TITLE: Trajectories on the $k$-dimensional torus
QUESTION [7 upvotes]: Let $r_1,\dots,r_k$ be irrational and linearly independent over $\mathbb Q$.
My intuition clearly tells me that the set 
$$\{(nr_1,\dots,nr_k)+\mathbb Z^k:n\in\mathbb N\}$$
is dense in $\mathbb R^k/\mathbb Z^k$.
I know a couple of sources of proofs of the two-dimensional case, which is not so hard, and it seems to me that the classification of subgroups of $\mathbb R^k$
gets me close to this, but does someone have a crisp proof of this fact, or a good reference?

REPLY [11 votes]: The result you want goes by the name "Kronecker's theorem". There is a proof
in Siegel's "Lecture on the Geometry of Numbers" and in Bump's book on Lie groups.