Expanding phenomena over matrix rings

Authors: Doowon Koh, Phạm Văn Thắng, Chun-Yen Shen, Lê Anh Vinh, Yesim Demiroglu Karabulut,

https://doi.org/10.1515/forum-2019-0032.

Publisher, magazine: ,

Publication year: 2019

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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 M2(Fq) and |A|≫q7/2, then |A(A+A)|,|A+AA|≫q4, •if A is a set of SL2(Fq) and |A|≫q5/2, then |A(A+A)|,|A+AA|≫q4. We also obtain similar results for the cases of A(B+C) and A+BC, where A,B,C are sets in M2(Fq).

Tags: Matrix rings; expander; sum-product; finite fields