Sumsets of dense sets and sparse sets

Abstract

R. Jin showed that whenever A and B are sets of integers having positive upper Banach density, the sumset A+B:= «a+b: a ∈ A, b ∈ B» is piecewise syndetic. This result was strengthened by Bergelson, Furstenberg, and Weiss to conclude that A+B must be piecewise Bohr. We generalize the latter result to cases where A has Banach density 0, giving a new proof of the previous results in the process.