Lower bound for the size of a Bohr set

For a Bohr set in chord-distance convention, with rank $d$ and width $0<\rho \le 2$,Partition each character circle into $M=\lceil 2\pi /\rho \rceil$ arcs. One of the $M^d$ resulting boxes contains at least $|G|/M^d$ images. Subtract a fixed group element in that box from all the others. Their character differences have chord distance at most $\rho$, giving that many distinct points in the Bohr set. Since $M\le 8/\rho$ in this width range, the displayed estimate follows.

For $\rho >2$ every character constraint is automatic and the Bohr set is all of $G$. One may therefore replace $\rho$ by $\min (\rho ,2)$ in a bound valid at every width. Without a width restriction the original right side could exceed $|G|$.