Bohr sets in triple products of large sets in amenable groups

TL;DR

This paper extends the understanding of Bohr-regularity in triple sums of large sets, providing effective estimates in integers and general discrete amenable groups, even with sparse densities.

Contribution

It offers new effective estimates for Bohr-regularity of triple sums in integers and amenable groups, relaxing density conditions on the sets involved.

Findings

Effective bounds for Bohr-regularity in integers.

Extension of results to arbitrary discrete amenable groups.

Relaxation of density requirements on one of the sets.

Abstract

We answer a question of Hegyv\'ari and Ruzsa concerning effective estimates of the Bohr-regularity of certain triple sums of sets with positive upper Banach densities in the integers. Our proof also works for any discrete amenable group, and it does not require all addends in the triple products we consider to have positive (left) upper Banach densities; one of the addends is allowed to only have positive upper asymptotic density with respect to a (possibly very sparse) ergodic sequence.