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Fourier analytic properties of Kakeya sets in finite fields

Jonathan M. Fraser

TL;DR

This work analyzes Kakeya-type sets in the finite field setting $\mathbb{F}_q^d$ through Fourier-analytic methods. It proves that any $(d,k,\Gamma)$-set supports a probability measure with Fourier decay $|\widehat{\mu}(\xi)| \le q^{-k}$ for all nonzero frequencies, and derives corresponding lower bounds on the set size, including a finite-field analogue of Oberlin's Fourier-dimension results. In particular, Kakeya sets ($k=1$) admit $|K| \gtrsim q^2$, and in dimension $d=2$ one obtains the sharp bound $|K| \ge q^2/2$, with a new, self-contained proof of the asymptotically sharp density in this case. The paper also proves sharpness of the bound via a product construction, showing the $q^{-1}$ decay cannot be improved in general for $d\ge 2$, and it extends the analysis to $(d,k)$-sets with restricted orientations.

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.

Fourier analytic properties of Kakeya sets in finite fields

TL;DR

This work analyzes Kakeya-type sets in the finite field setting through Fourier-analytic methods. It proves that any -set supports a probability measure with Fourier decay for all nonzero frequencies, and derives corresponding lower bounds on the set size, including a finite-field analogue of Oberlin's Fourier-dimension results. In particular, Kakeya sets () admit , and in dimension one obtains the sharp bound , with a new, self-contained proof of the asymptotically sharp density in this case. The paper also proves sharpness of the bound via a product construction, showing the decay cannot be improved in general for , and it extends the analysis to -sets with restricted orientations.

Abstract

We prove that a Kakeya set in a vector space over a finite field of size always supports a probability measure whose Fourier transform is bounded by 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 (which is asymptotically sharp). We also establish analogous results for sets containing -planes in a given set of orientations.
Paper Structure (8 sections, 5 theorems, 34 equations)

This paper contains 8 sections, 5 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Kakeya sets in finite fields
  3. Notation
  4. Fourier analysis in finite fields
  5. Main results
  6. Proofs
  7. Proof of Theorem \ref{['maindk']}
  8. Proof of Proposition \ref{['mainex']}

Key Result

Theorem 2.1

Let $d>k \geqslant 1$ be integers and $\Gamma \subseteq G(d,k)$. Suppose $K \subseteq \mathbb{F}_q^d$ is such that for all $V \in \Gamma$, there exists $u \in \mathbb{F}_q^d$ such that $u+V \subseteq K$. Then there exists a probability measure $\mu$ on $K$ satisfying for all non-zero $\xi \in \mathbb{F}_q^d$. In particular, $|K| \gtrsim \min\{q^d, |\Gamma|^2q^{-2k(d-k-1)}\}$ and more precise esti

Theorems & Definitions (5)

  • Theorem 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Corollary 2.5