Sticky Kakeya sets and the sticky Kakeya conjecture

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, 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.

69 pages, 6 figures. v2: typos corrected and exposition updated based on referees' suggestions. Results unchanged. To appear in J. Amer. Math. Soc