= Arithmetic progression in a cyclic Bohr set
Let $G=\mathbb Z/N\mathbb Z$ have prime order, and let $B(\Gamma,\rho)$ use chord width $0<\rho\leq2$ and positive rank $d=|\Gamma|\geq1$. Simultaneous character approximation gives a centered <arithmetic progression> inside it of <length> at least
$$
\frac18\rho N^{1/d}.
$$
The prime-order condition makes every nonzero step have order $N$, so the required short <arithmetic progression> can use distinct residues. If $\Gamma$ is empty, the <Bohr set> is the whole <group> and contains an <arithmetic progression> of $N$ distinct residues; the expression $N^{1/d}$ should not be used with $d=0$. For larger chord widths the <Bohr set> is again the whole <group>. The <nonwrapping progression in a cyclic Bohr set> supplies a weaker exponent that also works at composite orders.
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