Kakeya sets and dimension compression in compact Lie groups

Abstract

We study two Kakeya set problems in compact semisimple Lie groups. In the first, motivated by the group structure, a Kakeya set is required to contain a left coset of every closed one-dimensional torus. For a compact semisimple Lie group $G$ of dimension $d$ and rank $r$, we determine the optimal Minkowski dimension for this problem, proving that it is exactly $(d+r)/2$. The upper bound is obtained from a Lie-theoretic construction associated with a Chevalley involution, while the lower bound combines incidence geometry with geometry of numbers. We also consider a local Kakeya set problem, closer to the classical harmonic-analytic formulation, in which one requires a fixed-length one-parameter arc in every direction. For this problem we obtain lower bound $(d+1)/2$ and $(d+2)/2$ for odd and even $r$, and upper bound $(d+r)/2$. We conjecture that the upper bound should be sharp.