On directions determined by subsets of vector spaces over finite fields

TL;DR

This paper establishes thresholds for subsets of finite field vector spaces to determine all directions or a k-dimensional set of directions, with stronger results for random sets, extending geometric combinatorics in finite fields.

Contribution

It proves optimal bounds for the number of directions determined by large subsets of finite field vector spaces, generalizing previous Euclidean results to finite fields.

Findings

Subsets larger than q^{d-1} determine all directions.

Subsets larger than q^k determine a k-dimensional set of directions.

Results are optimal, as exemplified by hyperplanes.

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 ${\Bbb R}^d$, this question has been previously studied by Pach, Pinchasi and Sharir.