The Assouad dimension of Kakeya sets in $\mathbb{R}^3$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918009955680256 |
|---|---|
| author | Wang, Hong Zahl, Joshua |
| author_facet | Wang, Hong Zahl, Joshua |
| contents | 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 $0<δ<ρ\leq 1$ so that the $δ$ neighborhood of $K$ is almost as large as the $ρ$ neighborhood of $K$. As a consequence, every Kakeya set in $\mathbb{R}^3$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_12337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Assouad dimension of Kakeya sets in $\mathbb{R}^3$ Wang, Hong Zahl, Joshua Classical Analysis and ODEs 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 $0<δ<ρ\leq 1$ so that the $δ$ neighborhood of $K$ is almost as large as the $ρ$ neighborhood of $K$. As a consequence, every Kakeya set in $\mathbb{R}^3$ 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. |
| title | The Assouad dimension of Kakeya sets in $\mathbb{R}^3$ |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2401.12337 |