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 $SL_2(\mathbb{F}_q)$ 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 $M_2(\mathbb{F}_q)$.