On distinct perpendicular bisectors and pinned distances in finite fields

TL;DR

This paper investigates geometric properties of point sets in finite fields, establishing lower bounds on the number of distinct perpendicular bisectors and distances for large point sets, improving previous results.

Contribution

It proves new lower bounds on the number of perpendicular bisectors and distances determined by large point sets in finite fields, advancing understanding of finite field geometry.

Findings

Sets with at least q^{3/2} points determine Ω(q^2) perpendicular bisectors.

For sets with at least q^{4/3} points, a positive proportion of points have Ω(q) distinct distances.

Improves previous bounds on distances in finite field point sets.

Abstract

Given a set of points $P \subset \mathbb F_q^2$ such that $|P|\geq q^{3/2}$ it is established that $|P|$ determines $\Omega(q^2)$ distinct perpendicular bisectors. It is also proven that, if $|P| \geq q^{4/3}$, then for a positive proportion of points $a \in P$, we have $$|\{\| a- b\|: b \in P\}|=\Omega(q),$$ where $\|a- b\|$ is the distance between points $a$ and $b$. The latter result represents an improvement on a result of Chapman et al. (arxiv:0903.4218).