Definition:Nearest Integer Function

Definition

The nearest integer function $\| \cdot \| : \R \to [0,1/2]$ defined by one of the following equivalent properties:

• $\|\alpha \| = \min\{ |n - \alpha| : n \in \Z\}$
• $\|\alpha \| = \min\{ \{\alpha\},1-\{\alpha\}\}$ where $\{\alpha\}$ is the fractional part of $\alpha$.

The notation $\| \cdot \|_{\R/\Z}$ is also in use.