# Generalized arithmetic progressions contained in Bohr sets

Recall that a generalized arithmetic progression of dimension $d$ is, by definition, a set of the form $P = P_1+\dots+P_d$, where $P_j = \{lp_j\ \mid \ |l|\leq l_j\}\subset \mathbb Z$ is an ordinary arithmetic progression.

A Bohr set of rank $d$ is a set of the form $B = \{n\in \mathbb Z\ \mid \ {\rm dist}(n\alpha_j, \mathbb Z)\leq \varepsilon_j,\ j = 1,\dots, d\}$, where $\alpha_j\in \mathbb R$ and $\varepsilon_j>0$.

The question is the following: given a generalized arithmetic progression $P$ of dimension $d$ is it possible to find a Bohr set $B$ of rank $d-1$ such that $B\supset P$ and ${\rm card} (B\cap [-N,N]) \leq C\cdot {\rm card}(P)$ for some $C>0$ and $N$ such that $P\subset [-N,N]$?

$C$ may depend on $d$, but not on $P$.