 Solution

The quotient-circle norm is $\Vert x{\Vert }_{\mathbb{R}/\mathbb{Z}}={\min }_{m\in \mathbb{Z}}|\tilde{x}-m|$, independent of the chosen lift $\tilde{x}$. A finite set is $\delta$-well-spaced when every two distinct points satisfy $\Vert x_r-x_s\Vert \ge \delta$. This is separation in the circle metric, including the distance across the identified endpoints.