A new bound for finite field Besicovitch sets in four dimensions
Abstract
Description
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.
28 pages, no figures, to appear, Pacific J. Math. More exposition, less typos
28 pages, no figures, to appear, Pacific J. Math. More exposition, less typos