The Kakeya problem: a gap in the spectrum and classification of the smallest examples

Abstract

Kakeya sets in the affine plane \(\mathrm AG (2,q)\) are point sets that are the union of lines, one through every point on the line at infinity. The finite field Kakeya problem asks for the size of the smallest Kakeya sets and the classification of these Kakeya sets. In this article we present a new example of a small Kakeya set and we give the classification of the smallest Kakeya sets up to weight \(\frac{q(q+2)}{2}+\frac{q}{4}\), both in case \(q\) even.