Some sum-product estimates in matrix rings over finite fields

TL;DR

This paper investigates sum-product phenomena in matrix rings over finite fields, establishing new bounds and generalizations that improve previous results using spectral graph theory and linear algebra techniques.

Contribution

The paper introduces improved sum-product estimates in matrix rings over finite fields and generalizes prior results, providing new proofs with spectral graph theory methods.

Findings

Established lower bounds for sum sets involving matrix products.

Proved that large sets in matrix rings have either large sum or product sets.

Generalized previous sum-product estimates to higher-dimensional matrix rings.

Abstract

We study some sum-product problems over matrix rings. Firstly, for $A, B, C\subseteq M_n(\mathbb{F}_q)$, we have $$ |A+BC|\gtrsim q^{n^2}, $$ whenever $|A||B||C|\gtrsim q^{3n^2-\frac{n+1}{2}}$. Secondly, if a set $A$ in $M_n(\mathbb{F}_q)$ satisfies $|A|\geq C(n)q^{n^2-1}$ for some sufficiently large $C(n)$, then we have $$ \max\{|A+A|, |AA|\}\gtrsim \min\left\{\frac{|A|^2}{q^{n^2-\frac{n+1}{4}}}, q^{n^2/3}|A|^{2/3}\right\}. $$ These improve the results due to The and Vinh (2020), and generalize the results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a recent result due to The and Vinh (2020). Our method is based on spectral graph theory and linear algebra.