Ergodicity criterion for torus translations

Statement

On $\mathbb T^d=\mathbb R^d/\mathbb Z^d$ , $d\geq1$ , with Haar probability, translation $T_\omega(x)=x+\omega$ is ergodic if and only if

k\cdot\omega\notin\mathbb Z\qquad(0\ne k\in\mathbb Z^d).

Equivalently, $1,\omega_1,\ldots,\omega_d$ are linearly independent over $\mathbb Q$ .

Fourier proof

For $e_k(x)=e^{2\pi ik\cdot x}$ ,

U_{T_\omega}e_k=e^{2\pi ik\cdot\omega}e_k, \qquad \alpha_\omega(e_k)=e^{-2\pi ik\cdot\omega}e_k.

The first is forward Koopman composition; the second is inverse pullback on $L^\infty$ . Their fixed Fourier modes agree. Completeness of the characters and the fixed-function criterion prove the result.

Examples

For a circle, the increment must be irrational. In two dimensions, $(\sqrt2,\sqrt3)$ works, while $(\theta,\theta)$ fails even for irrational $\theta$ , because $k=(1,-1)$ gives zero. Rational independence is a joint condition, not separate irrationality of coordinates.

Every torus translation has discrete spectrum and, for $d\geq1$ , nonconstant eigenfunctions. Thus none is weakly mixing.