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 generalization of the sticky Kakeya theorem previously proved by the authors.
