Occurrence of Right Angles in Vector Spaces Over Finite Fields

Abstract

Here we examine some Erdos-Falconer-type problems in vector spaces over finite fields involving right angles. Our main goals are to show that a) a subset A of F_q^d of size >> q^[(d+2)/3] contains three points which generate a right angle, and b) a subset A of F_q^d of size >> q^[(d+2)/2] contains two points which generate a right angle with the vertex at the origin. We will also prove that b) is sharp up to constants and provide some partial results for similar problems related to spread and collinear triples.