Sticky Kakeya sets and the sticky Kakeya conjecture

A Kakeya set is a compact subset of R n \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 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 2000 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 n . We prove this conjecture in three dimensions.