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We prove that if G is a finite simple group of Lie type and S 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.doi:10.4171/ggd/208 fatcat:eugjcwk66naovnrflqjzpqeyvy