An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension

An improved bound on the Minkowski dimension of Besicovitch sets in R^3

A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in \R^3. In this paper we show that the Minkowski dimension must in fact be greater than 5/2 + εfor some absolute constant ε> 0. One observation arising from the argument is that Besicovitch sets of near-minimal dimension have to satisfy certain strong properties, which we call ``stickiness,'' ``planiness,'' and ``graininess.'' The purpose of this paper is to improve upon the known bounds for the Minkowski dim…

关于 $\boldsymbol{\rm \bb{R}^3}$ 中Besicovitch集Hausdorff维数的一个改进界

An improved bound on the Hausdorff dimension of Besicovitch sets in $\mathbb{R}^3$

该研究改进了三维欧氏空间中Besicovitch集的Hausdorff维数下界至$5/2+ε_0$，通过分析满足Wolff公理的管并体积得到结果，还提出了Kakeya猜想的新近乎反例$SL_2$示例供后续研究。

New bounds on Kakeya problems

We establish new estimates on the Minkowski and Hausdorff dimensions of Besicovitch sets and obtain new bounds on the Kakeya maximal operator.

On the arithmetic Kakeya conjecture of Katz and Tao

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 $\mathbb{R}^n$ is $n$. In this note we discuss this conjecture, giving a number of equivalent forms of it. We show that a natural finite field variant of it does hold. We also give some lower bounds.

A new bound on partial sum-sets and difference-sets, and applications to the Kakeya conjecture

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…