An energy decomposition theorem for matrices and related questions

Abstract

AbstractGiven $A\subseteq GL_2(\mathbb {F}_q)$ , we prove that there exist disjoint subsets $B, C\subseteq A$ such that $A = B \sqcup C$ and their additive and multiplicative energies satisfying $$\begin{align*}\max\{\,E_{+}(B),\, E_{\times}(C)\,\}\ll \frac{|A|^3}{M(|A|)}, \end{align*}$$ where $$ \begin{align*} M(|A|) = \min\Bigg\{\,\frac{q^{4/3}}{|A|^{1/3}(\log|A|)^{2/3}},\, \frac{|A|^{4/5}}{q^{13/5}(\log|A|)^{27/10}}\,\Bigg\}. \end{align*} $$ 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 $$\begin{align*}|AB+C|, ~|(A+B)C|\gg q^4,\end{align*}$$ whenever $|A||B||C|\gg q^{10 + 1/2}$ . These improve earlier results due to Karabulut, Koh, Pham, Shen, and Vinh ([2019], Expanding phenomena over matrix rings, $Forum Math.$ , 31, 951–970).