  Beiglboeck, Bergelson and Fish proved that if subsets A,B of a countable
discrete amenable group G have positive Banach densities a and b respectively,
then the product set AB is piecewise syndetic, i.e. there exists k such that
the union of k-many left translates of AB is thick. Using nonstandard analysis
we give a shorter alternative proof of this result that does not require G to
be countable, and moreover yields the explicit bound that k is not greater than
1/ab. We also prove with similar methods that if $\{A_i\}$ are finitely many
subsets of G having positive Banach densities $a_i$ and G is countable, then
there exists a subset B whose Banach density is at least the product of the
densities $a_i$ and such that the product $BB^{-1}$ is a subset of the
intersection of the product sets $A_i A_i^{-1}$. In particular, the latter set
is piecewise Bohr.
