Kronecker's approximation theorem

Remarks

For $k = 1$, fix $\varepsilon > 0$. The pigeonhole principle, as in Dirichlet's approximation theorem, gives an $m \ge 1$ with $0 < \{m\theta\} < \varepsilon$ or $1 - \varepsilon < \{m\theta\} < 1$. The value $\{m\theta\}$ is never $0$, since $\theta$ is irrational. The multiples of $m$ then move around the circle in steps shorter than $\varepsilon$, so they come within $\varepsilon$ of every point. In general dimension, the theorem follows from the $k$-dimensional form of Weyl's criterion, which gives even equidistribution in $[0,1]^k$. The independence hypothesis is necessary. Suppose $a_0 + a_1 \theta_1 + \cdots + a_k \theta_k = 0$ with integers $a_i$ not all $0$; then some $a_i$ with $i \ge 1$ is nonzero. All points then satisfy $a_1 \{n\theta_1\} + \cdots + a_k \{n\theta_k\} \in \Z$, so they lie on finitely many hyperplanes and are not dense. Example: $1, \sqrt{2}, \sqrt{3}$ are linearly independent over $\Q$, so the points $(\{n\sqrt{2}\}, \{n\sqrt{3}\})$ are dense in the unit square.

Going deeper

For a student

The theorem is the qualitative form of the equidistribution theorem and its higher-dimensional versions. It is a statement about the orbit of a rotation of the torus $\R^k / \Z^k$: under the independence condition, the orbit of $0$ is dense. Its higher-dimensional form is used in dynamics and in the theory of almost periodic functions.