# Spectrum of a real number

Whilst reading Concrete Mathematics, the authors mention something which they refer to as the "spectrum" of a real number (pg. 77):

We define the spectrum of a real number $\alpha$ to be an infinite multiset of integers, $$\operatorname{Spec}{(\alpha)}=\{\left\lfloor\alpha\right\rfloor, \left\lfloor2\alpha\right\rfloor,\left\lfloor3\alpha\right\rfloor,\cdots\}.$$

It then goes on to describe interesting properties of these "spectra", such as the inequality of any two spectra, i.e. given some $\alpha\in\mathbb{R},\space\not\exists\beta\in\mathbb{R}\setminus\{\alpha\}:\operatorname{Spec}{(\alpha)}=\operatorname{Spec}{(\beta)}$.

It also gives a proof regarding the partitioning of the set $\mathbb{N}$ into two spectra, $\operatorname{Spec}{(\sqrt{2})}$ and $\operatorname{Spec}{(\sqrt{2}+2)}$.

These properties intrigued me, so I wondered if there were any other properties of these multisets, however; a google search for "spectrum of a real number" doesn't appear to yield any relevant results, so I wondered if there was any research into these objects, and if there is whether "spectrum" is a non-standard name (if it is, I'd appreciate the common name for these objects).

This is non-standard (and does not agree well with any of the other meanings of "spectrum" in mathematics so I would not use it). The standard name is Beatty sequence (at least when $\alpha$ is irrational).