  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.
