes in the 𝑘planebrush. It will first be needed to give an estimate for the number of tubes contained in the 𝑘-planebrush. 29 Figure 3.1: A 𝑘-planebrush at a point 𝑥 Definition 26 (𝑘-planebrush). If 𝑝 P point 𝑝 and angle 𝜌 is a set of tubes Ð 𝑇 PT 𝑌 (𝑇) and 0 ă 𝜌 ă 1, a 𝑘-planebrush at Tplanebrush ( 𝑝, 𝜌) = {𝑇 P T : ∠(𝑇, 𝑉 ( 𝑝)) „ 𝜌 and D𝑇 1 P T( 𝑝) : 𝑇 X 𝑇 1 ≠ H}. Lemma 27 (Constant multiplicity refinement). Let T be a family of 𝛿-tubes pointing in 𝛿-separated directions. Assume that every tube 𝑇 P T has a subset 𝑌 (𝑇) and 0 ă 𝜆 ă 1 is a number such that |𝑌 (𝑇)| ě 𝜆|𝑇 | for every 𝑇 P T. Then there exists a refinement T1 Ă T and a new shading 𝑌 1 (𝑇) Ď 𝑌 (𝑇) for each 𝑇 P T1 , which satisfy the following properties: i There is a dyadic number 𝜇 such that for every point 𝑥 P T(𝑥) ă 2𝜇. ii 1 𝑇 PT1 𝑌 (𝑇), 𝜇 ď Ð Í 1 2 𝑇 PT1 |𝑌 (𝑇)| ≳ ( 𝑇 PT |𝑌 (𝑇)|) /log(1/𝛿) . Í iii 𝜇 ≳ 𝜆𝛿 𝑛´1 #T/| Ð 𝑇 PT 𝑌 (𝑇)|. Ð Proof. For every integer 𝑘 between 1 and log 𝛿1´𝑛 , let 𝐸 𝑘 Ď 𝑇 PT 𝑌 (𝑇) be the set of points with multiplicity around 2 𝑘 , i.e.   Ø 𝑘 𝑘+1 𝐸𝑘 = 𝑝 P 𝑌 (𝑇) : 2 ď #T( 𝑝) ă 2 . 𝑇 PT For each 𝑘, define a shading 𝑌𝑘 by 𝑌𝑘 (𝑇) := 𝑌 (𝑇) X 𝐸 𝑘 . For each integer ℓ between 0 and log(𝜆/𝛿), let T 𝑘,ℓ be the collection of tubes 𝑇 whose for which the shading 𝑌𝑘 (𝑇) has density around 2´ℓ 𝜆, i.e.   ´ℓ ´ℓ+1 T 𝑘,ℓ = 𝑇 P T : 2 𝜆|𝑇 | ď |𝑌𝑘 (𝑇)| ă 2 𝜆|𝑇 | . 30 From the definitions of 𝑘 and ℓ, one can observe that ∑︁ |𝑌 (𝑇)| „ #T ¨ 𝜆𝛿 𝑛 ´1 „ 𝑇 PT log(1/𝛿) ∑︁ log(1/𝛿) ∑︁ 𝑘=0 #T 𝑘,ℓ 2´ℓ 𝜆𝛿 𝑛´1 . ℓ=0 By the pigeonhole principle, there exists at least one pair (𝑘, ℓ) that satisfies #T 𝑘,ℓ 2´ℓ 𝜆𝛿´1 ≳ 1 #T ¨ 𝜆𝛿´1 . (log(1/𝛿)) 2 (3.10) Take 𝜇 = 2 𝑘 , define the shading 𝑌 1 by 𝑌𝑘 and define T1 as T 𝑘,ℓ for this choice of 𝑘 and ℓ. From here, points (i) and (ii) in the statement of the lemma follow immediately from (3.10). One can also observe that 𝜆 ¨ #T 𝑘,ℓ ¨ 2´ℓ 𝛿 𝑛´1 𝜆 Ð ⪆ Ð , 𝜇=2 ě | 𝑇 PT 𝑌 (𝑇)| | 𝑇 PT 𝑌 (𝑇)| 𝑘 □ which is statement (iii). Lemma 28 (Planebrush multiplicity). Let T be a family of direction-separated 𝛿tubes with shading 𝑌 that obeys the hypotheses of Proposition 22. In addition, assume that ∑︁ |𝑌 (𝑇)| ≳ 𝜆. 𝑇 PT Then there exist an angle 𝜌 ą 0 and a point 𝑝 such that 𝜆 3 𝜌 𝛿 ´1 . #Tplanebrush ( 𝑝, 𝜌) ⪆ Ð | 𝑇 PT 𝑌 (𝑇)| 2 Proof. Let T1 , 𝑌 1 be the constant multiplicity refinement of T, 𝑌 given by Lemma 27. Let 𝑝 be any point for which #T1 ( 𝑝) ≳ 𝜇. To see that such a point 𝑝 must exist, suppose to the contrary that at all points, #T1 ( 𝑝) ă 𝑐𝜇 for a fixed constant 𝑐 to be chosen later. Then it follows that Í ∑︁ Ø 𝑐 𝑇 PT |𝑌 (𝑇)| 1 1 . (3.11) |𝑌 (𝑇)| ď 𝑐𝜇 ¨ 𝑌 (𝑇) ă 𝑐𝜆 ≲ log(1/𝛿) 2 𝑇 PT1 𝑇 P T1 On the other hand, by point (ii) in Lemma 27, Í ∑︁ |𝑌 (𝑇)| 1 |𝑌 (𝑇)| ≳ 𝑇 PT . log(1/𝛿) 2 𝑇 PT1 (3.12) This contradicts (3.11) if the constant 𝑐 is chosen to be smaller than the implicit constant in (3.12). For such a point 𝑝, one can count the number of triples (𝑇1 , 𝑄 1 , 𝑇2 ) 31 where 𝑇1 is a tube whose shading intersects 𝑝 and 𝑇2 is a tube whose shading intersects the shading of 𝑇1 in a 𝛿-cube 𝑄 1 . More precisely, #{(𝑇1 , 𝑄 1 , 𝑇2 ) P T1 ˆ Q ˆ T : 𝑇1 P T1 ( 𝑝), 𝑄 1 Ď 𝑌 1 (𝑇1 ) X 𝑌 1 (𝑇2 )} ≳ 𝜇 ¨ 𝜆1 𝛿´1 ¨ 𝜇 ⪆ 𝜇2𝜆𝛿´1 . By the pigeonhole principle, there exists an angle 𝜌 with 𝛿 ď 𝜌 ď 1 such that #{(𝑇1 , 𝑄 1 , 𝑇2 ) P T1 ˆ Q ˆ T : 𝑇1 P T1 ( 𝑝), 𝑄 1 Ď 𝑌 1 (𝑇1 ) X 𝑌 1 (𝑇2 ), ∠(𝑇2 , 𝑉 ( 𝑝)) „ 𝜌} ⪆ 𝜇2𝜆𝛿´1 . Since the tube 𝑇1 must pass through both 𝑝 and 𝑄 1 , and since the tubes passing through 𝑝 are assumed to be (1, 𝛿𝜖 )-broad, it follows that the choice of 𝑄 1 determines 𝑇1 entirely. Thus, for each 𝑄 1 , there is precisely one 𝑇1 that is a triple counted in the previous step. #{(𝑄 1 , 𝑇2 ) P QˆT : 𝑇1 P T1 ( 𝑝), (𝑇1 , 𝑄 1 , 𝑇2 ) is a triple in the previous step } ≳ 𝜇2𝜆𝛿´1 . Since the tube 𝑇2 makes angle 𝜌 with 𝑉 ( 𝑝), there are „ 𝜌 ´1 𝛿-cubes contained in 𝑇2 that are also contained in a 𝛿-neighborhood of 𝑉 ( 𝑝). Thus, for each 𝛿-tube 𝑇2 that makes angle 𝜌 with 𝑉 ( 𝑝), there are potentially up to 𝜌 ´1 cubes 𝑄 1 for which (𝑄 1 , 𝑇2 ) is a pair counted in the previous statement, i.e. a pair for which 𝑄 1 Ď 𝑌 1 (𝑇2 ) X 𝑁 𝛿 (𝑉 ( 𝑝)) and 𝑄 1 Ď 𝑌 1 (𝑇1 ) for some 𝑇1 P T( 𝑝) . It follows that #{𝑇2 P T : D𝑇1 P T( 𝑝) such that 𝑌 1 (𝑇1 ) X 𝑌 1 (𝑇2 ) ≠ H and ∠(𝑇2 , 𝑉 ( 𝑝)) „ 𝜌} ⪆ 𝜇2𝜆𝛿´1 𝜌. This precisely counts the tubes in Tplanebrush ( 𝑝, 𝜌) and demonstrates the correct multiplicity of tubes in the planebrush Tplanebrush ( 𝑝, 𝜌). □ Lemma 29 (Volume of a single planebrush). Let Tplanebrush ( 𝑝, 𝜌) be a planebrush at point 𝑝 and angle 𝜌. Then Ø 𝑌 (𝑇) ≳ 𝐶𝜖 𝛿 (𝑛´2+𝑘 ´𝑑 (𝑘))/3+𝜖 𝜆 𝑑 (𝑘)/3+1 𝜌 1/3 . 𝑇 PTplanebrush ( 𝑝,𝜌) Proof. Choose a maximal set of 𝛿/𝜌-separated points {𝛼 𝑗 } 𝑗 in the unit sphere in the space 𝑉 ( 𝑝) K = {𝑣 P R𝑛 : ⟨𝑣, 𝑉 ( 𝑝)⟩ = 0}, which is an (𝑛 ´ 𝑘 + 1)-dimensional space. For each 𝛼 𝑗 , define a 𝑘-plane 𝑉 𝑗 by 𝑉 𝑗 = span(𝑉 ( 𝑝), 𝛼 𝑗 ). The resulting collection of 𝑘-planes {𝑉 𝑗 } 𝑗 satisfies the following two conditions. 32 (A) Every tube 𝑇 P Tplanebrush ( 𝑝, 𝜌) is contained in N𝐶2 𝛿𝑉 𝑗 , with 𝐶2 „ 1, for some 𝑗. (B) If a point 𝑥 P R𝑛 satisfies dist(𝑥, 𝑉 ( 𝑝)) ě 𝐶3 𝜌), then 𝑥 P N𝐶2 𝛿𝑉 𝑗 for at most „ (𝐶2 /𝐶3 ) 𝑛´ 𝑘+1 choices of 𝑘-plane 𝑉 𝑗 . To show that (A) holds, let 𝑇 P Tplanebrush ( 𝑝, 𝜌) and write dir(𝑇 1 ) = 𝑣 || + 𝑣 K , where 𝑣 || P 𝑉 ( 𝑝) and 𝑣 K P (𝑉 ( 𝑝)) K . Choose 𝛼 𝑗 such that ∠(𝛼 𝑗 , 𝑣 K ) ď 𝛽/𝜌. This is possible because of the maximality of the set {𝛼 𝑗 }. Write 𝑣 K = 𝑣 1 + 𝑣 2 , where 𝑣 1 lies in the direction of 𝛼 𝑗 and ⟨𝑣 2 , 𝛼 𝑗 ⟩ = 0. Since ∠(𝑇 1 , 𝑉 (𝑄 0 )) ď 𝜌,
|𝑣 K | = sin ∠(𝑇 1 , 𝑉 ( 𝑝)) ≲ 𝜌. Now  
sin ∠(dir(𝑇), 𝑉 𝑗 ) = |𝑣 2 | = sin ∠(𝑣 2 , 𝑣 K ) ¨ |𝑉 K | ≲ 𝛿, so (A) is proved. To show (B), let a point 𝑥 satisfy dist(𝑥, 𝑉 ( 𝑝)) ě 𝐶3 𝜌). If 𝑥 = 𝑥 || + 𝑥 K , where 𝑥 || lies in 𝑉 ( 𝑝) and ⟨𝑥 K , 𝑥 || ⟩ = 0, then |𝑥 K | ≳ 𝐶3 𝜌. Suppose that 𝑥 P N𝐶2 𝛿𝑉 𝑗 for some 𝑗. Then  K  𝑥 𝐶2 𝛽 . (3.13) , 𝛼𝑗 ≲ dist K 𝐶3 𝑝 |𝑥 | Since the directions {𝛼 𝑗 } are 𝛿/𝜌-separated, there are only at most „ (𝐶2 /𝐶3 ) 𝑛´ 𝑘+1 choices of 𝛼 𝑗 that can satisfy (3.13). This proves (B). For each 𝑘-plane 𝑉 𝑗 , let T 𝑗 be the family of tubes given by T 𝑗 = {𝑇 P T 𝑝 𝑏( 𝑝, 𝜌) : 𝑇 Ă N𝐶2 𝛿𝑉 𝑗 }. Define the set 𝐸 𝑗 by 𝐸𝑗 = Ø 𝑌 (𝑇)zN𝐶3 𝜌𝑉 (𝑄 0 ). 𝑇 PT 𝑗 By the conclusion of Lemma 21, we may assume that each tube 𝑇 P T 𝑗 satisfies |𝑌 (𝑇) X 𝐵(𝑥, 𝐶2 𝜌)| ≲ (𝐶2 𝜌) 𝜖1 |𝑌 (𝑇)| for every 𝜖1 ą 0. By choosing 𝜖1 small enough, |𝑌 (𝑇) X 𝐸 𝑗 | ≳ 𝜆 |𝑇 |. 2 Summing over all 𝑇 P T 𝑗 , we obtain ∫ ∑︁ 𝜆 𝜒𝑌 (𝑇) ě 𝛽𝑛´1 #T 𝑗 . 2 𝐸 𝑗 𝑇 PT 𝑗 (3.14) 33 On the other hand, by applying Hölder’s inequality we obtain ∫ ∑︁ ∑︁ 𝜒𝑌 (𝑇) ď || 𝜒𝑇 || 𝐿 𝑑 (𝑘 ) 1 (R𝑛 ) ¨ |𝐸 𝑗 | 1/𝑑 (𝑘) , 𝐸 𝑗 𝑇 PT (3.15) 𝑇 PT 𝑗 𝑗 where 𝑑 (𝑘) is chosen to be the highest dimension for which a Kakeya maximal estimate is known to hold in R 𝑘 , and 𝑑 (𝑘) 1 is the Hölder conjugate of 𝑑 (𝑘). Specifically,   𝑘 2 + 𝑘 + ℓ2 ´ ℓ . 𝑑 (𝑘) = max min 𝑘 ´ ℓ + 2, 2ďℓ ď 𝑘 2𝑘 Í We will now apply a Kakeya maximal estimate to || 𝑇 PT 𝑗 𝜒𝑇 || 𝐿 𝑑 (𝑘 ) 1 . Let 𝜌𝑉 𝑗 be the orthogonal projection function from R𝑛 onto 𝑉 𝑗 . Since each 𝑇 P T 𝑗 satisfies 𝑇 Ă N𝐶2 𝛿𝑉 𝑗 , we observe that ∠(𝑇, 𝑉 𝑗 ) ≲ 𝐶2 𝛿. This implies that 𝜌𝑉 𝑗 (𝑇) is a cylinder of radius 𝛽 and length ≳ 𝐶2´1 and is contained in a „ 𝐶2´1 -dilation of a 𝛿-tube in R 𝑘 . Thus, we may write 1/𝑑 (𝑘) 1 || ∫ ∑︁ ∫ ∑︁ 1ª © | 𝜒𝑇 | 𝑑 (𝑘) ® 𝜒𝑇 || 𝐿 𝑑 (𝑘 ) 1 (R𝑛 ) = ­ R 𝑛 ´ 𝑘 R 𝑘 𝑇 PT 𝑇 PT 𝑗 𝑗 ¬ « 1/𝑑 (𝑘) 1 ∫ ∫ ∑︁ 1 © ª =­ | 𝜒𝑇 | 𝑑 (𝑘) ® 𝑛 ´ 𝑘 [´𝐶2 𝛿,𝐶2 𝛿] 𝑉 𝑗 𝑇 PT 𝑗 « ¬ 1/𝑑 (𝑘) 1 ∫ ∑︁ 1 © 1ª ≲ 𝛿 (𝑛´ 𝑘)/𝑑 (𝑘) ­ | 𝜒𝐶2 𝜌𝑉 𝑗 (𝑇) | 𝑑 (𝑘) ® . 𝑉 𝑗 𝑇 PT 𝑗 ¬ « After applying the Kakeya maximal estimate, this gives ∑︁ 1 1 || 𝜒𝑇 || 𝐿 𝑑 (𝑘 ) 1 (R𝑛 ) ⪅ 𝛿 (𝑛´ 𝑘)/𝑑 (𝑘) 𝛿´ 𝑘/𝑑 (𝑘)+1´𝜖 (𝛿 𝑘 ´1 #T 𝑗 ) 1/𝑑 (𝑘) . 𝑇 PT 𝑗 Combining this estimate with (3.14) and (3.15), we obtain 1 1 𝜆 𝑛 ´1 𝛿 #T 𝑗 ⪅ 𝛿 (𝑛´ 𝑘)/𝑑 (𝑘) 𝛿´ 𝑘/𝑑 (𝑘)+1´𝜖 (𝛿 𝑘 ´1 #T 𝑗 ) 1/𝑑 (𝑘) ¨ |𝐸 𝑗 | 1/𝑑 (𝑘) , 2 which simplifies to |𝐸 𝑗 | ⪆ 𝛿 𝑛´1𝜆 𝑑 (𝑘) 𝛿 𝑘 ´𝑑 (𝑘) #T 𝑗 . (3.16) Summing (3.16) over all 𝑗 and using properties (A) and (B) above, we obtain Ø 𝑇 PTplanebrush ( 𝑝,𝜌) 𝑌 (𝑇) ⪆ 𝛿 𝑛´1𝜆 𝑑 (𝑘) 𝛿 𝑘 ´𝑑 (𝑘) #Tplanebrush ( 𝑝, 𝜌). (3.17) 34 Combining (3.17) with Lemma 28 gives 3 Ø 𝑌 (𝑇) ⪆ 𝛿 𝑛´2𝜆 𝑑 (𝑘)+3 𝛿 𝑘 ´𝑑 (𝑘) 𝜌. (3.18) 𝑇 PTplanebrush ( 𝑝,𝜌) □ 3.4 Proof of k-planebrush estimate The goal of this section is to prove Proposition 22. Below is described an algorithm that decomposes the set of 𝛿-tubes T into a collection of “planebrushes.” Algorithm: Input. The input to the algorithm consists of: (i) A number 𝛿 ą 0. (ii) A family of 𝛿-tubes T with shading 𝑌 satisfying |𝑌 (𝑇)| ě 𝜆|𝑇 | for every 𝑡 P T, for some number 0 ď 𝜆 ă 1. (iii) A shading 𝑌 1 of T satisfying 𝑌 1 (𝑇) Ď 𝑌 (𝑇) for every 𝑇 P T and ∑︁ |𝑌 1 (𝑇)| ě 𝑇 PT 𝜆 . 2 (iv) An index 𝑖, initially equal to 1. Output. The output of the algorithm will be: 𝑁 where 𝑝 is a point in R𝑛 and 0 ă 𝜌 ă 1 for (i) A sequence ( 𝑝𝑖 , 𝜌𝑖 )𝑖=1 𝑖 𝑖 every 1 ď 𝑖 ď 𝑁 𝑁 ´1 (ii) A collection of shadings {𝑌𝑖 }𝑖=0 of T together satisfying the following properties: (P1) The sets (“planebrushes”) joint, and Ð 𝑇 PTplanebrush ( 𝑝 𝑖 ,𝜌𝑖 ) 𝑌𝑖 ´1 (𝑇) are pairwise dis- (P2) The “planebrushes” together cover more than half the “incidences”, i.e. 𝑁 ∑︁ ∑︁ 𝜆 |𝑌𝑖´1 (𝑇)| ě . 2 𝑖=1 𝑇 PTplanebrush ( 𝑝 𝑖 ,𝜌𝑖 ) 35 The algorithm consists of the following steps: Step 1. Apply Lemma 28 to the family (T𝑖´1 , 𝑌𝑖´1 ) to obtain 𝑝𝑖 and 𝜌𝑖 such that 𝜆 3 𝜌 𝑖 𝛿 ´1 #Tplanebrush ( 𝑝𝑖 , 𝜌𝑖 ) ⪆ Ð . | 𝑇 PT 𝑌 (𝑇)| 2 Step 2. Define T𝑖 := T𝑖´1 zTplanebrush ( 𝑝𝑖 , 𝜌𝑖 ). Define the shading 𝑌𝑖 as follows: 𝑌𝑖 (𝑇) = 𝑌𝑖´1 (𝑇)z𝑁 𝜌𝑖 (𝑉 ( 𝑝𝑖 )). For any angle 𝜃, the number of tubes making angle 𝜃 or less to the (𝑘 ´ 1)plane 𝑉 ( 𝑝𝑖 ) is ≲ 𝜃 𝑖´𝑛+𝑘 ´1 𝛿1´𝑛 . If a tube 𝑇 P T𝑖 makes an angle 𝜃 to the (𝑘 ´ 1)-plane 𝑉 ( 𝑝𝑖 ), then a 𝜌𝑖 /𝜃-fraction of 𝑇 is contained in a 𝜌𝑖 neighborhood of 𝑉 ( 𝑝𝑖 ). Therefore the maximum number of incidences removed is ∑︁ ∑︁ ∑︁ 𝜌𝑖 |𝑌𝑖 (𝑇) X 𝑁 𝜌 (𝑉 ( 𝑝𝑖 ))| ď 𝜃 ´𝑛+𝑘 ´1 𝛿 𝑛´1 ď 2𝜌𝑖 𝛿 𝑘 ´1 . 𝜃 𝜃 𝜃 𝑇 PT,∠(𝑇,𝑉 ( 𝑝 𝑖 )) „𝜃 On the other hand, by the two ends condition,  𝜌 𝜖 ∑︁ ∑︁ ∑︁ 𝑖 𝜆𝛿 𝑛´1 ď 2𝜌𝑖𝜖 𝜆𝛿 𝑘 ´1 . |𝑌𝑖 (𝑇)X𝑁 𝜌 (𝑉 ( 𝑝𝑖 ))| ď 𝜃 ´𝑛+𝑘 ´1 𝜃 𝜃 𝜃 𝑇 PT,∠(𝑇,𝑉 ( 𝑝 𝑖 )) „𝜃 Taking the geometric mean of the two estimates above, ∑︁ ∑︁ |𝑌𝑖 (𝑇) X 𝑁 𝜌 (𝑉 ( 𝑝𝑖 ))| ď 2𝜌𝑖(1+𝜖)/2𝜆1/2 𝛿 𝑘 ´1 . 𝜃 𝑇 PT,∠(𝑇,𝑉 ( 𝑝 𝑖 )) „𝜃 Terminate the algorithm if 𝑖 ∑︁ 𝜌 (1+𝜖)/2 𝜆1/2 𝛿 𝑘 ´1 ě 𝑗 𝑗=1 𝜆 , 2 otherwise we observe that ∑︁ 𝑇 PT𝑖 |𝑌𝑖 (𝑇)| ě ∑︁ 𝑇 P T0 |𝑌0 (𝑇)| ´ 𝑖 ∑︁ 𝜌 (1+𝜖)/2 𝜆1/2 ě 𝑗 𝑗=1 so we return to Step 1 with 𝑖 increased by 1 and repeat. 𝜆 , 2 36 Proof of Proposition 22. Define T0 := T and 𝑌0 := 𝑌 . Then apply the above algorithm with beginning input (T0 , 𝑌0 ) and starting with 𝑖 = 1. 𝑁 is obtained, After applying the algorithm, a collection of “planebrushes” ( 𝑝𝑖 , 𝜌𝑖 )𝑖=1 and from (P1), it follows that | Ø 𝑌 (𝑇)| ě 𝑇 PT 𝑁 ∑︁ 𝑖=1 Ø | 𝑌𝑖´1 (𝑇)|. 𝑇 PTplanebrush ( 𝑝 𝑖 ,𝜌𝑖 ) By applying Lemma 29, | Ø 𝑌 (𝑇)| ě 𝐶𝜖 𝛿 (𝑛´2+𝑘 ´𝑑 (𝑘))/3+𝜖 𝜆 𝑑 (𝑘)/3+1 𝑇 PT Since ! 𝜌𝑖𝑑 (𝑘)/3 . 𝑖=1 𝑖 ∑︁ 𝑗=1 we have 𝑁 ∑︁ 𝑘 𝜌 ě 𝑖 ∑︁ 1 𝜌 (1+𝜖)/2 𝜆1/2 𝛿 𝑘 ´1 ě , 𝑗 2 𝑗=1 𝑑 (𝑘)/3 ě 21 , and this completes the proof. 𝑖=1 𝜌𝑖 Í𝑁 □ Proof of Proposition 18. This is given by combining Proposition 22 with Lemma 21 and Lemma 23. □ Proof of Proposition 17. Let 𝜀 ą 0 and let T, 𝑌 be as stated in the hypothesis of the proposition. By applying Proposition 18 with 𝑑 (𝑘) = (2 ´ ? 2)𝑘, which is the estimate given by Hickman-Rogers-Zhang, we obtain the estimate ? ? Ø ∑︁ | 𝑌 (𝑇)| ≳ 𝛿 (𝑛´2+( 2´1)𝑘)/3+𝜀 𝜆 (2´ 2)𝑘/3+1 |𝑇 |, 𝑇 PT which is the result needed. 𝑇 PT □ 37 Chapter 4 NEW KAKEYA ESTIMATE FOR STICKY SETS Definition 30 (Sticky set). A set T of 𝛿-tubes pointing in 𝛿-separated directions is sticky if, for any 𝜀 ą 0, whenever 𝛿1´𝜀 ≲ 𝜃 ă 1, there is a family T𝜃 of 𝜃-tubes pointing in 𝜃-separated directions so that T and T𝜃 satisfy the following conditions: i. #T ≳ 𝛿1´𝑛+𝜀 , ii. #T𝜃 ≳ 𝜃 1´𝑛+𝜀 , and iii. T can be partitioned into sets T[𝑇𝜃 ] := {𝑇 P T : 𝑇 Ď 𝑇𝜃 }, with #T[𝑇𝜃 ] ≳ (𝛿/𝜃) 1´𝑛+𝜀 for every 𝑇𝜃 P T𝜃 . The family T𝜃 in this definition will be called the 𝜃-parents of T. The major result of the this chapter is the following proposition. Proposition 31. For every 𝜀 ą 0, whenever T is a sticky family of 𝛿-tubes with shading 𝑌 , which satisfies |𝑌 (𝑇)| ≳ 𝛿 𝜀 |𝑇 | for all 𝑇 P T, Ø | 𝑌 (𝑇)| ≳ 𝛿 𝑛´𝑑+(𝑑+1)𝜀 , 𝑇 PT where 𝑑 ď 0.60376707287 𝑛 + O (1). Sections 4.1, 4.2 and 4.3 are devoted to the proof of this proposition. Section 4.4 in this chapter gives the relation between sticky sets as defined in this thesis and sticky Besicovitch sets as defined recently by [23]. Section 4.5 proves an x-ray transform estimate as a corollary of Zahl’s 𝑘-linear Kakeya estimate in [28]. 38 4.1 Shading and refinement of sticky sets Lemma 32 (Shading of sticky sets). Let 𝜖 ą 0 and let T be a sticky family of 𝛿-tubes. Suppose that every 𝑇 P T has a shading 𝑌 (𝑇) Ď 𝑇 such that |𝑌 (𝑇)| ě 𝐶𝜀 𝛿 𝜀 |𝑇 |. Then for any 𝛿 ď 𝜃 ď 𝛿1´𝜖 , there is a shading 𝑌 of the 𝜃-parent family T𝜃 such that for every 𝑇𝜃 P T𝜃 , |𝑌 (𝑇𝜃 )| ě 𝐶𝜀 𝜃 𝜀/(1´𝜀) |𝑇𝜃 | and 𝑌 (𝑇) Ď 𝑌 (𝑇𝜃 ) for every 𝑇 P T[𝑇𝜃 ]. Proof. For any 𝑇𝜃 P T𝜃 , define a shading by 𝑌 (𝑇𝜃 ) = Ø  ª © Ø 𝑌 (𝑇) ® ≠ H . 𝑄 𝜃 : 𝑄 𝜃 X 𝑇𝜃 X ­ 𝑇 P T[𝑇 ] 𝜃 ¬ « Since |𝑌 (𝑇)| ě 𝐶𝜀 𝛿 𝜖 |𝑇 | for every 𝑇 P T[𝑇𝜃 ], there are at least 𝐶𝜀 𝛿´1+𝜖 𝛿-cubes in each 𝑌 (𝑇). Thus, #{(𝑄, 𝑇) : 𝑇 P T[𝑇𝜃 ], 𝑄 Ď 𝑌 (𝑇)} „ 𝐶𝜀 𝜃 𝑛´1 𝛿´𝑛+𝜀 . Each 𝜃-cube 𝑄 𝜃 Ď 𝑇𝜃 contains at most (𝜃/𝛿) 𝛿-cubes in any one 𝛿-tube 𝑇, so #{(𝑄 𝜃 , 𝑇) : 𝑇 P T[𝑇𝜃 ], 𝑄 𝜃 X 𝑇 ≠ H} ď 𝐶𝜀 𝜃 ´1 (𝜃 𝑛´1 𝛿1´𝑛+𝜀 ). Since there are „ 𝜃 𝑛´1´𝜀 𝛿1´𝑛+𝜀 tubes in T[𝑇𝜃 ], we can conclude that there are at least „ 𝛿 𝜀 𝜃 ´1 𝜃-cubes 𝑄 𝜃 in 𝑌 (𝑇𝜃 ). Since 𝜃 ď 𝛿1´𝜀 , we have 𝛿 𝜀 ď 𝜃 𝜀 ď (𝛿1´𝜀 ) 𝜀 = 𝛿 𝑐𝜀 , where 𝑐 = 1 ´ 𝜀. This shows that |𝑌 (𝑇𝜃 )| ≳ 𝛿 𝜀/𝑐 . □ Remark 33. Lemma 32 will need to be applied inductively, by covering a family T of 𝛿-tubes by 𝛿1´𝜀 -tubes, and then covering these by (𝛿1´𝜀 ) 1´𝜀 -tubes, and so on, until we can work with a family of 𝛽-tubes, where 𝛽 „ 1. This means that Lemma 32 will be applied log(1/𝛿) many times. For small enough 𝛿, (1 ´ 𝜀) log(1/𝛿) ď (1 ´ 𝜀) 1/𝜀 ≲ 1 . 𝑒 39 Therefore 1 ≳ 𝑒, (1 ´ 𝜀) log(1/𝛿) and the family of 𝛽-tubes will have a shading of density 𝛽 𝑒𝜀 . A refinement of a set of 𝛿-tubes T with shading 𝑌 is a set of 𝛿-tubes T1 with shading 𝑌 1 such that T1 Ď T, 𝑌 (𝑇) Ď 𝑌 1 (𝑇) for every 𝑇 P T1 , and ∑︁ ∑︁ 1 1 |𝑌 (𝑇)| ≳ |𝑌 (𝑇)|. log(1/𝛿) 𝑇 PT 1 𝑇 PT Lemma 34. Let T be a sticky family of tubes with shading 𝑌 . Suppose that, for any 𝜀 ą 0, 𝑌 satisfies the condition |𝑌 (𝑇)| ≳ 𝛿 𝜀 |𝑇 | for every 𝑇 P T. If T1 , 𝑌 1 is a refinement of T, 𝑌 , then T1 is also sticky. Proof. Let 𝜀 ą 0 be given. Suppose T be a sticky family of tubes with shading 𝑌 satisfying |𝑌 (𝑇)| ≳ 𝛿 𝜀 |𝑇 | for every 𝑇 P T. Suppose that T1 , 𝑌 1 is a refinement of T, 𝑌 , i.e. T1 Ď T and 𝑌 1 (𝑇) Ď 𝑌 (𝑇) for every 𝑇 P T1 and ∑︁ ∑︁ 1 |𝑌 1 (𝑇)| ≳ |𝑌 (𝑇)|. log(1/𝛿) 𝑇 PT 1 𝑇 PT First, consider the cardinality of T1 . Since ∑︁ |𝑌 1 (𝑇)| ≳ 𝛿 𝑛´1+2𝜀 #T, 𝑇 P T1 we know that #T1 ≳ 𝛿2𝜀 #T ≳ 𝛿1´𝑛+3𝜀 . Next, let 𝛿1´𝜀 ě 𝜃 ă 1 be any number and let T𝜃 be the 𝜃-parent family of T. Since T1 is a subset of T, there is an inherited partition of T1 into sets T1 [𝑇𝜃 ] = {𝑇 P T1 : 𝑇 Ď 𝑇𝜃 }. Now define a refinement T1𝜃 Ď 𝑇𝜃 by T1𝜃 = {𝑇𝜃 P T𝜃 : T1 [𝑇𝜃 ] ≳ 2(𝛿/𝜃) 1´𝑛+3𝜀 }. Since #T1 ≳ 𝛿1´𝑛+3𝜀 , it follows that
1 # T𝜃 zT1𝜃 ≲ 𝜃 1´𝑛+3𝜀 , 2 so #T1𝜃 ≳ 12 𝜃 1´𝑛+3𝜀 , which makes T1𝜃 a valid set of 𝜃-parents for T1 . This proves that T𝜃 is sticky. □ 40 4.2 Extremal sticky sets The following definitions build up the concept of an “extremal sticky set,” which will be used in the following propositions to prove a Hausdorff dimension estimate for sticky sets of 𝛿-tubes. Define 𝑀 (𝛿) by Ø 𝑀 (𝛿) := inf T,𝑌 𝑌 (𝑇) 𝑇 PT where the infimum is taken over all sets T, 𝑌 satisfying the following conditions: i. T is a sticky set of 𝛿 tubes, and ii. 𝑌 is a shading of T satisfying |𝑌 (𝑇)| ≳ 𝛿 𝜀 |𝑇 | for every 𝑇 P T. Define 𝜎 by 𝜎 := lim sup 𝛿 Ñ0 log 𝑀 (𝛿) . log 𝛿 Definition 35 (Extremal sticky set). For any 𝜀 ą 0, if T is a sticky set of 𝛿-tubes with shading 𝑌 satisfying, firstly, |𝑌 (𝑇)| ≳ 𝛿 𝜀 |𝑇 | for every 𝑇 P T, and secondly, Ø | 𝑌 (𝑇)| ≲ 𝛿 𝑛´𝜎+𝜀 (𝛿 𝜀 ) 𝜎 , (˚) 𝑇 PT then T will be called an 𝜀-extremal sticky set, or just an 𝜀-extremal set. From the definition of 𝜎, it is immediate that 𝜀-extremal sets of 𝛿-tubes exist, if 𝛿 is small enough, for every 𝜀 ą 0. The following two lemmas will show that 𝜀-extremal sticky sets exhibit a kind of self-similarity, provided 𝜀 is small enough. The following notation will be used in the lemmas. Notation. For any point 𝑥 P R𝑛 , we define: T𝜃 (𝑥) := {𝑇𝜃 P T𝜃 : 𝑥 P 𝑌𝜃 (𝑇𝜃 )} and T[𝑇𝜃 ] (𝑥) := {𝑇 P T[𝑇𝜃 ] : 𝑥 P 𝑌 (𝑇)}. The following notation will be used to denote multiplicity: 𝜇 𝛿 (𝑥) := #T(𝑥), 𝜇𝜃 (𝑥) := #T𝜃 (𝑥), and 𝜇 𝛿 [𝑇𝜃 ] (𝑥) := #T[𝑇𝜃 ] (𝑥). 41 Lemma 36 (Multiplicities in extremal sticky sets). There exists 𝜀0 ą 0 such that, for every 0 ă 𝜀 ă 𝜀0 , if T is an 𝜀-extremal family of 𝛿-tubes with shading 𝑌 , then there exists a refined shading 𝑌 1 of T (i.e. 𝑌 1 (𝑇) Ď 𝑌 (𝑇) for every 𝑇 P T) such that |𝑌 1 (𝑇)| ≳ 𝛿2𝜀 |𝑇 | for every 𝑇 P T, and whenever 𝜃 is a number satisfying 𝛿 ă 𝜃 ď 𝛿1´𝜀 , there is a family of 𝜃-tubes T𝜃 with shading 𝑌𝜃 such that (i) (Fine multiplicity) 𝜇 𝛿 ( 𝑝) ≳ 𝛿´𝑛+𝜎´𝜀 . (ii) (Coarse multiplicity) 𝜇𝜃 ( 𝑝) ≳ 𝜃 ´𝑛+𝜎´𝜀 . (iii) (Inner multiplicity) 𝜇 𝛿 [𝑇𝜃 ] ( 𝑝) ≳ (𝜃/𝛿) ´𝑛+𝜎´𝜀 . Proof. Let 𝜀 ą 0 and suppose that T is an 𝜀-extremal set of 𝛿-tubes with shading 𝑌 . Let T1 , 𝑌1 be a constant multiplicity refinement of (T, 𝑌 ) with multiplicity 𝜇 𝛿 . The details of how to obtain this refinement are given below. For every positive integer 𝑘, let 𝐸 𝑘 be the set of points at which the multiplicity of 𝛿-tubes in T is „ 2 𝑘 , i.e. Ø 𝐸 𝑘 = {𝑥 P 𝑌 (𝑇) : 2𝑘 ´1 ă #T(𝑥) ď 2 𝑘 }. 𝑇 PT Since Í 𝑘 PN |𝐸 𝑘 | ¨ 2 𝑘 „ Í 𝜀 𝑇 PT |𝑌 (𝑇)| ≳ 𝛿 , there is an integer 𝑘 such that |𝐸 𝑘 | ¨ 2 𝑘 ≳ 1 𝛿𝜀 . log(1/𝛿) Set 𝜇 𝛿 := 2 𝑘 for this value of 𝑘. Now define a new shading 𝑌1 of T by 𝑌1 (𝑇) = 𝑌 (𝑇) X 𝐸 𝑘 , @𝑇 P T. For every 0 ă 𝜆 ă 1, define a new family T1,𝜆 by T1,𝜆 = {𝑇 P T : 𝜆|𝑇 | ď |𝑌1 (𝑇)| ă 2𝜆|𝑇 |}. By the pigeonhole principle, there exists some 𝜆 such that ∑︁ ∑︁ 1 1 |𝑌1 (𝑇)| ě |𝑌1 (𝑇)| ≳ 𝛿𝜀 . 2 log(𝜃/𝛿) log(1/𝛿) 𝑇 PT 𝑇 T P 1,𝜆 Let T1 := T1,𝜆 for this chosen value of 𝜆, and further observe from (4.1) that 𝜆≳ 1 𝛿𝜀 . (log(1/𝛿) 2 (4.1) 42 This shows how to obtain the constant multiplicity refinement, and we now need to show that this refinement obeys the multiplicity bound in part (i) in the statement of this lemma. Using the hypothesis that T is an 𝜀-extremal set, we see that Ø Ø | 𝑌1 (𝑇)| ď | 𝑌 (𝑇)| ≲ 𝛿 𝑛´𝜎+𝜖 (𝛿 𝜀 ) 𝜎 . 𝑇 PT1 𝑇 PT On the other hand, from the definition of 𝜎 and because T1 is also a sticky set by Lemma 34, Ø | 𝑌1 (𝑇)| ≳ 𝛿 𝑛´𝜎+𝜖 (𝛿 𝜀 ) 𝜎 . 𝑇 PT1 Since 𝜇𝛿 ¨ | Ø 𝑇 PT1 𝑌1 (𝑇)| „ ∑︁ |𝑌1 (𝑇)| ≳ 𝛿2𝜀 , 𝑇 PT1 we conclude that 𝜇 𝛿 ≳ 𝛿´𝑛+𝜎´𝜖 (𝛿 𝜀 ) 1´𝜎 . (4.2) Fix a tube 𝑇𝜃 P T𝜃 and consider the set of 𝛿-tubes T1 [𝑇𝜃 ] with shading 𝑌 1 . Let T1 [𝑇𝜃 ] be a constant multiplicity refinement of T1 [𝑇𝜃 ] with shading 𝑌1 and multiplicity 𝜇 𝛿 [𝑇𝜃 ]. (The details of how to obtain the constant multiplicity refinement are very similar to those given in detail for T above, so they are not repeated.) We will prove an upper bound on 𝜇 𝛿 [𝑇𝜃 ] by observing the structure of the set T1 [𝑇𝜃 ]. Let 𝑒 = dir(𝑇𝜃 ). Let 𝜙 : R𝑛 Ñ R𝑛 be the anisotropic rescaling defined by 𝜙(𝑥) = 𝑥 ¨ 𝑒 + 𝑥 ¨ 𝑒K . 𝜃 Observe that 𝜙(𝑇𝜃 ) equals a cylinder of radius 1 and length 1. Every 𝛿-tube 𝑇 in T1 [𝑇𝜃 ] no longer has the shape of a tube. Since ∠(dir(𝑇), 𝑒) ≲ 𝜃, we know that |dir(𝑇) ¨ 𝑒 K | ≲ 𝜃, and thus |𝜙(dir(𝑇))| ď 2. This implies that 𝜙(𝑇) is contained in the 2-fold dilate of a 𝛿/𝜃-tube 𝑇˜ which has the same center as 𝑇 but points in the direction as the vector 𝜙(dir(𝑇)). Similarly 𝑇˜ is contained in the 2-fold dilate of ˜ one may speak of 𝜙(𝑇) as a 𝛿/𝜃-tube, 𝜙(𝑇). Through the identification 𝜙(𝑇) ÞÑ 𝑇, which we will now do by an abuse of notation. Let T̃𝛿/𝜃 denote the collection of 𝛿/𝜃-tubes 𝜙(𝑇) contained in 𝜙(𝑇𝜃 ). Since T1 [𝑇𝜃 ] consists of tubes pointing in 𝛿-separated directions, 𝑇˜𝛿/𝜃 consists of (𝛿/𝜃)-tubes which are (𝛿/𝜃)-separated, It will be now shown that T̃𝛿/𝜃 is a sticky family of 𝛿/𝜃-tubes. To see this, consider the set T1 [𝑇𝜃 ] and let 𝜂 be an “intermediate” scale, i.e. let 𝛿 ă 𝜂 ă 𝜃. Since T is sticky, there is an 𝑒𝑡𝑎-parent family T𝜂 of 43 𝜂-tubes covering every 𝑇 P T. There are „ 𝜂1´𝑛+𝜖 𝜂-tubes in T𝜂 , which means that for all except at most 𝜃 ´𝜖 𝜃-tubes 𝑇𝜃 , there are 𝜃 𝑛´1 𝜂1´𝑛 𝜂-tubes covering all T(𝑇𝜃 ). Finally, observe that if an 𝜂-tube 𝑇𝜂 contains a 𝛿-tube 𝑇 P T1 [𝑇𝜃 ], then ∠(𝑇𝜂 , 𝑇𝜃 ) ≲ 𝜃, and thus 𝑇𝜂 is contained in a 2-fold dilate of 𝑇𝜃 . Applying the rescaling 𝜙 to each 𝑇𝜂 , one sees that 𝜙(𝑇𝜂 ) is an 𝜂/𝜃-tube containing « 𝜂 𝑛´1 (𝛿/𝜃) 1´𝑛 (𝛿/𝜃)-tubes in it. This shows that the family 𝑇˜𝛿/𝜃 is sticky. For every 𝑇˜ P T̃𝛿/𝜃 , let 𝑌˜ be the shading given by ˜ = 𝜙(𝑌1 (𝑇)). 𝑌˜ (𝑇) By the definition of 𝜎, and also because |𝑌 (𝑇)| ≳ 𝛿2𝜀 |𝑇 | for every 𝑇 P T[𝑇𝜃 ], we know that Ø ˜ ≳ (𝛿/𝜃) 𝑛´𝜎+𝜖 (𝛿2𝜀 ) 𝜎 . | 𝑌˜ (𝑇)| 𝑇˜ PT̃ 𝛿/𝜃 Since 𝜇 𝛿 [𝑇𝜃 ] ¨ | Ø ∑︁ ˜ « 𝑌˜ (𝑇)| 𝑇˜ PT̃ 𝛿/𝜃 ˜ „ 𝛿2𝜀 , |𝑌˜ (𝑇)| 𝑇˜ P𝑇˜𝛿/𝜃 we conclude that 𝜇 𝛿 [𝑇𝜃 ] ≲ (𝛿/𝜃) ´𝑛+𝜎´2𝜀 (𝛿2𝜀 ) 1´𝜎 . (4.3) Now consider the set of tubes T𝜃 , which is the 𝜃-parent family to the set T1 . Let 𝑌𝜃 be the shading for T𝜃 that is given by applying Lemma 32 to (T1 , 𝑌 1 ). Let T𝜃,1 , 𝑌𝜃,1 be a constant multiplicity refinement of (T𝜃 , 𝑌𝜃 ) with multiplicity 𝜇𝜃 . Then since T𝜃,1 is sticky, we have, using the definition of 𝜎 as above, 𝜇𝜃 ≲ 𝜃 ´𝑛+𝜎´𝜖 (𝜃 𝑒𝜖 ) 1´𝜎 . (4.4) Finally, by taking another refinment of T𝜃,2 of T𝜃,1 , it is possible to ensure that for all 𝑇𝜃 P T𝜃,2 , 𝜇 𝛿 [𝑇𝜃 ] „ 𝜇 𝛿 [T𝜃 ] for a constant 𝜇 𝛿 [T𝜃 ], which also satisfies 𝜇 𝛿 [T𝜃 ] ≲ (𝛿/𝜃) ´𝑛+𝜎´2𝜀 (𝛿2𝜀 ) 1´𝜎 . We will use the observation that at a point 𝑥, 𝜇 𝛿 (𝑥) ď 𝜇𝜃 (𝑥) ¨ 𝜇 𝛿 [𝑇𝜃 ] (𝑥). (4.5) 44 For this observation to be applicable, it is necessary to take a refinement T2 , 𝑌2 of T1 , 𝑌1 such that Ø Ø 𝑌2 (𝑇2 ) Ď 𝑌𝜃 (𝑇𝜃 ). 𝑇 P T2 𝑇𝜃 PT 𝜃 ,2 Now putting together (4.2), (4.4) and (4.5), and using the observation at any point 𝑥, and for any tube 𝑇𝜃 P T𝜃 , we can conclude the following statements: 𝜇 𝛿 ≳ 𝛿´𝑛+𝜎´𝜖 (𝛿 𝜀 ) 1´𝜎 𝜇𝜃 ≳ 𝜃 ´𝑛+𝜎´𝜖 (𝛿 𝜀 ) 1´𝜎 𝜇 𝛿 [T𝜃 ] (𝑥) ≳ (𝛿/𝜃) ´𝑛+𝜎´𝜀 (𝛿 𝜀 ) 1´𝜎 . □ Lemma 37 (Self-similarity of extremal sticky sets). There exists 𝜀0 ą 0 such that for any 0 ă 𝜀 ă 𝜀0 , if T is an 𝜀-extremal family of 𝛿-tubes with shading 𝑌 , then the 𝜃-parent family T𝜃 , with the induced shading from Lemma 32, is 𝜀(1 ´ 𝜀)-extremal, provided 𝛿 ă 𝜃 ď 𝛿1´𝜀 . Proof. This is a direct consequence of Lemma 36. Let 𝜀 ą 0 and suppose that T is an 𝜀-extremal family of 𝛿-tubes with shading 𝑌 . Let T𝜃 be the 𝜃-parent set of T and let 𝑌𝜃 be the shading given by Lemma 32. By Lemma 36, there is a refinement Ð T1 , 𝑌 1 of T, 𝑌 and a refinement T1𝜃 , 𝑌𝜃1 of T𝜃 , 𝑌𝜃 such that for all 𝑥 P 𝑇 PT1 𝑌 1 (𝑇), and any 𝑇𝜃 P T1𝜃 (𝑥), 𝜇(𝑥) „ 𝜇 ≳ 𝛿´𝑛+𝜎´𝜀 𝜇𝜃 (𝑥) „ 𝜇𝜃 ≳ 𝜃 ´𝑛+𝜎´𝜀 𝜇 𝛿 [𝑇𝜃 ] (𝑥) „ 𝜇 𝛿 [T1𝜃 ] ≳ (𝛿/𝜃) ´𝑛+𝜎´𝜀 , for some numbers 𝜇, 𝜇𝜃 and 𝜇 𝛿 [T1𝜃 ]. Since we have Ø ∑︁ ∑︁ 𝜇𝜃 ¨ | 𝑌 1 (𝑇)| „ |𝑌 1 (𝑇)| ≳ 𝛿 𝜀 |𝑌 (𝑇)| ≳ 𝛿2𝜀 , 𝑇 P T1 𝑇 PT1 𝑇 PT it follows that | Ø 𝑌 1 (𝑇)| ≲ 𝜃 2𝜀(1´𝜀) 𝜃 𝑛´𝜎+𝜀 , 𝑇 PT1 which is the desired conclusion. □ 45 Lemma 38 (Balanced cover). For any 𝜀 ą 0, if T is an 𝜀-extremal sticky set of 𝛿-tubes with shading 𝑌 and if T𝜃 is the 𝜃-parent set of T for some 𝛿1´𝜀 ď 𝜃 ď 𝛿1´2𝜀 , and if 𝑌𝜃 is the shading of T𝜃 induced by Lemma 32, then there are refinements Ð (T1 , 𝑌 1 ) of (T, 𝑌 ) and (T1𝜃 , 𝑌𝜃1 ) of (T𝜃 , 𝑌𝜃 ) such that every 𝜃-cube 𝑄 𝜃 P 𝑇𝜃 PT1 𝑌𝜃1 (𝑇𝜃 ) 𝜃 Ð contains „ (𝜃/𝛿) 𝜎´𝜀 𝛿-cubes in 𝑇 PT 𝑌 (𝑇). Proof. Let 𝜀 ą 0 be given. Let T, 𝑌 , T𝜃 and 𝑌𝜃 be as described in the hypothesis in the lemma statement. Since (T, 𝑌 ) is 𝜀-extremal, the number of 𝛿-cubes contained Ð Ð in 𝑇 PT 𝑌 (𝑇) is ≲ 𝛿´𝜎+𝜀 . By Lemma 37, the number of 𝜃-cubes in 𝑇𝜃 PT 𝜃 𝑌𝜃 (𝑇𝜃 ) Ð is ≲ 𝜃 ´𝜎+𝜀 . By dyadic pigeonholing, there is a set 𝐸 Ď 𝑇𝜃 PT 𝜃 𝑌𝜃 (𝑇𝜃 ) such that every 𝜃-cube 𝑄 𝜃 contained in 𝐸 contains „ 𝑁 𝛿-cubes and ∑︁ ∑︁ |𝑌𝜃 (𝑇𝜃 ) X 𝐸 | ≳ 𝜃 𝜀 |𝑌𝜃 (𝑇𝜃 )|. 𝑇𝜃 PT 𝜃 𝑇 PT 𝜃 Define a new shading of T𝜃 by 𝑌𝜃1 (𝑇𝜃 ) := 𝑌𝜃 (𝑇𝜃 ) X 𝐸 for every 𝑇𝜃 P T𝜃 . Define a subset T1𝜃 Ď T𝜃 so that |𝑌𝜃1 (𝑇𝜃 )| „ 𝜆 𝜃 |𝑇𝜃 | for all 𝑇𝜃 P T𝜃 and ∑︁ ∑︁ |𝑌𝜃1 (𝑇𝜃 )| ≳ 𝜃 2𝜀 |𝑌𝜃 (𝑇𝜃 )|. 𝑇𝜃 PT1𝜃 𝑇 PT 𝜃 It follows that (#T1𝜃 ) ¨ 𝜆 𝜃 ¨ |𝑇𝜃 | ≳ 𝜃 3𝜀 , which allows us to conclude that #T1𝜃 ≳ 𝜃 3𝜀 #T𝜃 and 𝜆 𝜃 ≳ 𝜃 3𝜀 . Now define a refined shading of T, 𝑌 by 𝑌 1 (𝑇) := 𝑌 (𝑇) X 𝐸, and take T1 Ď T such that |𝑌 1 (𝑇)| „ 𝜆|𝑇 | for all 𝑇 P T1 , all so that ∑︁ ∑︁ |𝑌 1 (𝑇)| ≳ 𝛿 𝜀 |𝑌 (𝑇)|. 𝑇 PT1 𝑇 PT Similarly, it follows that #T ≳ 𝛿 𝜀 #T and 𝜆 ≳ 𝛿 𝜀 . From the definition of 𝜎, we know Ð that 𝑇 PT1 𝑌 1 (𝑇) contains at least 𝛿´𝜎+𝜀 𝛿-cubes. But since T, 𝑌 was 𝜀-extremal, Ð the number of 𝛿-cubes in 𝑇 PT1 𝑌 1 (𝑇) must be „ 𝛿´𝜎+𝜀 . Similarly, the number Ð Ð of 𝜃-cubes in 𝑇𝜃 PT1 𝑌𝜃1 (𝑇𝜃 ) is „ 𝜃 𝜎𝜀 . Every 𝜃-cube in 𝑇𝜃 PT1 𝑌𝜃1 (𝑇𝜃 ) contains the 𝜃 𝜃 same number of delta cubes, so 𝑁 „ (𝛿/𝜃) ´𝑑+𝜖 . □ 4.3 Kakeya estimate for sticky sets Lemma 39. If T 𝛽 is a set of 𝛽-tubes that point in 𝛽-separated directions, with 𝛽 „ 1, and 𝑌𝛽 is a shading of T 𝛽 , then there is a refinement T1𝛽 , 𝑌𝛽1 such that one of the following cases holds: 46 (a) (T1𝛽 , 𝑌𝛽1 ) is (𝑘 ´ 1)-plany. (b) At every point 𝑥 P 1 𝑇𝛽 PT1𝛽 𝑌𝛽 (𝑇𝛽 ), Ð  # ((𝑇1 , . . . , 𝑇𝑘 ) P  T1𝛽 (𝑥)) 𝑘   : |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| ą 𝛽 𝑘 ≳ #(T1𝛽 (𝑥)) 𝑘 . (4.6) Proof. Let T 𝛽 be a set of 𝛽-tubes that point in 𝛽-separated directions, and let 𝑌𝛽 be Ð a shading of T 𝛽 . We will say that a point 𝑥 P 𝑇𝛽 PT𝛽 𝑌𝛽 (𝑇𝛽 ) satisfies the 𝑘-plane condition if there exists a (𝑘 ´ 1)-plane 𝑉 (𝑥) such that #{𝑇𝛽 P T 𝛽 (𝑥) : ∠(dir(𝑇𝛽 ), 𝑉 (𝑥)) ≲ 𝛽} ≳ 1 #T 𝛽 (𝑥), 𝑁 (4.7) where 𝑁 is any number greater than 𝑘. For example, one could use 𝑁 = 𝑛. Observe Ð that at any point 𝑥 P 𝑇𝛽 PT𝛽 𝑌𝛽 (𝑇𝛽 ) which does not satisfy the 𝑘-plany condition, the condition (4.6) is satisfied. To see this, let {𝑉𝑖 } be a maximal collection of „ 𝛽-separated (𝑘 ´ 1)-planes, all passing through the point 𝑥, in R𝑛 . We may partition T 𝛽 (𝑥) into sets T 𝛽 (𝑥) [𝑉𝑖 ] := {𝑇𝛽 P T 𝛽 (𝑥) : ∠(dir(𝑇𝛽 , 𝑉𝑖 ) ≲ 𝛽}. If 𝑇1 is chosen from T 𝛽 (𝑥) [𝑉𝑖1 ], . . ., and 𝑇𝑘 is chosen from T 𝛽 (𝑥) [𝑉𝑖 𝑘 ], and if 𝑖1 , . . . , 𝑖 𝑘 are pairwise distinct, then |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| ≳ 𝛽 𝑘 . Since 𝑥 does not satisfy the 𝑘-plany condition, there are at least (1 ´ 2 𝑘 1 ) ¨ (1 ´ ) ¨ ¨ ¨ (1 ´ ) ¨ (#T 𝛽 (𝑥)) 𝑘 𝑁 𝑁 𝑁 such 𝑘-tuples that can be chosen, and this shows (4.6). Denote Ø 𝐸 = {𝑥 P 𝑌𝛽 (𝑇𝛽 ) : 𝑥 satisfies the 𝑘-plany condition}. 𝑇𝛽 PT 𝛽 Now we will consider two cases separately. First, suppose it is the case that ∑︁ 𝑇𝛽 PT 𝛽 |𝑌𝛽 (𝑇𝛽 ) X 𝐸 | ě 1 ∑︁ |𝑌𝛽 (𝑇𝛽 )|. 2 𝑇 PT 𝛽 𝛽 Then we will define 𝑌𝛽1 (𝑇𝛽 ) = {𝑥 P 𝑌𝛽 (𝑇𝛽 ) X 𝐸 : ∠(dir(𝑇𝛽 ), 𝑉 (𝑥)) ≲ 𝛽} (4.8) 47 and define T1𝛽 = T 𝛽 , and now (T1𝛽 , 𝑌𝛽1 ) is (𝑘 ´ 1)-plany, so condition (a) holds. On the other hand, if (4.8) is not true, then it must be that 1 ∑︁ |𝑌𝛽 (𝑇𝛽 )z𝐸 | ě |𝑌𝛽 (𝑇𝛽 )|. 2 𝑇 PT 𝑇 PT ∑︁ 𝛽 𝛽 𝛽 𝛽 In this case we will define 𝑌𝛽1 (𝑇𝛽 ) = 𝑌𝛽 (𝑇𝛽 )z𝐸 and T1𝛽 = T 𝛽 , and now condition (b) is satisfied. □ Proposition 40. Let 3 ď 𝑘 ď 𝑛. For every 𝜀 ą 0, whenever T is a sticky family of 𝛿-tubes with shading 𝑌 , which satisfies |𝑌 (𝑇)| ≳ 𝛿 𝜀 |𝑇 | for all 𝑇 P T, there exists 𝐶 ą 0 such that Ø | 𝑌 (𝑇)| ≳ 𝛿 𝑛´𝑑+𝐶𝜀 , 𝑇 PT where ?  2𝑛 + 2 ´ ( 2 ´ 1)𝑘 𝑛2 + 𝑘 2 + 𝑛 ´ 𝑘 , 𝑑 ď max min . 3ď 𝑘 ď 𝑛 3 2𝑛  Proof. Fix 3 ď 𝑘 ď 𝑛. The proof is by induction on the scale 𝛿. In the base case, the hypothesis is that T is a family of 𝛽-tubes, where 𝛽 „ 1 is the constant appearing in the definition of 𝑘-linear tubes. By Lemma 39, T is either (𝑘 ´ 1)-plany, or else satisfies (4.6). If T is (𝑘 ´ 1)-plany, Proposition 18 gives the result. If T satisfies (4.6), the result is given by Corollary 12 with 𝐶 = 𝑑 + 1. The induction hypothesis is the following: for every 𝜀 ą 0 and every 𝜃 in the interval [𝛿/𝛽, 𝛽), if (i) T𝜃 is a sticky set of 𝜃-tubes with shading 𝑌𝜃 , and (ii) |𝑌𝜃 (𝑇𝜃 )| ≳ 𝜃 𝜀 |𝑇𝜃 | for all 𝑇𝜃 P T𝜃 , then there is a constant 𝐶𝜀 ą 0 such that Ø 𝑌𝜃 (𝑇𝜃 ) ě 𝐶𝜀 𝜃 𝑛´𝑑+(𝑑+1)𝜀 . 𝑇𝜃 PT 𝜃 Let 𝜀 ą 0 and assume that T is a sticky family of 𝛿-tubes, with shading 𝑌 , such that |𝑌 (𝑇)| ě 𝛿 𝜀 |𝑇 | for all 𝑇 P T. From the definition of 𝜎, it is enough to consider the case where (T, 𝑌 ) is an 𝜀extremal family of 𝛿-tubes. Since T is sticky, it is possible to cover T by a family 48 T𝜃 of 𝜃-tubes pointing in 𝜃-separated directions, with 𝜃 = 𝛿/𝛽. Let 𝑌𝜃 be the shading given by Lemma 32. By Lemma 37, the family (T𝜃 , 𝑌𝜃 ) admits a refinement (T1𝜃 , 𝑌𝜃1 ) which is 𝜀(1 ´ 𝜀)-extremal. Since (T, 𝑌 ) is 𝜀-extremal and (T1𝜃 , 𝑌𝜃1 ) is 𝜖 (1 ´ 𝜀)-extremal, Ø Ø | 𝑌 (𝑇)| ≲ 𝛿 𝑛´𝜎+(𝜎+1)𝜀 and | 𝑌𝜃1 (𝑇𝜃 )| ≲ 𝜃 𝑛´𝜎+(𝜎+1)𝜀(1´𝜀) . (4.9) 𝑇𝜃 PT1𝜃 𝑇 PT By Lemma 38, there is a refinement (T1 , 𝑌 1 ) of (T, 𝑌 ) and a refinement (T2𝜃 , 𝑌𝜃2 ) of (T1𝜃 , 𝑌𝜃1 ), and a number 𝑁 ą 0 such that Ø | 𝑌 1 (𝑇) X 𝑄 𝜃 | „ 𝑁𝛿 𝑛 (4.10) 𝑇 PT1 for all 𝜃-cubes 𝑄 𝜃 Ă 2 𝑇𝜃 PT2𝜃 𝑌𝜃 (𝑇𝜃 ). From the definition of 𝜎, Ð Ø | 𝑌𝜃2 (𝑇𝜃 )| ≳ 𝜃 𝑛´𝜎+(𝜎+1)𝜀 . 𝑇𝜃 PT2𝜃 This statement, together with (4.10) and (4.9), gives the conclusion that (𝛿/𝜃) ´𝜎+(𝜎+1)𝜀 „ 𝛽´𝜎+(𝜎+1)𝜀 ≲ 𝑁 ≲ 𝛽´𝜎+(𝜎+1)𝜀 𝜃 ´ (𝜎+1)𝜀 . 2 By the induction hypothesis, Ø | 𝑌𝜃2 (𝑇𝜃 )| ě 𝐶𝜀 𝜃 𝑛´𝑑+(𝑑+1)𝜀 . 𝑇𝜃 PT2𝜃 With (4.9), this implies that 𝜎 ´ (𝜎 ´ 1)𝜀 + (𝜎 ´ 1)𝜀 2 ě 𝑑 ´ (𝑑 ´ 1)𝜀 and thus 2 𝑁 ≳ 𝛽´𝑑+(𝑑+1)𝜀´ (𝜎+1)𝜀 . Together with (4.10) and the induction hypothesis, this gives Ø 2 | 𝑌 (𝑇)| ≳ 𝐶𝜖 𝛿 𝑛´𝑑+(𝑑+1)𝜖 𝛽 (𝜎+1)𝜀 . 𝑇 PT If 𝛽 is chosen to be small enough that 𝛽 (𝜎+1)𝜀 times the implicit constant in the statement above is smaller than 1, this closes the induction. □ 2 The proof of Proposition 31 follows immediately from this result. Proof of Proposition 31. Let 𝜀 ą 0 and let T, 𝑌 be as in the hypothesis of the proposition. The result follows immediately by finding the value of 𝑘 for which the maximum is obtained in the following expression: ?   2𝑛 + 2 ´ ( 2 ´ 1)𝑘 𝑛2 + 𝑘 2 + 𝑛 ´ 𝑘 max min , . 3ď 𝑘 ď 𝑛 3 2𝑛 49 The optimal value of 𝑘 is given by a ? ? 1´ 2+ 6´2 2 𝑘= 𝑛 + O (1) « 0.45555915723 𝑛 + O (1). 3 This gives 𝑑 ě 0.60376707287 𝑛 + O (1). In the worst case scenario, given any 𝜀 ą 0, there are infinitely many dimensions for which 𝑑 ď 0.60376707287 𝑛 + O (1) ´ 𝜀. □ 4.4 Relation to sticky sets in Wang-Zahl Recently, Wang and Zahl introduced the concept of a sticky Besicovitch set using the packing definition of Besicovitch sets ([23]). In this section of the paper, the relation between sticky sets of 𝛿-tubes and sticky Besicovitch sets as defined by Wang and Zahl is outlined. A version of this result is proved in [23]. Definition 41. A sticky Besicovitch set is a compact set 𝐾 Ď R𝑛 obeying the following hypotheses: there is a set of lines 𝐿 in R𝑛 such that i. 𝐿 has a line in every direction, ii. 𝐾 X ℓ contains a unit line segment for every ℓ P L, and iii. 𝐿 has packing dimension 𝑛 ´ 1. Let L be the set of all lines in R𝑛 . Every non-vertical line ℓ P L can be represented as ℓ = (a, 0) + R𝑒 where 𝑒 P S𝑛´1 and a P R𝑛´1 . In this section ℓ will be represented as ℓ = (aℓ , 𝑒 ℓ ). The distance between two lines ℓ and ℓ 1 in L is defined by dist(ℓ, ℓ 1 ) = |aℓ ´ aℓ1 | + ∠(𝑒 ℓ , 𝑒 ℓ1 ). A measure 𝜆 is defined on L to be the product of the Lebesgue measure on R𝑛´1 and the normalized surface measure 𝜈 on S𝑛´1 . With these definitions, the packing dimension of a set 𝐿 Ď L agrees with its upper modified box dimension. 50 Proposition 42. For any 𝜀 ą 0, if 𝐾 is a sticky Besicovitch set in R𝑛 , then there exists 𝛿0 ą 0 such that (1) for any 𝛿 ă 𝛿0 , there is a set of 𝛿-tubes T which is sticky; and (2) if dim𝐻 (𝐾) = 𝑑, then there is a shading𝑌 (𝑇) satisfying𝑌 (𝑇)| ≳ log(1/𝛿) for every 𝑇 P T, such that Ø 𝑌 (𝑇) ≲ 𝛿 𝑛´𝑑+𝜀 .
´2 |𝑇 | 𝑇 PT Proof. Let 𝜖 ą 0 be given and let 𝐾 be a sticky Besicovitch set. Since the packing dimension of 𝐿 coincides with its upper modified box dimension, 𝐿 can be written as a countable union of closed sets {𝐿 𝑖 } such that dim 𝑀 (𝐿 𝑖 ) ď 𝑛 ´ 1 + 𝜀 4
for all 𝑖. There exists an index 𝑖 for which 𝜈 dir(𝐿 𝑖 ) ą 0. From the upper Minkowski dimension of 𝐿 𝑖 , there eixsts 𝛿1 ą 0 such that for all 𝛿 ă 𝛿1 , 𝜆(𝑁 𝛿 (𝐿 𝑖 )) ď 𝛿 (2𝑛´2) ´ (𝑛´1) ´𝜀/2 . We will say that a direction 𝑒 P S𝑛´1 is over-represented at scale 𝛿 if |{a P R𝑛´1 : (a, 𝑒) P 𝑁 𝛿 (𝐿 𝑖 )}| ą 𝛿 𝑛´1´𝜀 . By Fubini’s theorem, it follows that
𝜈 {𝑒 P S𝑛´1 : 𝑣 is over-represented at scale 𝛿} ď 𝛿 𝜀/2 . By making 𝛿1 smaller if necessary, we can then ensure that ∑︁
1 𝜈 {𝑒 P S𝑛´1 : 𝑣 is over-represented at scale 𝛿} ď 𝜈(dir(𝐿 𝑖 )). 2 dyadic 𝛿:𝛿ă𝛿 1 Now define a subset 𝐿 1 Ď 𝐿 𝑖 by Ø
𝐿 1 = 𝐿𝑖 z dir´1 {𝑒 P S𝑛´1 : 𝑣 is over-represented at scale 𝛿} . dyadic 𝛿:𝛿ă𝛿0 By the definition of Hausdorff dimension, there exists a number 𝛿0 ą 0 (which we can ensure is smaller than 𝛿1 above) and a cover Q of 𝐾, where Ø Q= Q𝛿 , dyadic𝛿:𝛿ă𝛿0 51 where Q𝛿 is a collection of 𝛿-cubes and #Q 𝑘 ≲ 𝛿´𝑑+𝜀/2 . For each ℓ P 𝐿 1 , since |𝐾 X ℓ| ě 1, there exists at least one 𝛿 such that Ø
1 ℓX 𝑄 ě
2 . 100 log(1/𝛿) 𝑄 PQ 𝛿 (4.11) For each dyadic number 𝛿, let 𝐿 𝛿 Ď 𝐿 1 be the set of lines for which (4.11) holds.
2 Then there exists a dyadic scale 𝛿 ă 𝛿0 for which 𝜈(𝐿 𝛿 ) ą 𝜈(𝐿 1 )/100 log(1/𝛿) . Fix this value of 𝛿 and rename 𝐿 𝛿 as 𝐿 2 . Now we define E to be a 𝛿-separated subset of dir(𝐿 2 ), then 𝜈(𝐿 1 )𝛿1´𝑛 #E ě
2 . 100 log(1/𝛿) Define T be a set of 𝛿-tubes pointing in directions in E, with centers and shading 𝑌 (𝑇) chosen so that, using (4.11), Ø Ø 𝑌 (𝑇) = 𝑄 𝑇 PT and |𝑌 (𝑇)| ≳ |𝑇 |/ log(1/𝛿)
2 𝑄 PQ 𝛿 for every 𝑇 P T. Since this implies that Ø | 𝑌 (𝑇)| ≲ 𝛿 𝑛´𝑑+𝜀 , 𝑇 PT conclusion (2) in the statement of the proposition is shown. It now remains to show that T is sticky. Let 𝜌 be a number such that 𝛿 ă 𝜌 ă 1. If 𝜌 ě 𝛿0 : since 𝛿0 may be considered an admissible constant, if T 𝜌 is any set of maximal 𝜌-separated tubes, it is trivial that #T 𝜌 « 𝜌 1´𝑛 and #T[𝑇𝜌 ] « (𝛿/𝜌) 1´𝑛 . If 𝜌 ă 𝛿0 : We will greedily cover 𝐿 2 with 𝜌-balls, so that 𝜌/3-balls at the same centers would be disjoint. Define T 𝜌 to be the set of tubes pointing in the directions of the lines in the cover and with centers inside the set on the left-hand-side of (4.11). From this definition, it is clear that every 𝑇 P T is contained in some 𝑇𝛽 P T 𝜌 . Since 𝛿0 ă 𝛿1 , and since dir(𝐿 2 ) has no directions over-represented at any scale less than 𝛿1 , there are at ≲ 𝛿´𝜀 tubes 𝑇𝜌 P T 𝜌 that point in the same direction. To make the directions 𝜌-separated, choose any one tube out of the possible 𝛿´𝜀 tubes in the cover pointing in a given direction, and remove the other tubes pointing in that direction from T 𝜌 . Then #T 𝜌 « 𝜌 1´𝑛 . To ensure T is still covered by T 𝜌 , we will take a refinement T1 Ă T, losing at most a 𝜌 ´𝜀 ď 𝛿´𝜀 -fraction of T. Then T1 will be a sticky set, and it can be easily seen that conclusion (2) still holds for this refinement. □ 52 Corollary 43. The Hausdorff dimension of every sticky Besicovitch set in R𝑛 must exceed 0.60376707287 𝑛 + O (1). 4.5 An x-ray estimate derived as a corollary of Zahl’s k-linear estimate In this section, an 𝑥-ray estimate is obtained as a corollary of Zahl’s 𝑘-linear estimate. The proof is similar to that of Proposition 4.2 in [12]. For convenience, the definition of an x-ray estimate is repeated here. An x-ray estimate at dimension d holds in R𝑛 if there exist 0 ď 𝛼, 𝛾 ă 1 such that the following statement is true. For any 𝜀 ą 0, if E is a set of 𝛿-separated directions in S𝑛´1 and T is a set of 𝛿-tubes obeying the following properties: i. No tube in T is contained in the 10-fold dilate of another one. ii. Every tube in T points in a direction in E and no more than 𝛿´ 𝛽 tubes in T point in any given direction, then there is a constant 𝐶𝜀 ą 0 such that ∑︁ 𝑛 || 𝜒𝑇 || 𝑑 1 ď 𝐶𝜀 𝛿1´ 𝑑 ´𝜀 (𝛿´ 𝛽 ) 1´𝛼 (𝛿 𝑛´1 #Ω) 𝛾 𝑇 PT Proposition 44. An x-ray estimate holds in R𝑛 at dimension 𝑑 = max2ď 𝑘 ď𝑛 min{𝑛2 + 𝑛 + 𝑘 2 ´ 𝑘)/2𝑛, 𝑛 ´ 𝑘 + 2}. Lemma 45. Suppose that 𝛿0 ą 0 is a small number (e.g. 𝛿0 ă 𝑛/109 ). Let 2 ď 𝑘 ď 𝑛 and let 𝑑 ď 𝑛 ´ 𝑘 + 2 + 𝛿0 . Suppose that for every 𝜀 ą 0, whenever T is a set of 𝛿-tubes pointing in 𝛿-separated directions, then there is a constant 𝐶𝜖 ą 0 such that ∑︁ ||( 1𝑇1 ¨ ¨ ¨ 1𝑇𝑘 |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| 𝑘/𝑑 ) 1/𝑘 || 𝑑/(𝑑 ´1) 𝑇1 ,...,𝑇𝑘 PT 𝑛 ď 𝐶 𝜖 𝛿 1´ 𝑑 ´ 𝜖 ( ∑︁ |𝑇 |) 𝑛(𝑑 ´1)/(𝑛´1)𝑑 . (4.12) 𝑇 PT Then an x-ray estimate holds at any dimension 𝑑0 := 𝑑 ´ 𝛿0 . Proof of Lemma 45. Fix 2 ď 𝑘 ď 𝑛 and let 𝑑 ď 𝑛 ´ 𝑘 + 2. Assume that the 𝑘-linear estimate in the lemma hypothesis holds. Let 𝜀 ą 0 be given. The argument uses induction on scale 𝛿. The induction hypothesis is that there exist 0 ă 𝛼, 𝛾 ă 1 and 𝐶𝜀1 ą 0, whose values will be specified later, such that the following statement holds: 53 If T̃ is a set of (𝛿/𝑠)-tubes obeying properties i and ii in the definition of an x-ray estimate, then there exists 𝐶𝜀 ą 0 such that ∑︁ 𝑛 || 𝜒𝑇˜ || 𝑑 1 ď 𝐶𝜀 (𝛿/𝑠) 1´ 𝑑 ´𝜀 (𝛿/𝑠) ´ 𝛽 ) 1´𝛼 ((𝛿/𝑠) 𝑛´1 #E) 𝛾 . 𝑇˜ P𝑇˜ Let E be a set of 𝛿-separated directions in S𝑛´1 and let T be a set of 𝛿-tubes obeying properties i and ii in the definition of the x-ray estimate. Let 𝑠 „ 1 be an admissible constant whose value will be specified later. Ð Define a set Narrow𝑘 Ď 𝑇 PT 𝑇 as a set of all points 𝑥 for which there exists a (𝑘 ´ 1)-plane 𝑉 (𝑥) such that 1 T(𝑥). 10𝑛 Ð Ð Define another set Broad 𝑘 Ď 𝑇 PT 𝑇 as the set of all points 𝑥 P 𝑇 PT at which this condition is not satisfied. Observe that for every point 𝑥 P Broad 𝑘 , this leads to the conclusion   𝑘 𝑘 # (𝑇1 , . . . , 𝑇𝑘 ) P (T(𝑥)) : |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| ą 𝑠 ≳ (#T1 (𝑥)) 𝑘 . (4.13) #{𝑇 P T(𝑥) : ∠(dir(𝑇), 𝑉 (𝑥)) ≲ 𝑠} ≳ # (For a more detailed justification of the last claim, see the proof of Lemma 39, where the same claim is explained in detail.) Now ∫ ∫ ∑︁ ∑︁ ∑︁ 𝑑/(𝑑 ´1) 𝑑/(𝑑 ´1) || 𝜒𝑇 || 𝑑/(𝑑 ´1) ď | 𝜒𝑇 | + | 𝜒𝑇 | 𝑑/(𝑑 ´1) . 𝑇 PT Broad 𝑘 𝑇 PT Narrow 𝑘 𝑇 PT In the right-hand-side above, we will call the first term the “𝑘-broad contribution” and the second term the “𝑘-narrow contribution”. If the 𝑘-broad contribution dominates, then it can be estimated by using the the 𝑘-linear estimate on 𝛿´ 𝛽 sets of direction-separated tubes partitioning T, as follows. ∫ ∫ ∑︁ ∑︁
1/𝑘 𝑑/(𝑑 ´1) 𝑑/(𝑑 ´1) | 𝜒𝑇 | = 𝜒𝑇1 ¨ ¨ ¨ 𝜒𝑇𝑘 . Broad 𝑘 𝑇 PT Broad 𝑘 𝑇1 ,...,𝑇𝑘 PT From property 4.13 above, we can ignore the contribution of 𝑘-tuples (𝑇1 , . . . , 𝑇𝑘 ) unless |dir(𝑇1 ) ^ ¨ ¨ ¨ ^ dir(𝑇𝑘 )| ą 𝑠 𝑘 . Then, by using the 𝑘-linear estimate on the norm applied to these 𝑘-tuples alone, ∑︁ 𝑛(𝑑 ´1)
1/𝑘 𝑛 || 𝜒𝑇1 ¨ ¨ ¨ 𝜒𝑇𝑘 || 𝐿 𝑑/(𝑑´1) (Broad𝑘 ) ď 𝐶𝜖 𝛿 (1´ 𝑑 ´𝜖) (𝛿´ 𝛽 )(𝛿 𝑛´1 #E) (𝑛´1) 𝑑 . 𝑇1 ,...,𝑇𝑘 PT 54 Using Riesz-Thorin interpolation between this estimate and a known x-ray estimate at dimension (𝑛 + 2)/2, from Theorem 1.2 of [{\L }aba1999xray], we obtain ∑︁ 𝑛(𝑑 ´1)
1/𝑘 (1´ 𝑛 ´𝜖) || 𝜒𝑇1 ¨ ¨ ¨ 𝜒𝑇𝑘 || 𝐿 𝑑0 /(𝑑0 ´1) (Broad𝑘 ) ď 𝐶𝜖 𝛿 𝑑0 (𝛿´ 𝛽 ) 𝛼 (𝛿 𝑛´1 #E) (𝑛´1) 𝑑 𝑇1 ,...,𝑇𝑘 PT provided that 𝛼 and 𝛾 have the following values: ! 1 1 (𝑛 + 1) 𝑑0 ´ 1 𝑑0 ´ 𝑑 and 𝛾 = . 𝛼= 2 1 (𝑛 ´ 1)(𝑛 + 2) 𝑑 ´ 0 𝑛+2 𝑑 This shows the conclusion if the 𝑘-broad contribution dominates. On the other hand, if the 𝑘-narrow contribution dominates, we will use the following tiling of S𝑛´1 : let R𝑛´1 be tiled by 𝑠-cubes given by translates (0, 𝑠] 𝑛´1 placed end-to-end and let a “tile” 𝜃 on the sphere be given by the image of one of these 𝑠-cubes via the map a 𝜉 ÞÑ (𝜉, 1 ´ |𝜉 | 2 ) sending R𝑛´1 to the northern hemisphere. Let T be partitioned into sets T[𝜃] = {𝑇 P T1 : dir(𝑇) P 𝜃}. Then we may write ∫ ∑︁ 1 ∫ 𝑌 (𝑇) „ Narrow 𝑘 𝑇 PT1 | Narrow 𝑘 ∑︁ ∑︁ 𝜒𝑌 1 (𝑇) (𝑥)| 𝑑𝑥. 𝜃 𝑇 PT 𝜃 Consider that at any point 𝑥 P Narrow 𝑘 , the following statements hold, first by the definition of Narrow 𝑘 , and then by Hölder’s inequality, ∑︁ ∑︁ ∑︁ ∑︁ 𝜒𝑇 (𝑥) = 𝜒𝑇 (𝑥) 𝜃:∠(𝜃,𝑉 (𝑥))≲𝑠 𝑇 PT[𝜃] 1/𝑑0 𝜃 𝑇 PT 𝜃 ∑︁ © ª ď­ 1® «𝜃:∠(𝜃,𝑉 (𝑥))≲𝑠 ¬ (𝑑0 ´1)/𝑑0 ©∑︁ ∑︁ ª ¨­ | 𝜒𝑇 (𝑥)| 𝑑0 /(𝑑0 ´1) ® « 𝜃 𝑇 PT[𝜃] ¬ . For any (𝑘 ´1)-plane, the number of caps 𝜃 making angle ≲ 𝑠 with the (𝑘 ´1)-plane is „ 𝑠´ 𝑘+2 . Thus, ∫ ∑︁ ∑︁ ∑︁ ∑︁ /(𝑑0 ´1) | 𝜒𝑌 1 (𝑇) (𝑥)| 𝑑0 /(𝑑0 ´1) ď 𝑠 (´ 𝑘+2)/(𝑑0 ´1) || 𝜒𝑇 || 𝑑𝑑0 /(𝑑 ´1) Narrow 𝑘 0 𝜃 𝑇 PT 𝜃 𝜃 0 𝑇 PT[𝜃] and (𝑑0 ´1)/𝑑0 ©∑︁ ∑︁ /(𝑑0 ´1) ª || 𝜒𝑇 || 𝑑0 /(𝑑0 ´1) ď 𝑠 (´ 𝑘+2)/𝑑0 ­ || 𝜒𝑇 || 𝑑𝑑0 /(𝑑 ® 0 0 ´1) 𝜃 𝑇 P T[𝜃] 𝑇 PT1 ¬ « ∑︁ . (4.14) 55 To estimate the quantity on the right hand side above, we will use the inductive hypothesis. For any tile 𝜃, let 𝑒 𝜃 P S𝑛´1 be the center of 𝜃. Perform an anisotropic re-scaling of all tubes T[𝑇𝜃 ] via the map 𝜉 ÞÑ (𝜉 ¨ 𝑒 𝜃 ) + 𝑠´1 ¨ 𝑠´1 (𝜉 ´ 𝜉 ¨ 𝑒 𝜃 ). ˜ The image of every 𝛿-tube 𝑇 is contained in a 2𝛿/𝑠-tube which will be denoted 𝑇. To ensure that the set of directions in which the tubes 𝑇˜ are (𝛿/𝑠)-separated, we will take a subset E 1 Ă E such that E 1 is 𝛿/𝑠-separated. Then #E 1 „ 𝑠𝑛´1 #E. ˜ where 𝑇 P T and dir(𝑇) P E 1 . Now it is immediate Let T̃ be the set of all tubes 𝑇, that 𝑇˜ satisfies properties i and ii in the hypothesis of the x-ray transform, and that 𝑇˜ has the following relation to T[𝜃]: ∑︁ ∑︁ || 𝜒𝑇 || 𝑑0 /(𝑑0 ´1) = 𝑠 (𝑛´1) (𝑑0 ´1)/𝑑0 || 𝜒𝑇˜ || 𝑑0 /(𝑑0 ´1) . 𝑇 PT[𝜃] 𝑇˜ PT̃ By the induction hypothesis, the right-hand-side above is majorized by  𝛾 1´ 𝑑𝑛 ´𝜀 ´ 𝛽 𝛼 (𝑛´1) (𝑑0 ´1)/𝑑0 𝑛´1 1 0 𝑠 (𝛿/𝑠) (𝛿 ) (𝛿/𝑠) (#E X 𝜃)  𝛾 1´ 𝑛 ´𝜀 „ 𝑠 (𝑛´1) (𝑑0 ´1)/𝑑0 (𝛿/𝑠) 𝑑0 (𝛿´ 𝛽 ) 𝛼 𝛿 𝑛´1 (#E X 𝜃) . After putting this together with (4.14), and our choice of 𝛾 = (𝑑0 ´ 1)/𝑑0 , we find Í that || 𝑇 PT 𝜒𝑇 || 𝑑0 /(𝑑0 ´1) is majorized by 𝑠 𝜀+(𝑛´ 𝑘+2´𝑑0 )/𝑑0 𝛿 ´1+ 𝑑𝑛 ´𝜀 0 (𝛿´ 𝛽 ) 𝛼 𝑠𝑛´1 ∑︁  𝛿 𝑛´1 #(#E X 𝜃) !  𝛾𝑑0 /(𝑑0 ´1) (𝑑0 ´1)/𝑑0 𝜃 =𝑠 𝑛 𝜀+(𝑛´ 𝑘+2´𝑑0 )/𝑑0 ´1+ 𝑑0 ´𝜀 𝛿 (𝛿 ´𝛽 𝛼 )  𝛿 𝑛 ´1 𝛾 #E . By hypothesis, 𝑑0 ď 𝑛 ´ 𝑘 + 2, and the exponent of 𝑠 is positive. Thus 𝑠 may be chosen to satisfy 1 𝑠 𝜀+(𝑛´ 𝑘+2´𝑑0 )/𝑑0 ď 𝐶𝜀1 . 2 1 If 𝐶𝜀 is chosen such that 𝐶𝜀 ą 2 max{𝐶𝜀 , 𝐶𝜀2 }, then the induction hypothesis closes after adding the estimates obtain above for the 𝑘-broad and the 𝑘-narrow contributions. □ 56 Proof of Proposition 44. Let 𝛿0 ą 0 be a small number (e.g. smaller than 𝑛/109 ). Let 𝑘 0 be the value of 𝑘 for which the maximum   2 𝑛 + 𝑛 + 𝑘2 ´ 𝑘 , 𝑛 ´ 𝑘 + 2 =: 𝑑0 max min 2ď 𝑘 ď 𝑛 2𝑛 is achieved. Let 𝑘 1 be the value of 𝑘 for which the maximum  2  𝑛 + 𝑛 + 𝑘2 ´ 𝑘 max min , 𝑛 ´ 𝑘 + 2 + 𝛿0 =: 𝑑1 . 2ď 𝑘 ď 𝑛 2𝑛 is achieved. Note that 𝑑0 ă 𝑑1 ă 𝑑0 + 𝛿0 . By Theorem 11, the 𝑘 1 -linear estimate (4.12) holds for 𝑑= 𝑛2 + 𝑛 + 𝑘 12 ´ 𝑘 1 ď 𝑑1 . 2𝑛 Now, by Lemma 45, an x-ray estimate holds at dimension 𝑑0 , since 𝑑0 ě 𝑑1 ´𝛿0 . □ The following proposition is essentially the same as Proposition 3.3 from [17]. It has been provided here for completeness since the definition of stickiness is slightly different from the definition appearing in that paper. Proposition 46. Suppose that there is an x-ray estimate at dimension 𝑑 and that there exists a Besicovitch set 𝐵 with dim 𝑀 (𝐸) ď 𝑑 + 𝜀, for some 𝜀 ě 10´10 . Then, for any sufficiently small 𝛿 ą 0, there is a sticky collection T of 𝛿-tubes satisfying Ø | 𝑇 | ≲ 𝛿 𝑛´𝑑+𝜀 . 𝑇 PT In the following proof, the same notation as in previous section will be used to parametrize lines in R𝑛 . Proof. Let 𝐵 be a Besicovitch set with dim 𝑀 (𝐸) ď 𝑑 + 𝜀. From the definition of upper Minkowski dimension, this means |𝑁𝜃 (𝐵)| ≲ 𝜃 𝑛´𝑑+𝜀 for all 𝛿 ď 𝜃 ă 1. Let 𝐿 be a set of lines such that i. 𝐿 contains a line in every direction (except the vertical direction), and (4.15) 57 ii. 𝐵 X ℓ contains a unit line segment for every ℓ P 𝐿. Every line in ℓ can be parametrized as ℓ = (a, 0) + R𝑒 where a P R𝑛´1 is a point and 𝑒 P S𝑛´1 is the direction in which ℓ points; and thus ℓ will be denoted (𝑎, 𝑒). Call a direction 𝑒 P S𝑛´1 sticky at scale 𝜃 if |{a P R𝑛´1 : (a, 𝑒) P 𝑁𝜃 (𝐿)}| ≲ 𝜃 𝑛´1´𝐶1 𝜀 . (4.16) Let E Ď S𝑛´1 be the set of directions given by Ù E= {𝑒 P S𝑛´1 : 𝑒 is sticky at scale 2´ 𝑘 }. 𝑘 PZ:1≲𝑘 ≲log(1/𝛿) For each 𝑒 P S𝑛´1 zE, there exists a scale 𝜃 𝑒 ą 0 such that (4.16) fails. By dyadic pigeonholing, there is a set E0 Ď S𝑛´1 zE and a number 𝜃 ą 0 such that 𝜃 𝑒 „ 𝜃 for all 𝑒 P E0 and 1 𝜈(S𝑛´1 zE). 𝜈(E0 ) ≳ log(1/𝛿) Let E1 be a maximal 𝜃-separated subset of E0 . Since every 𝑒 P E1 is fails to be sticky at scale 𝜃, there are ≳ 𝜃 ´𝐶1 𝜀 𝜃-separated points a for which the line (a, 𝑒) P 𝑁𝜃 (𝐿). Let T0 be the set of 𝜃-tubes given by taking the 𝜃-neighborhoods of the unit line segments contained in 𝐵Xℓ for each of these lines ℓ = (a, 𝑒). By the x-ray transform estimate, there exist numbers 0 ă 𝛼, 𝛾 ă 1 such that ∑︁ 𝑛 || 𝜒𝑇 || 𝑑 1 ≲ 𝜃 1´ 𝑑 ´𝜀 (𝜃 ´𝐶1 𝜀 ) 1´𝛼 (𝜃 𝑛´1 #E1 ) 1´𝛾 . 𝑇 PT0 By an application of Hölder’s inequality, this implies that Ø | 𝑇 | ≳ 𝜃 𝑛´𝑑+𝜀 𝜃 ´𝑑𝐶1 𝜀𝛼 (𝜃 𝑛´1 #E1 ) 𝑑𝛾 . 𝑇 PT0 Since | Ð 𝑇 PT0 𝑇 | Ď 𝑁2𝜃 (𝐵), it follows from (4.15) that #E1 ≲ 𝜃 1´𝑛 𝜃 𝐶1 𝜀𝛼/𝛾 . Since E1 is a maximal 𝜃-separated subset of E0 , this implies that 𝜈(E0 ) ≲ 𝜃 𝐶1 𝜀𝛼/𝛾 and thus 𝜈(E) ≳ 1 ´ log(1/𝛿)𝜃 𝐶1 𝜀𝛼/𝛾 ≳ 1, 58 provided 𝐶1 is chosen to be sufficiently large. Now let E 𝛿 be a maximal 𝛿-separated subset of E. Let T be the set of 𝛿-tubes given by the 𝛿-neighborhoods of the unit line segments contained in 𝐵 X ℓ for every ℓ P 𝐿 pointing in a direction 𝑒 P E 𝛿 , then T „ 𝛿1´𝑛 . It will be shown that a subset T1 of T is sticky. Choose a set of 𝜃-tubes 𝑇𝜃 so every tube 𝑇 P T is contained in at least one 𝑇𝜃 P T𝜃 . It can be ensured that T𝜃 is minimal, i.e. every 𝑇𝜃 P T𝜃 contains at least one 𝑇 P T which is not contained in any other 𝜃-tube in T𝜃 . Consider the set of coaxial lines of tubes in 𝑇𝜃 . Suppose that there are several of these lines ℓ1 , ¨ ¨ ¨ , ℓ𝑁 that are “almost parallel,”, i.e. all the lines ℓ1 , ¨ ¨ ¨ , ℓ𝑁 point in directions making at most angle 𝜃 with a single direction 𝑒 P S𝑛´1 . For each 1 ď 𝑖 ď 𝑁, the line ℓ𝑖 = (a𝑖 , 𝑒𝑖 ) is contained in 𝑁𝜃 (dir´1 (E)). Thus, the 𝜃-tubes with coaxial lines ℓ1 , . . . , ℓ𝑁 all lie in dir´1 (𝑁2𝜃 ({𝑒})) X 𝑁2𝜃 (dir´1 (E)). It follows that  ´1 𝜆 dir ´1 (𝑁2𝜃 ({𝑒}) X 𝑁2𝜃 (dir  (E)) ≳ 𝑁𝜃 2(𝑛´1) and then, by Fubini’s theorem, there must exist 𝑒 1 P 𝑁2𝜃 ({𝑒}) X E such that |{a P R𝑛´1 : (a, 𝑒 1 ) P 𝑁𝜃 (𝐿)}| ≳ 𝑁𝜃 𝑛´1 . From (4.16), 𝑁 ≲ 𝜃 ´𝐶1 𝜀 . Thus, it is possible to choose a 𝜃-direction-separated subset T1𝜃 Ď T𝜃 such that #T1𝜃 ≳ 𝜃 𝐶1 𝜀 #T𝜃 . After taking another refinement T2𝜃 , it is possible to ensure that every 𝑇𝜃 P T2𝜃 contains ≳ (𝜃/𝛿) 1´𝑛+𝐶2 𝜀 𝛿-tubes for a suitable constant 𝐶2 ą 0. If T1 is chosen to be T1 := {𝑇 P T : D𝑇𝜃 P T2𝜃 such that 𝑇 Ď 𝑇𝜃 }, it follows that T1 is sticky and that #T1 ≳ 𝛿1´𝑛+𝐶3 𝜀 . □ Together with Proposition 46 and Proposition 31, Proposition 44 gives the following corollary. Corollary 47. The Minkowski dimension of every Besicovitch set in R𝑛 must exceed ? 𝑑 ě (2 2 + 𝜀)𝑛, for a certain value of 𝜀 greater than 10´10 . Though this estimate is not as strong as one previously proved in [18], it provides a new proof of the result. 59 BIBLIOGRAPHY [1] Jonathan Bennett, Anthony Carbery, and Terence Tao. “On the multilinear restriction and Kakeya conjectures”. In: Acta Mathematica 196.2 (2006), pp. 261–302. doi: 10.1007/s11511-006-0006-4. url: https://doi. org/10.1007/s11511-006-0006-4. [2] A.S Besicovitch. “On Kakeya’s problem and a similar one”. In: Math. Zeitschrift 27 (1928), pp. 312–320. 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