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

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.