A bound for plany Kakeya sets in $\mathbb{F}_q^4$ using the planebrush method

TL;DR

This paper discusses a new lower bound for plany Kakeya sets in four-dimensional finite fields using the planebrush method, extending prior work on Hausdorff dimension bounds.

Contribution

It provides a nontechnical exposition of the planebrush argument for plany Kakeya sets in finite fields, improving the lower bound on their size.

Findings

Establishes a new lower bound for plany Kakeya sets in $ extbf{F}_q^4$

Adapts the planebrush method to finite field setting

Improves understanding of geometric structure of Kakeya sets

Abstract

Katz and Zahl used a planebrush argument to prove that Kakeya sets in $\mathbb{R}^4$ have Hausdorff dimension at least 3.059. In the special case when the Kakeya set is plany, their argument gives a better lower bound of 10/3. We give a nontechnical exposition of the Katz-Zahl argument for plany Kakeya sets in the finite field setting.