Related papers: The Spherical Kakeya Problem in Finite Fields

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…

We derive Maximal Kakeya estimates for functions over $\mathbb{Z}/N\mathbb{Z}$ proving the Maximal Kakeya conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$ as stated by Hickman and Wright [HW18]. The proof involves using polynomial…

Given a sphere of any radius $r$ in an $n$-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which defines the average…

This paper is the third part of the series "Spherical higher order Fourier analysis over finite fields", aiming to develop the higher order Fourier analysis method along spheres over finite fields, and to solve the geometric Ramsey…

We prove that every Kakeya set in $\mathbb{R}^3$ formed from lines of the form $(a,b,0) + \operatorname{span}(c,d,1)$ with $ad-bc=1$ must have Hausdorff dimension $3$; Kakeya sets of this type are called $SL_2$ Kakeya sets. This result was…

We show that the problem of counting collinear points in a permutation (previously considered by the author and J. Solymosi in "Collinear Points in Permutations", 2005) and the well-known finite plane Kakeya problem are intimately…

This paper studies the structure of core sets under different similarity classes. We investigate the influence of factors of the minimal polynomial with different degrees on the structure of core sets. When $F$ is a finite field of prime…

We improve the previously best known upper bounds on the sizes of $\theta$-spherical codes for every $\theta<\theta^*\approx 62.997^{\circ}$ at least by a factor of $0.4325$, in sufficiently high dimensions. Furthermore, for sphere packing…

We give theorems that can be used to upper bound the densities of packings of different spherical caps in the unit sphere and of translates of different convex bodies in Euclidean space. These theorems extend the linear programming bounds…

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…

If the n-dimensional unit sphere is covered by finitely many spherically convex bodies, then the sum of the inradii of these bodies is at least {\pi}. This bound is sharp, and the equality case is characterized.

We prove that the largest convex shape that can be placed inside a given convex shape $Q \subset \mathbb{R}^{d}$ in any desired orientation is the largest inscribed ball of $Q$. The statement is true both when "largest" means "largest…

The arithmetic Kakeya conjecture, formulated by Katz and Tao in 2002, is a statement about addition of finite sets. It is known to imply a form of the Kakeya conjecture, namely that the upper Minkowski dimension of a Besicovitch set in…

We extend the "method of multiplicities" to get the following results, of interest in combinatorics and randomness extraction. (A) We show that every Kakeya set (a set of points that contains a line in every direction) in $\F_q^n$ must be…

In this paper we study the boundedness of extension operators associated with spheres in vector spaces over finite fields.In even dimensions, we estimate the number of incidences between spheres and points in the translated set from a…

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…

We prove that the Kakeya maximal conjecture is equivalent to the $\Omega$-Kakeya maximal conjecture. This completes a recent result in [2] where Keleti and Math{\'e} proved that the Kakeya conjecture is equivalent to the $\Omega$-Kakeya…

A finite subset $Y$ on the unit sphere $S^{n-1} \subseteq \mathbb{R}^n$ is called a spherical design of harmonic index $t$, if the following condition is satisfied: $\sum_{\mathbf{x}\in Y}f(\mathbf{x})=0$ for all real homogeneous harmonic…

We establish that if $d \geq 2k + 6$ and $q$ is odd and sufficiently large with respect to $\alpha \in (0,1)$, then every set $A\subseteq \mathbf{F}_q^d$ of size $|A| \geq \alpha q^d$ will contain an isometric copy of every spherical…

This paper is the first part of the series "Spherical higher order Fourier analysis over finite fields", aiming to develop the higher order Fourier analysis method along spheres over finite fields, and to solve the geometric Ramsey…