Definition 2.25 (Bohr set). Let $\mathrm{\Gamma }\subseteq \widehat{G}$ and $\rho >0$. By the Bohr set $B(\mathrm{\Gamma },\rho )$ we mean the set

$$B(\mathrm{\Gamma },\rho )=\{x\in G:|\gamma (x)-1|<\rho \forall \gamma \in \mathrm{\Gamma }\}.$$

We call $|\mathrm{\Gamma }|$ the rank of $B(\mathrm{\Gamma },\rho )$, and $\rho$ its width or radius.