NONSTANDARD ANALYSIS AND THE SUMSET PHENOMENON IN ARBITRARY AMENABLE GROUPS

摘要

Beiglboeck, Bergelson and Fish proved that if subsets A, B of a countable discrete amenable group G have positive Banach densities α and β respectively, then the product set AB is piecewise syndetic, that is, 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 k ≤ 1/αβ. We also prove with similar methods that if {A_i}_i~n=1 are finitely many subsets of G having positive Banach densities α_i and G is countable, then there exists a subset B whose Banach density is at least Π+(i=1)~n α_1 and such that BB~(-1) (∈) ∩_(i=1)~n A_iA_i~(-1). In particular, the latter set is piecewise Bohr.