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.