Bohr neighborhoods in three-fold difference sets

TL;DR

This paper proves that for sets of integers with positive upper Banach density, the three-fold difference set contains Bohr neighborhoods around many elements, with size depending on the density.

Contribution

It establishes a link between positive density sets and the presence of Bohr neighborhoods in their three-fold difference sets, answering a question by Hegyvári and Ruzsa.

Findings

Three-fold difference sets contain Bohr neighborhoods around many elements.

The size and dimension of these neighborhoods depend only on the set’s density.

The result applies to sets with positive upper Banach density.

Abstract

Answering a question of Hegyv\'ari and Ruzsa , we show that if A is a set of integers having positive upper Banach density, then the set A+A-A:= {a+b-c: a, b, c are in A} contains Bohr neighborhoods of many elements of A, where the radius and dimension of the Bohr neighborhood depend only the upper Banach density of A.