nt.number theory - Independence over Q and Kroneckers result

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.