On the determinants of matrices with elements from arbitrary sets

TL;DR

This paper explores bounds on the number of n×n matrices with elements from arbitrary sets, leveraging additive combinatorics to improve upon trivial estimates of matrices with a fixed determinant.

Contribution

It introduces a novel approach using additive combinatorics to derive stronger bounds for matrices with elements from arbitrary sets, surpassing previous trivial bounds.

Findings

Stronger bounds achieved for matrices with fixed determinants

Utilizes additive combinatorics techniques

Improves upon trivial upper bounds

Abstract

Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial upper bound $O(X^{n^2-1})$, where $X$ is the cardinality of $\mathcal X$. Here we show that even for arbitrary sets $\mathcal X\subseteq \mathbb R$,some recent results from additive combinatorics enable us to obtain a stronger bound with a power saving.