Fourier analytic properties of Kakeya sets in finite fields

TL;DR

This paper proves that Kakeya sets in finite fields support a measure with bounded Fourier transform, providing sharp bounds and new proofs for size estimates, and extends results to sets containing k-planes.

Contribution

It introduces a Fourier analytic approach to Kakeya sets in finite fields, establishing sharp bounds and size estimates, and generalizes to k-plane configurations.

Findings

Kakeya sets support a measure with Fourier transform bounded by q^{-1}

The bound is sharp in all dimensions at least 2

A Kakeya set in dimension 2 has size at least q^2/2

Abstract

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 $q^2/2$ (which is asymptotically sharp). We also establish analogous results for sets containing $k$-planes in a given set of orientations.