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Consider the system $$ \dot{\mathbf{x}}(t)=\mathbf{a} $$ where $t\in\mathbb{R},\mathbf{x}\in\mathbb{R}^n,$ and $\mathbf{a}\in\mathbb{R}^n.$ It is well known that the flow on the n-torus $\mathbb{T}^n$ generated by $$\varphi_t:\mathbb{T}^n\to\mathbb{T}^n $$ given by $$\varphi_t\left(\mathbf{x}_0\right):=t\mathbf{a}+\mathbf{x}_0 $$ is transitive if and only if it is incommensurate. This paper

http://fe.math.kobe-u.ac.jp/FE/FE_pdf_with_bookmark/FE01-10-en_KML/fe07-091-102/fe07-091-102.pdf

references several papers early on as a proof of this. However, I'm looking for a direct proof that doesn't rely on any measure theoretic notions or ergodic theory for pedagogical purposes. I think I have a proof, yet I haven't been able find one anywhere else despite this being an important example in integrable dynamical systems. Has anyone come across a direct proof of this well-known result?

garserdt216
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    in dimension 2 there is Kronecker's theorem, in Diophantine Approximations by Niven. In the chapter notes he sys that , for $1, \alpha, \beta$ linearly independent over the rationals, the images of $m \alpha, m \beta$ in the unit square are both dense and uniformly distributed. Here we go, dimension $n$ is in Cassels (1957), chapter 4. – Will Jagy Jan 07 '18 at 21:16
  • I would also like to see an answer to this question. You mentioned that you have a direct proof: can you share? – Willie Wong Apr 26 '19 at 15:28

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