Pattern Problems related to the Arithmetic Kakeya Conjecture

TL;DR

This paper explores problems related to the Kakeya conjecture, establishing equivalences with the arithmetic Kakeya conjecture, and extends results to polytopes, number fields, and different dimensions, advancing understanding of geometric measure theory.

Contribution

It demonstrates the equivalence of various homothets problems to the arithmetic Kakeya conjecture and extends known implications to packing dimension, also generalizing number theoretic methods to new contexts.

Findings

Many problems are equivalent to the arithmetic Kakeya conjecture.

The arithmetic Kakeya conjecture implies the Kakeya conjecture for packing dimension.

Provides bounds for polytopes and extends methods to number fields.

Abstract

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 arbitrary polytopes. We give a lower bound for polytopes in $\mathbb{R}^n$, and show that this is sharp for simplices. Finally, we generalize number theoretic methods of Green and Ruzsa to study patterns in number fields and thereby provide upper bounds for several of these homothet problems.