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 clog|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.