Expanding phenomena over higher dimensional matrix rings

TL;DR

This paper investigates expansion properties in higher dimensional matrix rings, establishing sum-product estimates and demonstrating that certain algebraic operations act as moderate or strong expanders, thus generalizing previous results.

Contribution

It introduces new sum-product estimates and expansion results for matrix rings, extending prior work to higher dimensions and more complex algebraic expressions.

Findings

x+yz and x(y+z) are moderate expanders over matrix rings

xy + z + t is a strong expander over matrix rings

Results generalize recent findings in the field

Abstract

In this paper, we study the expanding phenomena in the setting of higher dimensional matrix rings. More precisely, we obtain a sum-product estimate for large subsets and show that x+yz, x(y+z) are moderate expanders over the matrix ring, and xy + z + t is strong expander over the matrix rings. These results generalize recent results of Y.D. Karabulut, D. Koh, T. Pham, C-Y. Shen, and the second listed author.