  We establish in this paper a new form of Pl\"unnecke-type inequalities for
ergodic probability measure-preserving actions of any countable abelian group.
Using a correspondence principle for product sets, this allows us to deduce
lower bounds on the upper and lower Banach densities of any product set in
terms of the upper Banach density of an iterated product set of one of its
addends. These bounds are new already in the case of the integers. We also
introduce the notion of an ergodic basis, which is parallel, but significantly
weaker than the analogous notion of an additive basis, and deduce Pl\"unnecke
bounds on their impact functions with respect to both the upper and lower
Banach densities on any countable abelian group.
