LAUSR

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 ⁢ ( ???? q ) {M_{2}(\mathbb{F}_{q})} and | A | ≫ q 7 / 2 {\lvert A\rvert\gg q^{7/2}} , then | A ⁢ ( A + A ) | , | A + A ⁢ A | ≫ q 4 {\lvert A(A+A)\rvert,\lvert A+AA\rvert\gg q^{4}} , • if A is a set of SL 2 ⁢ ( ???? q ) {\mathrm{SL}_{2}(\mathbb{F}_{q})} and | A | ≫ q 5 / 2 {\lvert A\rvert\gg q^{5/2}} , then | A ⁢ ( A + A ) | , | A + A ⁢ A | ≫ q 4 {\lvert A(A+A)\rvert,\lvert A+AA\rvert\gg q^{4}} . We also obtain similar results for the cases of A ⁢ ( B + C ) {A(B+C)} and A + B ⁢ C {A+BC} , where A , B , C {A,B,C} are sets in M 2 ⁢ ( ???? q ) {M_{2}(\mathbb{F}_{q})} .