Logarithmic bounds for the diameters of some Cayley graphs

Let $S\subset {\text{GL}}_n(\mathbb{Z})$ be a finite symmetric set. We show that if the Zariski closure of $\mathrm{\Gamma }=⟨S⟩$ is a product of ${\text{SL}}_d$ or a special affine linear group, then the diameter of the Cayley graph $\text{Cay}(\mathrm{\Gamma }\mathrm{/}\mathrm{\Gamma }(q),{\pi }_q(S))$ is $O(\log q)$, where $q$ is an arbitrary positive integer, ${\pi }_q:\mathrm{\Gamma }\to \mathrm{\Gamma }\mathrm{/}\mathrm{\Gamma }(q)$ is the canonical projection induced by the reduction modulo $q$, and the implied constant depends only on $S$.