On directions determined by subsets of vector spaces over finite fields

Abstract

We prove that if a subset of a $d$-dimensional vector space over a finite field with $q$ elements has more than $q^{d-1}$ elements, then it determines all the possible directions. If a set has more than $q^k$ elements, it determines a $k$-dimensional set of directions. We prove stronger results for sets that are sufficiently random. This result is best possible as the example of a $k$-dimensional hyperplane shows. We can view this question as an Erd\H os type problem where a sufficiently large subset of a vector space determines a large number of configurations of a given type. For discrete subsets of ${\mathbb{R}}^d$, this question has been previously studied by Pach, Pinchasi and Sharir.