On directions determined by subsets of vector spaces over finite fields

2010/10/05 by Iosevich, Alex, Morgan, Hannah, Pakianathan, Jonathan · 1 citation
#42B05 #52C10 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1010.0749

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.

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