An improvement on the number of simplices in $\mathbb{F}_q^d$

TL;DR

This paper improves bounds on the number of simplices determined by Cartesian product sets in finite fields, showing near-complete enumeration under certain size conditions.

Contribution

It extends previous results by providing sharper bounds for Cartesian product sets, approaching optimality in counting congruence classes of simplices.

Findings

Number of simplices approaches total possible in certain conditions

Improved bounds for Cartesian product sets

Results are sharp in some cases

Abstract

Let $\mathcal{E}$ be a set of points in $\mathbb{F}_q^d$. Bennett, Hart, Iosevich, Pakianathan, and Rudnev (2016) proved that if $|\mathcal{E}|\gg q^{d-\frac{d-1}{k+1}}$ then $\mathcal{E}$ determines a positive proportion of all $k$-simplices. In this paper, we give an improvement of this result in the case when $\mathcal{E}$ is the Cartesian product of sets. More precisely, we show that if $\mathcal{E}$ is the Cartesian product of sets and $q^{\frac{kd}{k+1-1/d}}=o(|\mathcal{E}|)$, the number of congruence classes of $k$-simplices determined by $\mathcal{E}$ is at least $(1-o(1))q^{\binom{k+1}{2}}$, and in some cases our result is sharp.