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I am writing my thesis on quasi-periodic oscillations, which are signals containing two frequencies (let's leave it by that for now) with an incommensurable (irrational) ratio. However, I am a trained engineer and need a mathematical sharpening of my language:

The problem of quasi-periodic oscillations is that the frequencies constitute a Cantor set, since rationally dependent frequencies lead to periodic oscillations.

I know that the irrational ratios or - in general - irrational numbers constitute a Cantor set of positive measure within the real numbers. I just do not understand why it has positive measure. What about the density of the set?

Thank you very much!

SimonB
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  • It must have positive measure because its complement, $\mathbb Q$, has measure zero. – hmakholm left over Monica May 07 '19 at 11:26
  • That is a surprisingly simple but yet plausible and undestandable answer :-) I hadn't thought of that - thanks. Can you give me an interpretation apart from that? – SimonB May 07 '19 at 11:55
  • Afraid I don't have any intuition to offer. That $\mathbb Q$ has measure zero already feels pretty mysterious to me (even though I know how to prove that it does). – hmakholm left over Monica May 07 '19 at 12:08
  • I kinda like the formulation of Strogatz in "Nonlinear Dynamics and Chaos", where he just interprets "mearsure zero" has the total length of 0, since all intervals shrink to zero during the construction.

    Has anyone any clues about the density?

     – SimonB May 07 '19 at 12:38

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