Combinatorial properties of Nil-Bohr sets

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.