Translations on the Gaussian-integer torus

For $K=\mathbb Q(i)$ , the number-field torus is $\mathbb C/\mathbb Z[i]$ . Write $\omega=u+iv$ . Translation by $\omega$ is ergodic exactly when $1,u,v$ are linearly independent over $\mathbb Q$ .

Fourier calculation

Here $\mathcal O_K^\vee=\tfrac12\mathbb Z[i]$ and $\operatorname{Tr}_\infty z=2\operatorname{Re}z$ . With $a=(p-iq)/2$ ,

\operatorname{Tr}_\infty(a\omega)=pu+qv,

recovering the ordinary two-torus criterion. The increment $\sqrt2+i\sqrt3$ gives an ergodic translation.

One translation can fail while the joint action succeeds

For $\omega=\sqrt2$ , translation fixes the imaginary coordinate modulo one, so it is not ergodic. Nevertheless $k\sqrt2\notin\mathbb Z[i]$ for every nonzero $k\in\mathbb Z[i]$ .

The joint action includes translation by $\sqrt2$ and by $i\sqrt2$ . A character $e^{2\pi i(p\operatorname{Re}z+q\operatorname{Im}z)}$ is fixed by both only if $p\sqrt2,q\sqrt2\in\mathbb Z$ , forcing $p=q=0$ . The action is therefore ergodic although neither generator is individually ergodic.