Distinct Triangle Areas in a Planar Point Set over Finite Fields

Abstract

Let $\mathcal{P}$ be a set of $n$ points in the finite plane $\mathbb{F}_q^2$ over the finite field $\mathbb{F}_q$ of $q$ elements, where $q$ is an odd prime power. For any $s \in \mathbb{F}_q$, denote by $A (\mathcal{P}; s)$ the number of ordered triangles whose vertices in $\mathcal{P}$ having area $s$. We show that if the cardinality of $\mathcal{P}$ is large enough then $A (\mathcal{P}; s)$ is close to the expected number $|\mathcal{P}|^3/q$.