Nonstandard analysis and the sumset phenomenon in arbitrary amenable groups

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 B(B^−1) is a subset of the intersection of the product sets A_i(A_i^−1). In particular, the latter set is piecewise Bohr.