2024/01/22 by Hong Wang, Joshua Zahl · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #advanced mathematical theories #Mathematical Dynamics and Fractals
paper · pdf · doi:10.1007/s00222-025-01336-x
This paper studies the structure of Kakeya sets in ℝ3. We show that for every Kakeya set K⊂ℝ3, there exist well-separated scales 0<δ<ρ≤ 1 so that the δ neighborhood of K is almost as large as the ρ neighborhood of K. As a consequence, every Kakeya set in ℝ3 has Assouad dimension 3 and every Ahlfors-David regular Kakeya set in ℝ3 has Hausdorff dimension 3. We also show that every Kakeya set in ℝ3 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.