Kakeya sets in $\mathbb{F}_p^d$

A Kakeya set in the finite vector space $\mathbb{F}_p^d$ is a set containing a full line in every direction.

Precisely, a finite subset $K \subseteq \mathbb{F}_p^d$ is a Kakeya set when for every nonzero direction $v \in \mathbb{F}_p^d$ there exists a base point $x \in \mathbb{F}_p^d$ such that the entire line through $x$ with direction $v$,

is contained in $K$.

The base point is allowed to depend on the direction, which is what makes the notion nontrivial: a Kakeya set must accommodate $p^{d-1}$-ish many directions at once while staying small. The central question is how small such a set can be. Dvir's polynomial method shows $|K| \ge c_d q^d$, so Kakeya sets occupy a positive proportion of the space; determining the optimal constant is open even for $d = 3$.

The direction $v$ ranges over nonzero vectors rather than over projective classes. This is harmless for the notion itself, since $v$ and $\lambda v$ for $\lambda \ne 0$ trace out the same set of lines, so requiring a line for every nonzero $v$ is equivalent to requiring one for every projective direction.