An energy decomposition theorem for matrices and related questions

Given $A\subseteq GL_2(\mathbb{F}_q)$, we prove that there exist disjoint subsets $B, C\subseteq A$ such that $A = B \sqcup C$ and their additive and multiplicative energies satisfying $$\max\{\,E_{+}(B),\, E_{\times}(C)\,\}\ll \frac{|A|^3}{M(|A|)},$$ where \begin{equation*} \label{eqn:MAminBVPolyLSSS} M(|A|) = \min\Bigg\{\,\frac{q^{4/3}}{|A|^{1/3}(\log|A|)^{2/3}},\, \frac{|A|^{4/5}}{q^{13/5}(\log|A|)^{27/10}}\,\Bigg\}. \end{equation*} We also study some related questions on moderate expanders over matrix rings, namely, for $A, B, C\subseteq GL_2(\mathbb{F}_q)$, we have $$|AB+C|, ~|(A+B)C|\gg q^4,$$ whenever $|A||B||C|\gg q^{10 + 1/2}$. These improve earlier results due to Karabulut, Koh, Pham, Shen, and Vinh (2019).