Sharp density bounds on the finite field Kakeya problem

TL;DR

This paper establishes a lower bound on the density of Kakeya sets in finite fields, matching known constructions, and advancing understanding of their minimal size in finite field geometry.

Contribution

It provides a sharp density lower bound for Kakeya sets in finite fields, confirming the optimality of existing constructions.

Findings

Kakeya sets in finite fields have density at least 1/2^{n-1}

The bound matches the known explicit constructions

Advances understanding of geometric configurations in finite fields

Abstract

A Kakeya set in $\mathbb{F}_q^n$ is a set containing a line in every direction. We show that every Kakeya set in $\mathbb{F}_q^n$ has density at least $1/2^{n-1}$, matching the construction by Dvir, Kopparty, Saraf and Sudan.