Sumsets of dense sets and sparse sets

TL;DR

This paper extends the understanding of sumsets in additive number theory by proving that the sum of a set with Banach density zero and a dense set is piecewise Bohr, generalizing previous results for dense sets.

Contribution

It introduces a new proof that sumsets involving a set with Banach density zero can still be piecewise Bohr, broadening the scope of earlier theorems.

Findings

Sumsets of dense and sparse sets can be piecewise Bohr.

Generalization of previous results to sets with Banach density zero.

Abstract

R. Jin showed that whenever A and B are sets of integers having positive upper Banach density, the sumset A+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.