= Lower bound for the size of a Bohr set
For a <Bohr set> in chord-distance convention, with rank $d$ and width $0<\rho\leq2$,
$$
|B(\Gamma,\rho)|\geq\left(\frac{\rho}{8}\right)^d|G|.
$$
Partition each character <circle> into $M=\lceil2\pi/\rho\rceil$ 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 $M\leq8/\rho$ 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 $|G|$.
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