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

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 $0<\delta <\rho \le 1$ 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.