On the volume set of point sets in vector spaces over finite fields

TL;DR

This paper proves that large enough subsets of finite field vector spaces determine all possible volumes of parallelepipeds, with bounds that are nearly optimal, advancing understanding of geometric configurations over finite fields.

Contribution

It establishes a near-optimal threshold for the size of point sets in finite field vector spaces to ensure all volumes are realized, extending finite field geometric combinatorics.

Findings

Sets of size at least (d-1)q^{d-1} determine all volumes

The bound is sharp up to a factor of (d-1)

Volume set covers the entire finite field

Abstract

We show that if $\mathcal{E}$ is a subset of the $d$-dimensional vector space over a finite field $\mathbbm{F}_q$ ($d \geq 3$) of cardinality $|\mathcal{E}| \geq (d-1)q^{d - 1}$, then the set of volumes of $d$-dimensional parallelepipeds determined by $\mathcal{E}$ covers $\mathbbm{F}_q$. This bound is sharp up to a factor of $(d-1)$ as taking $\mathcal{E}$ to be a $(d - 1)$-hyperplane through the origin shows.