An improved lower bound on the size of Kakeya sets over finite fields

Abstract

In a recent breakthrough, Dvir showed that every Kakeya set in ${\mathbb{F}}^n$ must have cardinality at least $c_n|\mathbb{F}{|}^n$, where $c_n\approx 1∕n!$. We improve this lower bound to ${\beta }^n|\mathbb{F}{|}^n$ for a constant $\beta >0$. This pins down the correct growth of the constant $c_n$ as a function of $n$ (up to the determination of $\beta$).