= Nonwrapping progression in a cyclic Bohr set
In a <cyclic group> of arbitrary order $q$, a <Bohr set> with $d$ frequencies, defined by $\|rx/q\|\leq1/16$, contains an <arithmetic progression> of distinct residues of length at least $q^{1/(d+1)}/32$. Partition the $d$-dimensional cube into $Q^d$ boxes, with $Q=\lfloor q^{1/(d+1)}\rfloor$. The <pigeonhole principle> supplies $1\leq t\leq Q^d$ with $\|rt/q\|\leq1/Q$ simultaneously. The residues $jt$, $0\leq j\leq\lfloor Q/16\rfloor$, remain in the <Bohr set> and do not wrap around the cyclic group. The weaker exponent permits composite moduli without assuming that every nonzero step has large order.
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