Bohr set in phase-distance convention ($B(R,ϵ)=\{x:\Vert rx/N{\Vert }_{\mathbb{R}/\mathbb{Z}}\le ϵ(r\in R)\}$)

A Bohr set defined by distance of each character phase to the nearest integer. For $0\le ϵ\le 1/2$, it corresponds to the chord-distance radius $2\sin (\pi ϵ)$. Numerical radius constants depend on the convention. In a group of prime order and with $d=|R|\ge 1$, simultaneous torus approximation yields a nonzero element when $ϵ>N^{-1/d}$ and an arithmetic progression of length at least $\min (N,\lceil ϵN^{1/d}\rceil )$.