We will now apply the machinery already developed to a few concrete problems.

Proposition 1Let be a compact abelian group and the rotation by . Then is uniquely ergodic (with the Haar measure invariant) if is dense in .

The proof is straightforward. Suppose is invariant with respect to rotations by . Then for , we have

and hence

for any , which means that must be Haar measure (which is unique).

Corollary 2An irrational rotation of the unit circle is uniquely ergodic.

**Application: Equidistribution **

Theorem 3Let be irrational and let be continuous and -periodic. Then (more…)