The Assouad dimension of Kakeya sets in $\mathbb{R}^3$

Abstract

This paper studies the structure of Kakeya sets in $\mathbb{R}^3$. We show that for every Kakeya set $K\subset\mathbb{R}^3$, there exist well-separated scales $0<\delta<\rho\leq 1$ so that the $\delta$ neighborhood of $K$ is almost as large as the $\rho$ neighborhood of $K$. As a consequence, every Kakeya set in $\mathbb{R}^3$ has Assouad dimension 3 and every Ahlfors-David regular Kakeya set in $\mathbb{R}^3$ has Hausdorff dimension 3. We also show that every Kakeya set in $\mathbb{R}^3$ that has "stably equal" Hausdorff and packing dimension (this is a new notion, which is introduced to avoid certain obvious obstructions) must have Hausdorff dimension 3. The above results follow from certain multi-scale structure theorems for arrangements of tubes and rectangular prisms in three dimensions, and a mild generalization of the sticky Kakeya theorem previously proved by the authors.