Definition:Distance to Nearest Integer Function

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The distance to nearest integer function $\left\Vert{\cdot}\right\Vert : \R \to \left[{0 \,.\,.\, \dfrac 1 2}\right]$ is defined in the following ways:

Definition 1

$\left\Vert{\alpha}\right\Vert := \min \left\{{ \left\vert{n - \alpha}\right\vert : n \in \Z}\right\}$

Definition 2

$\left\Vert{\alpha}\right\Vert := \min \left\{{ \left\{{\alpha}\right\}, 1 - \left\{{\alpha}\right\} }\right\}$

where $\left\{{\alpha}\right\}$ is the fractional part of $\alpha$.

Also denoted as

The notation $\left\Vert{\cdot}\right\Vert_{\R / \Z}$ is also in use.

Also see