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It is well known (Kronecker's Theorem) that "irrational rotations" are dense on $[0,1)$. That is, the set $$ \{ x+nr\mod 1 : n \in \mathbb{N} \} $$ is dense on $[0,1)$, provided that $r$ is irrational. This theorem is relatively easy to prove.

On the two dimensional torus $\mathbb{T}=[0,1)\times[0,1)$ (with opposite edges identified), the following result is true. The set $$ \{ (x+nr \mod 1,x+nr' \mod 1) \in \mathbb{T} : n \in \mathbb{N} \} $$ is dense in $\mathbb{T}$ if and only if $\{r, r', 1\}$ are rationally independent (i.e., if there exist integers $a$ and $b$ such that $ar+br'$ is an integer, then $a=b=0$). I have seen a very complicated proof of this. Is there an "easy" proof? That is, something that one could assign for reading to an undergraduate (say a senior)?

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    I took the liberty of adding a IMHO more appropriate tag. I have only seen this result in number-theoretic contexts. Actually I'm a bit curious to see how this is used in dynamical systems!2012-06-06
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    @Jyrki: For a completely integrable Hamiltonian system, the Arnold–Liouville theorem says that the phase space (of dim $2n$, say) is foliated into invariant tori (of dim $n$) such that the motion on each torus is just a straight line (in suitable coordinates). Whether such a trajectory is periodic or fills the torus densely depends on the rational depencence or independence of the components of the direction vector of the line.2012-06-07
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    @Hans, thanks for that bit. Can't say I'm familiar with the result, but at least I can sort of see what's going on there. But here it looks like "time" (or whatever is the parameter of the motion) is ticking in discrete steps?2012-06-07
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    @Jyrki: True... Let's put it this way then: translation on a torus is a simple example of a discrete-time dynamical system, and it's interesting as an illustration of what type of global behaviour that orbits can exhibit. See for example Section 1.2 in [Zehnder's book](http://books.google.com/books?id=qva7p5NCayUC&printsec=frontcover&hl=en#v=onepage&q&f=false).2012-06-07
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    @Hans, I'm afraid I get a "no eBook available".2012-06-07
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    @Jyrki: Funny. I can preview almost the whole book here. However, I'm automatically redirected to books.google.se, so maybe the permissions depend on which country you're in?2012-06-07
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    @Hans, it may also be related to the fact that I will not allow google.analytics to set cookies on my computer.2012-06-07
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    @Jyrki: You'll have to borrow someone else's computer then. ;-) It's a really nice book, by the way.2012-06-07

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I have seen IMHO quite accessible proofs of this fact in number theory books. When I was an advanced high school kid (I had found my calling), I saw a proof of this with hints in Joe Roberts' lovely book, typeset in calligraphic font, Elementary Number Theory - A Problem Oriented Approach. IIRC I managed to follow the proof given there, but this was among the more taxing problems. As an undergraduate I had the pleasure of giving a talk about this at a seminar going through Apostol's Modular Functions and Automorphic Forms in Number Theory. It is in one of the late chapters, and I recall enjoying that chapter and the exercises therein immensely.

I don't know if this is helpful to you. This stuff is certainly not too demanding for an undergraduate in that it doesn't rely on any deep theory. But I wouldn't assign this to someone who hasn't shown a real interest in thinking things through for him/herself. You know your clients better.

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    Thanks Jyrki. I saw the (difficult) proof of this in a dynamical systems book, that's why I tagged it so. I'll look up the books you mentioned. Thanks!2012-06-06