On Diameters of Cayley Graphs over Matrix Groups

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