Definition:Distance to Nearest Integer Function

Definition

The distance to nearest integer function $\norm \cdot: \R \to \closedint 0 {\dfrac 1 2}$ is defined in the following ways:

Definition 1

$\norm \alpha:= \min \set {\size {n - \alpha}: n \in \Z}$

Definition 2

$\norm \alpha:= \min \set {\set \alpha, 1 - \set \alpha}$

where $\set \alpha$ is the fractional part of $\alpha$.

Also denoted as

The notation $\norm \cdot_{\R / \Z}$ is also in use.