The Assouad dimension of Kakeya sets in R3

• 成果类型: Article
• 署名作者: Wang, Hong; Zahl, Joshua
• 署名单位: New York University; University of British Columbia
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01336-x
• 发表日期: 2025
• 页码: 153-206
• 关键词: projections

摘要

This paper studies the structure of Kakeya sets in R-3. We show that for every Kakeya set K subset of R-3, there exist well-separated scales 0 < delta < rho <= 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 R-3 has Assouad dimension 3 and every Ahlfors-David regular Kakeya set in R-3 has Hausdorff dimension 3. We also show that every Kakeya set in 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.