A pointwise ergodic theorem for the group of rational rotations

Let f be a bounded, measurable function defined on the multiplicative group $\Omega$ of complex numbers of absolute value 1, and define

${{f_n}(\omega ) = \frac{1} {n}\sum\limits_{i = 1}^n {f(z_n^i\omega )} ,} \qquad \omega \in \Omega ,$

($(1)$)

where ${z_n}$ is a primitive nth root of unity. The present paper generalizes this result of Jessen [1934]: if $n(k)$ is an increasing sequence of positive integers with $n(k)$ dividing $n(k')$ whenever $k < k'$, then ${f_{n(k)}}$ converges almost surely as $k \to \infty$.