On the arithmetic Kakeya conjecture of Katz and Tao

TL;DR

This paper explores the arithmetic Kakeya conjecture, providing equivalent formulations, confirming a finite field version, and establishing some lower bounds, thereby advancing understanding of the conjecture's implications in additive combinatorics.

Contribution

It presents equivalent forms of the conjecture, proves a finite field variant, and offers new lower bounds, contributing to the study of the arithmetic Kakeya problem.

Findings

Finite field variant of the conjecture holds.

Multiple equivalent formulations of the conjecture are provided.

Lower bounds are established for related additive sets.

Abstract

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.