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.
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Griesmer, J.T. Sumsets of dense sets and sparse sets. Isr. J. Math. 190, 229–252 (2012). https://doi.org/10.1007/s11856-012-0008-1
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DOI: https://doi.org/10.1007/s11856-012-0008-1