Fourier analytic properties of Kakeya sets in finite fields

We prove that a Kakeya set in a vector space over a finite field of size $q$ always supports a probability measure whose Fourier transform is bounded by $q^{-1}$ for all non-zero frequencies. We show that this bound is sharp in all dimensions at least 2. In particular, this provides a new and self-contained proof that a Kakeya set in dimension 2 has size at least $q^2\mathrm{/}2$ (which is asymptotically sharp). We also establish analogous results for sets containing $k$-planes in a given set of orientations.