Bohr set ($B(\Gamma ,\rho )$)

For a finite abelian group $G$, frequencies $\Gamma \subseteq \hat{G}$, and $\rho >0$, the Bohr set isIts rank is $|\Gamma |$ and its width is $\rho$.

A Bohr set of rank $d$ and width $\rho$ satisfiesThe proof partitions each circle coordinate into arcs and applies translation averaging and the pigeonhole principle.

When $N$ is prime, simultaneous approximation of the frequencies shows that $B(\Gamma ,\rho )\subseteq \mathbb{Z}/N\mathbb{Z}$ contains a centered arithmetic progression of length at least