A new bound for finite field Besicovitch sets in four dimensions

Let $F$ be a finite field with characteristic greater than two. Define a \emph{Besicovitch set} in $F^4$ to be a set $P\subseteq F^4$ containing a line in every direction. The \emph{Kakeya conjecture} asserts that $|P| \approx |F|^4$\mathrm{.}AresultofWolffestablishesthat$|P| \gtrsim |F|^3$. In this paper we improve this to $\mathrm{\mid }P\mathrm{\mid }⪆\mathrm{\mid }F{\mathrm{\mid }}^{3+\text{gain}}$. On the other hand, we show that the bound of $\mathrm{\mid }F{\mathrm{\mid }}^3$ is sharp if we relax the assumption that the lines point in different directions. One new feature in the argument is the introduction of a small amount of basic algebraic geometry.