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arXiv · 2210.09581

Sticky Kakeya sets and the sticky Kakeya conjecture

Hong Wang·Joshua Zahl
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Abstract

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, {\L}aba, 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.

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Hong Wang, Joshua Zahl. 2022-10-18. Sticky Kakeya sets and the sticky Kakeya conjecture. https://doi.org/10.1090/jams/1067

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