Some spectral results on Kakeya sets

The finite field Kakeya problem asks both the minimum size of a point set inAG(2, q)which contains a line in every direction, as well as a characterization of the examples. Blokhuis and Mazzocca [2] solved this problem, and a subsequent paper [1] addresses the stability of this solution for even order planes, i.e. the spectrum of sizes near the minimum size of a Kakeya set for which non-minimum Kakeya sets exist. In this paper we provide some computational results in small order planes to determine the full spectrum of sizes of Kakeya sets. We then address some spectrum issues on the upper end of possible sizes, providing some bounds and new constructions.We also address the question of minimality, i.e.whether a given Kakeya set contains any smaller Kakeya set.

On the Size of Kakeya Sets in Finite Vector Spaces

For a finite field $\mathbb{F}_q$, a Kakeya set $K$ is a subset of $\mathbb{F}_q^n$ that contains a line in every direction. This paper derives new upper bounds on the minimum size of Kakeya sets when $q$ is even.