Beiglboeck, Bergelson and Fish proved that if subsets A,B of a countablediscrete 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 thatthe union of k-many left translates of AB is thick. Using nonstandard analysiswe give a shorter alternative proof of this result that does not require G tobe countable, and moreover yields the explicit bound that k is not greater than1/ab. We also prove with similar methods that if ${A_i}$ are finitely manysubsets of G having positive Banach densities $a_i$ and G is countable, thenthere exists a subset B whose Banach density is at least the product of thedensities $a_i$ and such that the product $BB^{-1}$ is a subset of theintersection of the product sets $A_i A_i^{-1}$. In particular, the latter setis piecewise Bohr.
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