A sum–product theorem in matrix rings over finite fields

Abstract

In this note, we study a sum–product estimate over matrix rings Mn(Fq). More precisely, for A⊂Mn(Fq), we have•if |A∩GLn(Fq)|≤|A|/2, thenmax⁡{|A+A|,|AA|}≫min⁡{|A|q,|A|3q2n2−2n};•if |A∩GLn(Fq)|≥|A|/2, thenmax⁡{|A+A|,|AA|}≫min⁡{|A|23qn23,|A|3/2qn22−14}. We also will provide a lower bound of |A+B| for A⊂SLn(Fq) and B⊂Mn(Fq).