On the product decomposition conjecture for finite simple groups

We prove that if $G$ is a finite simple group of Lie type and $S$ is a subset of $G$ of size at least two, then $G$ is a product of at most $c\log|G|/\log|S|$ conjugates of $S$, where $c$ depends only on the Lie rank of $G$. This confirms a conjecture of Liebeck, Nikolov and Shalev in the case of families of simple groups of bounded rank. We also obtain various related results about products of conjugates of a set within a group.