Combinatorial properties of Nil-Bohr sets

TL;DR

This paper explores the relationship between two notions of largeness in sets of positive integers, showing that Nil-Bohr sets are contained within SG_k sets with bounds depending on the order d, thus partially confirming a conjecture.

Contribution

It establishes a bound linking Nil-Bohr sets and SG_k sets, advancing understanding of their combinatorial properties and resolving part of a conjecture by Host and Kra.

Findings

Nil_d-Bohr sets are contained in SG_k sets with k bounded by d

Partially resolves a conjecture of Host and Kra

Provides bounds relating different notions of largeness

Abstract

In this paper we study the relation between two notions of largeness that apply to a set of positive integers, namely $\mathrm{Nil}_d{-}\mathrm{Bohr}$ and $\mathrm{SG}_k$, as introduced by Host and Kra. We prove that any $\mathrm{Nil}_d{-}\mathrm{Bohr}_0$ set is necessarily $\mathrm{SG}_k$ where ${k}$ is effectively bounded in terms of $d$. This partially resolves a conjecture of Host and Kra.