  We prove that if $G$ is a finite simple group of Lie type and $S_1,\dots,
S_k$ are subsets of $G$ satisfying $\prod_{i=1}^k|S_i|\geq|G|^c$ for some $c$
depending only on the rank of $G$, then there exist elements $g_1,\dots, g_k$
such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$. This theorem generalizes an
earlier theorem of the authors and Short. We also propose two conjectures that
relate our result to one of Rodgers and Saxl pertaining to conjugacy classes in
$SL_n(q)$, as well as to the Product Decomposition Conjecture of Liebeck,
Nikolov and Shalev.
