On rapid generation of \(SL_2(\mathbb{Z}_q)\)

We prove that if \( A \subset \mathbb{Z}_q \setminus \{0\} \), \( A \neq \langle p \rangle \), \( q = p^\ell \), \( \ell \geq 2 \) with \( |A| > C \sqrt[3]{\sqrt{\ell}^2 q^{(1-\frac{1}{4\ell})}} \), then
\[
|P(A) \cdot P(A)| \geq C’ q^3
\]
where
\[
P(A) = \left\{ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \in SL_2(\mathbb{Z}_q) : a_{11} \in A \cap \mathbb{Z}_q^\times, a_{12}, a_{21} \in A \right\}.
\]

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 \( |\mathcal{R}(t)| \leq \ell |A|^2 q^{(1-\frac{1}{2\ell})} \).