Difference between revisions of "Distance to integers"
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Revision as of 19:31, 22 January 2016
Define the function $\mathrm{dist}_{\mathbb{Z}} \colon \mathbb{R} \rightarrow \mathbb{R}$ by $$\mathrm{dist}_{\mathbb{Z}}(x)=\inf_{n \in \mathbb{Z}} |x-n|,$$ where $\inf$ denotes the infimum. This function can be computed using the floor and ceiling functions: $$\mathrm{dist}_{\mathbb{Z}}(x)=\min \left(2^n x - \lfloor 2^n x \rfloor, \lceil 2^n x \rceil - x \right).$$
