Definition:Distance to Nearest Integer Function
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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.