Sticky Kakeya sets and the sticky Kakeya conjecture
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915657195454464 |
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| author | Wang, Hong Zahl, Joshua |
| author_facet | Wang, Hong Zahl, Joshua |
| contents | A Kakeya set is a compact subset of $\mathbb{R}^n$ that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have Hausdorff and Minkowski dimension $n$. There is a special class of Kakeya sets, called sticky Kakeya sets. Sticky Kakeya sets exhibit an approximate multi-scale self-similarity, and sets of this type played an important role in Katz, Łaba, and Tao's groundbreaking 1999 work on the Kakeya problem. We propose a special case of the Kakeya conjecture, which asserts that sticky Kakeya sets must have Hausdorff and Minkowski dimension $n$. We prove this conjecture in three dimensions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_09581 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sticky Kakeya sets and the sticky Kakeya conjecture Wang, Hong Zahl, Joshua Classical Analysis and ODEs A Kakeya set is a compact subset of $\mathbb{R}^n$ that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have Hausdorff and Minkowski dimension $n$. There is a special class of Kakeya sets, called sticky Kakeya sets. Sticky Kakeya sets exhibit an approximate multi-scale self-similarity, and sets of this type played an important role in Katz, Łaba, and Tao's groundbreaking 1999 work on the Kakeya problem. We propose a special case of the Kakeya conjecture, which asserts that sticky Kakeya sets must have Hausdorff and Minkowski dimension $n$. We prove this conjecture in three dimensions. |
| title | Sticky Kakeya sets and the sticky Kakeya conjecture |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2210.09581 |