On the Fourier dimension of (d,k)-sets and Kakeya sets with restricted directions

Abstract

A (d, k)-set is a subset of ℝd containing a k-dimensional unit ball of all possible orientations. Using an approach of D. Oberlin we prove various Fourier dimension estimates for compact (d, k)-sets. Our main interest is in restricted (d, k)-sets, where the set only contains unit balls with a restricted set of possible orientations Γ. In this setting our estimates depend on the Hausdorff dimension of Γ and can sometimes be improved if additional geometric properties of Γ are assumed. We are led to consider cones and prove that the cone in ℝd+1 has Fourier dimension d−1, which may be of interest in its own right.