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. 



Independence over Q and Kroneckers result
This is on wikipedia. See [Kronecker's theorem](http://en.wikipedia.org/wiki/Kronecker%27s_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 2-torus. 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.


