Some sum-product estimates in matrix rings over finite fields

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 $|A|\ge C(n)q^{n^2-1}$ 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.

18 pages

References in corpus (2)

• On higher energy decompositions and the sum-product phenomenon
• Expanding phenomena over higher dimensional matrix rings