The deep theory of approximate subgroups was developed by Helfgott, Hrushovski, Pyber and Szab{\'o}, and by Breuillard, Green and Tao. It establishes 3-step growth in various situations, most notably in finite simple groups of Lie type of bounded rank. Gowers' theory of quasi-random groups, further developed and applied by Nikolov, Pyber and Babai, provides 3-step covering results. Can we replace 3 by 2?

In recent joint works with Larsen and Tiep, we study similar problems for two normal subsets of finite simple groups, with applications to word maps and to permutation groups. In particular we show that every element of a sufficiently large finite simple transitive permutation group is a product of two fixed-point-free permutations.

We also replace subsets of a group by its representations, and subset products by tensor products of representations, obtaining stronger growth results, such as 2-step growth for representations of finite simple groups of Lie type, including those of unbounded rank.