Expanding phenomena over matrix rings

TL;DR

This paper investigates expanding phenomena in matrix rings over finite fields, establishing size conditions under which certain algebraic sum-product sets grow significantly, thus extending additive combinatorics to matrix settings.

Contribution

It proves new expansion results for sets in matrix rings and special linear groups over finite fields, extending sum-product phenomena to these algebraic structures.

Findings

Sets of size > q^{7/2} in M_2(F_q) expand under A(A+A) and A+AA to size > q^4.

Sets of size > q^{5/2} in SL_2(F_q) expand similarly, achieving size > q^4.

Results also apply to sets involving B+C and BC in 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 $M_2(\mathbb{F}_q)$.