A sum-product theorem in matrix rings over finite fields

In this note, we study a sum-product estimate over matrix rings M-n(F-q). More precisely, for A subset of M-n(F-q), we have if vertical bar A boolean AND GL(n)(F-q)vertical bar <= vertical bar A vertical bar/2, then max{vertical bar A + A vertical bar, vertical bar AA vertical bar} >> min {vertical bar A vertical bar q, vertical bar A vertical bar(3)/q(2n2-2n)}; if vertical bar A boolean AND GL(n)(F-q)vertical bar >= vertical bar A vertical bar/2, then max{vertical bar A + A vertical bar, vertical bar AA vertical bar} >> min {vertical bar A vertical bar(2/3)q(n2/3), vertical bar A vertical bar(3/2)/q(n2/2-1/4)}. We also will provide a lower bound of vertical bar A + B vertical bar for A subset of SLn(F-q) and B subset of M-n(F-q). (C) 2019 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.