Expanding phenomena over matrix rings

Abstract

In this paper, we study expanding phenomena in the setting of matrix rings. More precisely, we will prove that If $A$ is a set of $M_2({\mathbb{F}}_q)$ and $|A|\gg q^{7/2}$, then we have \[|A(A+A)|, ~|A+AA|\gg q^4.\] If $A$ is a set of $S{L}_2({\mathbb{F}}_q)$ and $|A|\gg q^{5/2}$, then we have \[|A(A+A)|, ~|A+AA|\gg q^4.\] We also obtain similar results for the cases of $A(B+C)$ and $A+BC$, where $A,B,C$ are sets in $M_2({\mathbb{F}}_q)$.

31 pages