We investigate analogues for curves of the Kakeya problem for straight lines. These arise from H"ormander-type oscillatory integrals in the same way as the straight line case comes from the restriction and Bochner-Riesz problems. Using some of the geometric and arithmetic techniques developed for the straight line case by Bourgain, Wolff, Katz and Tao, we are able to prove positive results for families of parabolas whose coefficients satisfy certain algebraic conditions.
Here we show some results related with Kakeya conjecture which says that for any integer $n\geq 2$, a set containing line segments in every dimension in $\mathbb{R}^n$ has full Hausdorff dimension as well as box dimension. We proved here that the Kakeya books, which are Kakeya sets with some restrictions on positions of line segments have full box dimension. We also prove here a relation between the projection property of Kakeya sets and the Kakeya conjecture. If for any Kakeya set $K\subset\mathbb{R}^n$, the Hausdorff dimension of orthogonal projections on $k\leq n$ subspaces is independent o…
We develop a notion of finite order lacunarity for direction sets in $\mathbb R^{d+1}$. Given a direction set $Ω$ that is sublacunary according to this definition, we construct random examples of Euclidean sets that contain unit line segments with directions from $Ω$ and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. This generalizes to higher dimensions a planar result due to Bateman. Combined with earlier work of Alfonseca, Bateman, Parcet and Rogers, this notion of lacunarity and Kakeya-type sets also yields a characterization in all…
Kakeya and Furstenberg problems for sticky sets of tubes
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Department of Mathematics, Massachusetts Institute of Technology（麻省理工学院数学系）
Given a Cantor-type subset $Ω$ of a smooth curve in $\mathbb R^{d+1}$, we construct examples of sets that contain unit line segments with directions from $Ω$ and exhibit analytical features similar to those of classical Kakeya sets of arbitrarily small $(d+1)$-dimensional Lebesgue measure. The construction is based on probabilistic methods relying on the tree structure of $Ω$, and extends to higher dimensions an analogous planar result of Bateman and Katz. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set $Ω$ is unbounded …
The Assouad dimension of Kakeya sets in $\mathbb{R}^3$
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Department of Mathematics, The University of British Columbia（不列颠哥伦比亚大学数学系）
We define Kakeya sets in the Heisenberg group and show that the Heisenberg Hausdorff dimension of Kakeya sets in the first Heisenberg group is at least 3. This lower bound is sharp since, under our definition, the $\{xoy\}$-plane is a Kakeya set with Heisenberg Hausdorff dimension 3.
A Study on Kakeya Needle Problem for $(n-1)$-Rectifiable Sets
School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；School of Mathematics and Statistics, Henan University（河南大学数学与统计学院）
A K^n_2-set is a set of zero Lebesgue measure containing a translate of every plane in an (n-2)-dimensional manifold in Gr(n,2), where the manifold fulfills a curvature condition. We show that this is a natural class of sets with respect to the Kakeya problem and prove that dim_H(E)\ge 7/2 for all K^4_2-sets E. When the underlying field is replaced by the complex numbers C, we get dim_H(E)\ge 7 for all K^4_2-sets over C, and we construct an example to show that this is sharp. Thus K^4_2-sets over C do not necessarily have full Hausdorff dimension.
Let $\frak T$ be a set of cylindrical tubes in $\mathbb{R}^3$ of length $N$ and radius 1. If the union of the tubes has volume $N^{3 - σ}$, and each point in the union lies in tubes pointing in three quantitatively different directions, and if a technical assumption holds, then at scale $N^σ$, the tubes are clustered into rectangular slabs of dimension $1 \times N^σ\times N^σ$. This estimate generalizes the graininess estimate proven by Katz-Laba-Tao. The proof is based on modeling the union of tubes with a high-degree polynomial.
We prove that the Kakeya maximal conjecture is equivalent to the $Ω$-Kakeya maximal conjecture. This completes a recent result in [2] where Keleti and Math{é} proved that the Kakeya conjecture is equivalent to the $Ω$-Kakeya conjecture. Moreover, we improve concrete bound on the Hausdorff dimension of a $Ω$-Kakeya set : for any Bore set $Ω$ in S n--1 , we prove that if X $\subset$ R n contains for any e $\in$ $Ω$ a unit segment oriented along e then we have dX $\ge$ 6 11 d$Ω$ + 1 where dE denotes the Hausdorff dimension of a set E.
I show that $L^{p}-L^{q}$ estimates for the Kakeya maximal function yield lower bounds for the conformal dimension of Kakeya sets, and upper bounds for how much quasisymmetries can increase the Hausdorff dimension of line segments inside Kakeya sets. Combining the known $L^{p}-L^{q}$ estimates of Wolff and Katz-Tao with the main result of the paper, the conformal dimension of Kakeya sets in $\mathbb{R}^{n}$ is at least $\max\{(n + 2)/2,(4n + 3)/7\}$. Moreover, if $f$ is a quasisymmetry from a Kakeya set $K \subset \mathbb{R}^{n}$ onto any at most $n$-dimensional metric space, the $f$-image of …
In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. It was proved only in the plane, whereas in higher dimensions both geometric and arithmetic combinatorial methods were used to obtain partial results. In the first part of the thesis we define generalized Kakeya sets in metric spaces satisfying c…
We present a construction of a measure-zero Kakeya-type set in a finite-dimensional space $K^d$ over a local field with finite residue field. The construction is an adaptation of the ideas appearing in [12] and [13]. The existence of measure-zero Kakeya-type sets over discrete valuation rings is also discussed, giving an alternative construction to the one presented in [4] over $\mathbb{F}_q[[t]]$.
A Besicovitch set is a subset of $\R^d$ that contains a unit line segment in every direction and the famous Kakeya conjecture states that Besicovitch sets should have full dimension. We provide a number of results in support of this conjecture in a variety of contexts. Our proofs are simple and aim to give an intuitive feel for the problem. For example, we give a very simple proof that the packing and lower box-counting dimension of any Besicovitch set is at least $(d+1)/2$ (better estimates are available in the literature). We also study the `generic validity' of the Kakeya conjecture in the …
Given a family G of rectangles, to which one associates a tree [G], one defines a natural number $λ$ [G] called its analytic split and satisfying, for all 1 < p < $\infty$ log($λ$ [G]) p MG p p where MG is the Hardy-Littlewood type maximal operator associated to the family G. As an application, we completely characterize the boundeness of planar rarefied directional maximal operators on L p for 1 < p < $\infty$. Precisely, if $Ω$ is an arbitrary set of angles in [0, $π$ 4), we prove that any rarefied basis B of the directional basis R $Ω$ yields an operator MB that has the same L p-behavior th…
A $(d,k)$-set is a subset of $\mathbb{R}^d$ containing a $k$-dimensional unit ball of all possible orientations. Using an approach of D.~Oberlin we prove various Fourier dimension estimates for compact $(d,k)$-sets. Our main interest is in restricted $(d,k)$-sets, where the set only contains unit balls with a restricted set of possible orientations $Γ$. In this setting our estimates depend on the Hausdorff dimension of $Γ$ and can sometimes be improved if additional geometric properties of $Γ$ are assumed. We are led to consider cones and prove that the cone in $\mathbb{R}^{d+1}$ has Fourier d…
The Fourier restriction phenomenon and the size of Kakeya sets are explored in the setting of the ring of integers modulo $N$ for general $N$ and a striking similarity with the corresponding euclidean problems is observed. One should contrast this with known results in the finite field setting.
We compute the Minkowski dimension for a family of self-affine sets on the plane. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this class where we allow overlapping, and do not impose any conditions on the norms of the linear maps. The family under consideration was inspired by the theory of Kakeya sets.
We say that a planar set $A$ has the Kakeya property if there exist two different positions of $A$ such that $A$ can be continuously moved from the first position to the second within a set of arbitrarily small area. We prove that if $A$ is closed and has the Kakeya property, then the union of the nontrivial connected components of $A$ can be covered by a null set which is either the union of parallel lines or the union of concentric circles. In particular, if $A$ is closed, connected and has the Kakeya property, then $A$ can be covered by a line or a circle.
We completely characterize the boundedness of planar directional maximal operators on L^p. More precisely, if Omega is a set of directions, we show that M_Omega, the maximal operator associated to line segments in the directions Omega, is unbounded on L^p, for all p < infinity, precisely when Omega admits Kakeya-type sets. In fact, we show that if Omega does not admit Kakeya sets, then Omega is a generalized lacunary set, and hence M_Omega is bounded on L^p, for p>1.
A Kakeya set is a subset of F^n, where F is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least C_n * q^n, where C_n depends only on n. This improves the previously best lower bound for general n of ~q^{4n/7}.
Let $A, B$, be finite subsets of an abelian group, and let $G \subset A \times B$ be such that $# A, # B, # \{a+b: (a,b) \in G \} \leq N$. We consider the question of estimating the quantity $# \{a-b: (a,b) \in G \}$. Recently Bourgain improved the trivial upper bound of $N^2$ to $N^{2-1/13}$, and applied this to the Kakeya conjecture. We improve Bourgain's estimate further to $N^{2-1/6}$, and obtain the further improvement of $N^{2-1/4}$ if we also know that $# \{a+2b: (a,b) \in G\} \leq N$. We conclude that Besicovitch sets in $\R^n$ have Hausdorff dimension at least 6n/11+5/11 and Minkowski…
We construct a union of N parallelograms of dimensions approximately 1/N x 1 in the plane, with the slope of their long sides in the standard Cantor set. The union has area 1/log N but the union of the doubles has area log log N/ log N. In particular, this implies unbounded of the associated maximal operator in L^p for any p different from infinity. The construction is by randomizing an earlier construction of the second author for the L^2 case. The proof that the construction satisfies the desired conditions is by elementary estimates in the theory of percolation on trees as developed by R. L…
A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle $\Z_n$. A hunter and a rabbit move on the nodes of $\Z_n$ without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al (2003) provide strategies for hunter and rabbit that are optimal up to constant factors and achieve probability of capture in the first $n$ steps of order $1/\log n$. We show these strategies yield a Kakeya set consisting of $4n$ triangles with…
This paper considers the problem of establishing $L^p$-improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension $n$ of the ambient space and the dimension $k$ of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang and…
We consider unions of $SL(2)$ lines in $\mathbb{R}^{3}$. These are lines of the form $$L = (a,b,0) + \mathrm{span}(c,d,1),$$ where $ad - bc = 1$. We show that if $\mathcal{L}$ is a Kakeya set of $SL(2)$ lines, then the union $\cup \mathcal{L}$ has Hausdorff dimension $3$. This answers a question of Wang and Zahl. The $SL(2)$ lines can be identified with horizontal lines in the first Heisenberg group, and we obtain the main result as a corollary of a more general statement concerning unions of horizontal lines. This statement is established via a point-line duality principle between horizontal …
We prove the discrete analogue of Kakeya conjecture over $\mathbb{R}^n$. This result suggests that a (hypothetically) low dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization which completely solves the discrete analogue of the Furstenberg set problem in all dimensions. The difference between our theorems and the (true) problems is only the (still difficult) issue of continuity since no transversality-at-incidences assumptions are imposed. The main tool of the proof is a theorem of Wongkew \cite{wongkew2003volumes} which states that…
Given a discrete group $\G$ and an orthogonal action $γ: \G \to O(n)$ we study $L_p$ convergence of Fourier integrals which are frequency supported on the semidirect product $\R^n \rtimes_γ\G$. Given a unit $u \in \R^n$ and $1 < p \neq 2 < \infty$, our main result shows that the twisted (directional) Hilbert transform $H_u \rtimes_γid_\G$ is $L_p$-bounded iff the orbit $\mathcal{O}_γ(u)$ is finite. This is in sharp contrast with twisted Riesz transforms $R_u \rtimes_γid_\G$, which are always bounded. Our result characterizes Fourier summability in $L_p$ for this class of groups. We also extend…
We study a variety of problems about homothets of sets related to the Kakeya conjecture. In particular, we show many of these problems are equivalent to the arithmetic Kakeya conjecture of Katz and Tao. We also provide a proof that the arithmetic Kakeya conjecture implies the Kakeya conjecture for packing dimension, as this implication was previously only known for Minkowski dimension. We consider several questions analogous to the classical results of Stein and Bourgain about the Lebesgue measure of a set containing a sphere centered at every point of $[0,1]^n$, where we replace spheres by ar…
We extend a result of the second author \cite[Theorem 1.1]{soggekaknik} to dimensions $d \geq 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<\frac{2(d+1)}{d-1}$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee \cite{leebilinear} and a variable coefficient variant of an "$\veps$ removal lemma" of Tao and Vargas \cite{tv1}. We also use Hörmander's \cite{HorOsc} $L^2$ oscillatory integral theorem and the Cartan-Hadamard theorem to show that, under the assumption of nonpositive curvature, …
In this paper, we study continuous Kakeya line and needle configurations, of both the oriented and unoriented varieties, in connected Lie groups and some associated homogenous spaces. These are the analogs of Kakeya line (needle) sets (subsets of $\mathbb{R}^n$ where it is possible to turn a line (respectively an interval of unit length) through all directions {\bf continuously, without repeating a "direction"}.) We show under some general assumptions that any such continuous Kakeya line configuration set in a connected Lie group must contain an open neighborhood of the identity, and hence mus…
This thesis investigates two problems that are discrete analogues of two harmonic analytic problems which lie in the heart of research in the field. More specifically, we consider discrete analogues of the maximal Kakeya operator conjecture and of the recently solved endpoint multilinear Kakeya problem, by effectively shrinking the tubes involved in these problems to lines, thus giving rise to the problems of counting joints and multijoints with multiplicities. In fact, we effectively show that, in $\mathbb{R}^3$, what we expect to hold due to the maximal Kakeya operator conjecture, as well as…
