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Dense orbits of irrational n-tuples in n-Torus

  1. Apr 14, 2012 #1
    Hey all, this is my first post! (Although I've found a lot of useful answers here during the past).
    I have been trying to prove this fact, which is widely stated in literature and relatively well-known, about density of orbits of irrational n-tuples in the n-torus. My question is this: If
    [itex]a=(a_1,...,a_n)[/itex], with a_i irrational, and all rationally independent, show that the orbit {qa}_{q \in Z} is dense in the n-torus. Here qa = (qa_1,...,qa_n).

    For some background, if n=1, then it is not hard to show that (qa)mod1 (as q moves through the integers) is dense in [0,1). I can also show a similar result when n is 2, but I want to extend this and it's driving me nuts since it's referenced everywhere but I can't find a solid proof!
  2. jcsd
  3. Apr 15, 2012 #2


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    It is in Thierry Aubin's book A Course in Differential Geometry !
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