Circle group ($S^1$)

The circle group is the unit complex numbers under multiplication, equivalently planar rotations or the quotient $\mathbb{R}/(2\pi \mathbb{Z})$.

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.