# 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).

Thanks in advance!

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## 1 Answer

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).

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Ahh thank you! This is exactly what I was looking for! –  Shaktal Jul 2 '12 at 20:06
@Qiaochu : oddly, the book does not use the term "Beatty sequence", and there is an exercise which is essentially to prove Rayleigh's Theorem, and the term "Rayleigh's Theorem" is not mentioned. It is strange that this excellent book would use the overworked word "spectrum" for something that already had a name. –  Stefan Smith Dec 3 '13 at 21:34
@Qiaochu: come to think of it, maybe they did have a reason. Wikipedia, at least, defines a Beatty sequence as, well, a sequence, and maybe the authors of "Concrete Mathematics" had a good reason why they wanted to use a multiset rather than a sequence (it seems to me that they are not equivalent concepts, because the order of elements in a multiset should not matter). Even if they had a good reason, using the overworked word "spectrum" is questionable. –  Stefan Smith Dec 3 '13 at 21:44