Expanding phenomena over higher dimensional matrix rings

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(y+z), x+yz, xy+z+t are moderate expanders over the matrix ring Mn(Fq).These results generalize recent results of Y.D. Karabulut, D. Koh, T. Pham, C-Y. Shen, and the second listed author.