A new bound for finite field Besicovitch sets in four dimensions

Abstract

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 $|P|\approx |F{|}^4$. A result of Wolff establishes that $|P|\gtrsim |F{|}^3$. In this paper we improve this to $|P| \gtrapprox |F|^{3+\gain}$. On the other hand, we show that the bound of $|F{|}^3$ 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.

arXiv comment28 pages, no figures, to appear, Pacific J. Math. More exposition, less typos