# 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.

## Key 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.