On Diameters of Cayley Graphs over Matrix Groups

We establish for the matrix group $G=\mathrm{SL}_{n}\left(\mathbb{F}_{p}\right)$ that there exist absolute constants $c\in\left(0,1\right)$ and $C>0$ such that any symmetric generating set $A$, with $\left|A\right|\geq\left|G\right|^{1-c}$ has a covering number $\leq Cn^{2}.$ This result is sharp up to the value of the constant $C>0$.