Generalized Arithmetic Kakeya

TL;DR

This paper extends the arithmetic approach to Kakeya problems to higher dimensions, providing new bounds and strengthening sum-difference inequalities, with implications for Minkowski dimensions of Besicovitch sets.

Contribution

It introduces a generalized arithmetic Kakeya problem in higher dimensions and proves an upper bound using an innovative iterative method, strengthening existing inequalities.

Findings

Established an upper bound for the higher-dimensional arithmetic Kakeya problem.

Developed a new method to strengthen sum-difference inequalities.

Derived a lower bound for the Minkowski dimension of Besicovitch sets.

Abstract

Around the early 2000-s, Bourgain, Katz and Tao introduced an arithmetic approach to study Kakeya-type problems. They showed that the Euclidean Kakeya conjecture follows from a natural problem in additive combinatorics, now referred to as the `Arithmetic Kakeya Conjecture'. We consider a higher dimensional variant of this problem and prove an upper bound using a certain iterative argument. The main new ingredient in our proof is a general way to strengthen the sum-difference inequalities of Katz and Tao which might be of independent interest. As a corollary, we obtain a new lower bound for the Minkowski dimension of $(n, d)$-Besicovitch sets.