Combinatorial properties of Nil-Bohr sets

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