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 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.

  • $\begingroup$ That must be the result Alexei wanted, but density is not necessary for the condition. As long as the tangent space to the closure intersects the positive orthant, there will be multiples which are all slightly greater than lattice points. Specifically, this is true for $r_1 = r_2 = \sqrt 2$. $\endgroup$ – Douglas Zare Mar 14 '10 at 20:18
  • $\begingroup$ I fixed the link. Replace each ' in a URL by %27. $\endgroup$ – Douglas Zare Mar 14 '10 at 21:21
  • $\begingroup$ Fair enough: asking for the fractional parts of nr_1 and nr_2 to both be arbitrarily small for some n, we don't have to insist the integral multiples of (r_1,r_2) in the torus are a dense subset, e.g., if r_1 and r_2 are the same irrational number. That example doesn't even require r_1 and r_2 to be linearly independent over the rationals, which was one of Alexei's hypotheses. (Извините, Алексей!) $\endgroup$ – KConrad Mar 14 '10 at 21:35

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