Finite field Kakeya and Nikodym sets in three dimensions

TL;DR

This paper improves lower bounds on the size of Kakeya and Nikodym sets in three-dimensional finite fields and proposes a conjecture linking line configurations to bounds on Nikodym sets.

Contribution

It provides new lower bounds for Kakeya and Nikodym sets in three dimensions and introduces a conjecture that implies optimal bounds for Nikodym sets.

Findings

Enhanced lower bounds for Kakeya sets in 3D finite fields

Enhanced lower bounds for Nikodym sets in 3D finite fields

Proposed a conjecture connecting line unions to Nikodym set bounds

Abstract

We give improved lower bounds on the size of Kakeya and Nikodym sets over $\mathbb{F}_q^3$. We also propose a natural conjecture on the minimum number of points in the union of a not-too-flat set of lines in $\mathbb{F}_q^3$, and show that this conjecture implies an optimal bound on the size of a Nikodym set.