Circle metric ($d(x,y)=\Vert x-y{\Vert }_{\mathbb{R}/\mathbb{Z}}$)

The quotient metric of circumference one uses the shorter distance around the circle: $\Vert x\Vert ={\min }_{m\in \mathbb{Z}}|\tilde{x}-m|$. It also measures separation across the identified endpoints. This is the metric used by circular spacing in the analytic large sieve inequality.