Size of local finite field Kakeya sets

TL;DR

This paper investigates the size of local Kakeya sets over finite fields, providing bounds on their minimum size relative to arbitrary subsets of the vector space.

Contribution

It introduces bounds for the minimum size of local Kakeya sets with respect to any subset of finite field vector spaces, extending previous understanding.

Findings

Established upper bounds for local Kakeya set sizes.

Derived lower bounds for the minimum size of such sets.

Applicable to arbitrary subsets of finite field vector spaces.

Abstract

Let $\mathbb{F}$ be a finite field consisting of $q$ elements and let $n \geq 1$ be an integer. In this paper, we study the size of local Kakeya sets with respect to subsets of $\mathbb{F}^{n}$ and obtain upper and lower bounds for the minimum size of a (local) Kakeya set with respect to an arbitrary set ${\mathcal T} \subseteq \mathbb{F}^{n}.$