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

Abstract

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