An energy decomposition theorem for matrices and related questions

Given A⊆GL2(Fq), we prove that there exist disjoint subsets B,C⊆A such that A=B⊔C and their additive and multiplicative energies satisfying max{E+(B),E×(C)}≪|A|3/M(|A|), where M(|A|)=min{q4/3/|A|1/3(log|A|)2/3,|A|4/5/q13/5(log|A|)27/10}. We also study some related questions on moderate expanders over matrix rings, namely, for A,B,C⊆GL2(Fq), we have |AB+C|, |(A+B)C|≫q4, whenever |A||B||C|≫q10+1/2. These improve earlier results due to Karabulut, Koh, Pham, Shen, and Vinh ([2019], Expanding phenomena over matrix rings, ForumMath., 31, 951–970).