An optimal bound on the number of determinants generated by a subset in $\mathbb{F}_q^d$

TL;DR

This paper establishes a near-optimal bound on the size of a subset in a finite field vector space that guarantees the set of determinants generated covers the entire field, extending previous results.

Contribution

It proves a nearly optimal size condition for subsets in finite fields to generate all possible determinants, generalizing earlier work.

Findings

Set of determinants equals the entire field for large enough subsets

Bound is nearly optimal in the context of finite field geometry

Generalizes previous determinant set results

Abstract

In this short note, we prove that for $\mathcal{E} \subset \mathbb{F}_q^d$ with $|\mathcal{E}| \geq q^{d-1} + O(q^2)$ then the set of determinants generated by $\mathcal{E}$ is $\mathbb{F}_q.$ This result is nearly optimal and generalizes the previous results of Vinh (2013) and Iosevich, Rudnev and Zhai (2015).