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Math Help - Spectrum of a real number

  1. #1
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    Spectrum of a real number

    Hello!

    I have a test this week and some examples where I have no idea! These are some bonus problems, but I want to solve them.

    The problem is:

    Do there exist real numbers a,b,c such that Spec(a), Spec(b), Spec(c) together partition the set of positive integers?

    Spec....spectrum of a real number
    Its defined as: Spec(a)={[a],[2a],.....} BUT [ ] means the greatest integer less than or equal to a. - (the Gauss brackets)

    I hope someone can help me
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  2. #2
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    Do there exist real numbers a,b,c such that Spec(a), Spec(b), Spec(c) together partition the set of positive integers?
    No, the spectra of any three real numbers cannot be pairwise disjoint. See this document (p. 1) that uses something like this lemma.

    People in the Number theory forum may be better equipped to answer such questions.
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