= Bohr set in phase-distance convention
{c}
{title2=$B(R,\epsilon)=\{x:\|rx/N\|_{\mathbb R/\mathbb Z}\leq\epsilon\ (r\in R)\}$}
A <Bohr set> defined by distance of each character phase to the nearest integer. For $0\leq\epsilon\leq1/2$, it corresponds to the chord-distance radius $2\sin(\pi\epsilon)$. Numerical radius constants depend on the convention. In a group of prime order and with $d=|R|\geq1$, simultaneous torus approximation yields a nonzero element when $\epsilon>N^{-1/d}$ and an <arithmetic progression> of length at least $\min(N,\lceil\epsilon N^{1/d}\rceil)$.
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