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

Abstract

A $(d,k)$-set is a subset of $\mathbb{R}^d$ 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 $\Gamma$. In this setting our estimates depend on the Hausdorff dimension of $\Gamma$ and can sometimes be improved if additional geometric properties of $\Gamma$ are assumed. We are led to consider cones and prove that the cone in $\mathbb{R}^{d+1}$ has Fourier dimension $d-1$, which may be of interest in its own right.