On rapid generation of $S{L}_2({\mathbb{Z}}_q)$

Abstract

We prove that if $A\subset {\mathbb{Z}}_q\setminus \{0\}$, $A\ne ⟨p⟩$, $q=p^{\ell }$, $\ell \ge 2$ with $|A|>C\sqrt[3]{{\sqrt{\ell }}^2{q}^{(1-\frac{1}{4\ell })}}$, then
$$|P(A)\cdot P(A)|\ge C^{\prime }{q}^3$$
where
$$P(A)=\{\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\in S{L}_2({\mathbb{Z}}_q):a_{11}\in A\cap {\mathbb{Z}}_q^{\times },a_{12},a_{21}\in A\}.$$

The proof relies on a result in $[4]$ previously established by D. Covert, A. Iosevich, and J. Pakianathan, which implies that if $|A|$ is much larger than $\sqrt{\ell }{q}^{(1-\frac{1}{4\ell })}$, then
$$|\{(a_{11},a_{12},a_{21},a_{22})\in A\times A\times A\times A:a_{11}{a}_{22}+a_{12}{a}_{21}=t\}|=|A{|}^4{q}^{-1}+\mathcal{R}(t)$$
where $|\mathcal{R}(t)|\le \ell |A{|}^2{q}^{(1-\frac{1}{2\ell })}$.