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 (F q ) and |A| ≫ q 7/2 , then we have• If A is a set of SL 2 (F q ) and |A| ≫ q 5/2 , then we haveWe also obtain similar results for the cases of A(B + C) and A + BC, where A, B, C are sets in M 2 (F q ).

An energy decomposition theorem for matrices and related questions

“…We also study some related questions on moderate expanders over matrix rings, namely, for A, B, C ⊆ GL 2 (F q ), we have |AB + C|, |(A + B)C| ≫ q 4 , whenever |A||B||C| ≫ q 10+1/2 . These improve earlier results due to Karabulut, Koh, Pham, Shen, and Vinh (2019).…”

Expanding phenomena over higher dimensional matrix rings

“…In this section, we mimic the study of sum-product digraph over M 2 (F q ) in [3] to extend results over M n (F q ).…”

“…In [3], Y. D. Karabulut, D. Koh, T. Pham, C-Y. Shen, and the second listed author constructed some families of moderate expanders over M 2 (F q ).…”

“…Corollary 1.4. Let f (x, y, z) = x + yz be a function from SL n (F q ) 3 to M n (F q ). For A ⊂ |f (A, A, A)| ≫ q n 2 .…”

“…In [3], Karabulut, Koh, Pham, Shen, and the second listed author obtained the following sum-product estimate over the matrix ring M 2 (F q ).…”