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 M n (F q ).These results generalize recent results of Y.

Some sum-product estimates in matrix rings over finite fields

“…For every positive integer n, there exists C(n) such that the following holds. 13]). For A, B ⊆ M n (F q ) and C ⊆ GL n (F q ), we have…”