# The Kakeya problem in finite fields

The kakeya problem, in its best known formulation, is the following. Let $E\subset\mathbf{R}^n$ be set which contains a translate of every unit segment; equivalently, for every direction $e\in S^{n-1}$, $E$ contains a unit line segment parallel to $e$. An $n$-dimensional ball of radius $1/2$ is a simple example of  a set with this property, but there are many other such sets, some of which have $n$-dimensional measure $0$.

Can $E$ be even smaller than that and have Hausdorff dimension strictly smaller that $n$?