  We consider the question of when sets definable in first-order expansions of
groups contain the product of two infinite sets (we refer to this as the
"productset property"). We first show that the productset property holds for
any definable subset $A$ of an expansion of a discrete amenable group such that
$A$ has positive Banach density and the formula $x\cdot y\in A$ is stable. For
arbitrary expansions of groups, we consider a "$1$-sided" version of the
productset property, which is characterized in various ways using coheir
independence. For stable groups, the productset property is equivalent to this
$1$-sided version, and behaves as a notion of largeness for definable sets,
which can be characterized by a natural weakening of model-theoretic
genericity. Finally, we use recent work on regularity lemmas in distal theories
to prove a definable version of the productset property for sets of positive
Banach density definable in certain distal expansions of amenable groups.
