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Math Help - Discrete set

  1. #1
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    Discrete set

    Hi, I want to know whether the set S={a+bt|a,b are rational integers} is discrete, where 0<t<1 is an irrational number. Namely, does the set S intersect each interval (-r,r) with only finite elements? Thank you very much!
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    No, the set is dense in \mathbb{R}. To prove it, reduce the set \{bt : b \in \mathbb{Z}\} modulo 1, and show that all of its members are distinct modulo 1 (show that otherwise t would be rational). A consequence is that the set is dense in (0,1), and thus by translation it is dense everywhere.
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