Independence over Q and Kroneckers result 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.    
 A: 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 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.
