Occurrence of Right Angles in Vector Spaces Over Finite Fields

TL;DR

This paper investigates the occurrence of right angles in subsets of vector spaces over finite fields, establishing size thresholds for the existence of such angles and proving the sharpness of these bounds.

Contribution

It introduces new size bounds for subsets of finite field vector spaces that guarantee the presence of right angles, extending Erdos-Falconer problems.

Findings

Subsets larger than q^[(d+2)/3] contain three points forming a right angle.

Subsets larger than q^[(d+2)/2] contain two points with a right angle at the origin.

The bound for the second case is proven to be sharp up to constants.

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.