Congruence classes of triangles in $\mathbb{F}_p^2$

TL;DR

This paper establishes a lower bound on the number of congruence classes of triangles determined by small subsets of points in the finite plane over a prime field, advancing combinatorial geometry in finite fields.

Contribution

It provides a new lower bound on the number of triangle congruence classes for small point sets in finite fields, specifically when the set size is up to p^{2/3}.

Findings

Lower bound of |A|^{7/2} for triangle classes

Applicable for sets with size up to p^{2/3}

Advances understanding of geometric configurations in finite fields

Abstract

In this short note, we give a lower bound on the number of congruence classes of triangles in a small set of points in $\mathbb{F}_p^2$. More precisely, for $\mathcal{A}\subset \mathbb{F}_p^2$ with $|\mathcal{A}|\le p^{2/3}$, we prove that the number of congruence classes of triangles determined by points in $\mathcal{A}\times \mathcal{A}$ is at least $|\mathcal{A}|^{7/2}$. This note is not intended for journal publication.