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)?