Nil-Bohr Sets of Integers

TL;DR

This paper introduces higher order Bohr sets of integers, exploring their properties and connections to additive structure and Fourier analysis, advancing understanding in additive combinatorics and ergodic theory.

Contribution

It generalizes classical Bohr sets to higher order versions, linking abelian and non-abelian Fourier analysis in additive combinatorics.

Findings

Introduction of higher order Bohr sets

Generalization of properties of classical Bohr sets

Connections to Fourier analysis and additive structure

Abstract

We study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr sets are fundamentally abelian in nature and are linked to Fourier analysis. Recently it has become apparent that a higher order, non-abelian, Fourier analysis plays a role in both additive combinatorics and in ergodic theory. Here we introduce a higher order version of Bohr sets and give various properties of these objects, generalizing results of Bergelson, Furstenberg, and Weiss.