= Progression in a Bohr set from successive minima
{title2=$|P|\ge c(d,\rho)\,|G|$}
For a cyclic group $G$, a rank-$d$ phase-radius-$\rho$ <Bohr set> contains a symmetric proper generalized arithmetic progression of rank at most $d+1$ and cardinality at least $c(d,\rho)|G|$. The constant can be taken to depend only on $d$ and $\rho$. This geometry-of-numbers consequence follows by applying successive minima to the lattice recording the simultaneous character congruences; any cyclic subgroup factor can be represented by one additional progression coordinate. It is stronger than finding just one long ordinary progression.
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