Products of normal subsets and derangements

In recent years there has been significant progress in the study of products of subsets of finite groups and of finite simple groups in particular. In this paper we consider which families of finite simple groups $G$ have the property that for each $\epsilon > 0$ there exists $N > 0$ such that, if $|G| \ge N$ and $S, T$ are normal subsets of $G$ with at least $\epsilon |G|$ elements each, then every non-trivial element of $G$ is the product of an element of $S$ and an element of $T$. We show that this holds in a strong sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form ${\mathrm{PSL}}_n(q)$ where $q$ is fixed and $n$ tends to infinity. Our second main result is that any element in a transitive permutation representation of a sufficiently large finite simple group is a product of two derangements.