Everyone knows the result by Kronecker: if $r$ is a real number not rational and $\epsilon>0$ then there exist a natural number $N$ such that $\{Nr\}<\epsilon$. There must be such a result for pairs (and even for any other quantity) of real numbers: let $r_1$, $r_2$ ` be real numbers independent over $\mathbb{Q}$ and $\epsilon>0$ then there exist a natural number $N$ such that $\{Nr_1\}<\epsilon$ and $\{Nr_2\}<\epsilon$. I heard this result more than 10 years ago but i still don't know the proof. I just guess this problem is related to everywhere density of trajectory on torus.
This is on wikipedia. See Kronecker's theorem. It was proved by Kronecker in 1884. The necessary and sufficient condition for integral multiples of a point $(r_1,\dots,r_n)$ in the $n$torus $(\mathbf R/\mathbf Z)^n$ to be dense is not that the $r_i$'s are all irrational: that is necessary but far from sufficient. Consider, for example, integral multiples of $(\sqrt{2},1+\sqrt{2})$ in the 2torus. The correct necessary and sufficient condition is that $1, r_1, \dots,r_n$ are linearly independent over $\mathbf Q$. (For $n = 1$, this recovers the irrationality condition as being necessary and sufficient for denseness of integral multiples on a circle.) A proof of this theorem can be found in Hardy and Wright's Introduction to the Theory of Numbers (first in one dimension and then in general; see Chapter 23). It can also be proved by ideas from ergodic theory: the hypothesis that $1,r_1,\dots,r_n$ are linearly independent over $\mathbf Q$ implies translation on the $n$torus by $(r_1,\dots,r_n)$ is ergodic and the orbit of any point in a compact topological group under a left or right translation that's ergodic is dense in the group. (Initially one can say only that almost every point in a compact group  in the sense of its Haar measure  has a dense orbit under an ergodic transformation, but left and right translation by a fixed element is a pretty special transformation: if such a translation has one dense orbit then all the orbits of that translation are dense.) Note: The linear independence of $1, r_1,\dots,r_n$ over $\mathbf Q$ is actually equivalent to the ergodicity of translation by $(r_1,\dots,r_n)$ on the $n$torus. Weyl's equidistribution theorem strengthens Kronecker's theorem: that sequence of integral multiples isn't just dense in the $n$torus but in fact is uniformly distributed in the $n$torus. This quantifies Kronecker's theorem in the same way the ergodic theorem quantifies the Poincare Recurrence Theorem. 

