A pointwise ergodic theorem for the group of rational rotations

Abstract

Let f be a bounded, measurable function defined on the multiplicative group Ω \Omega of complex numbers of absolute value 1, and define ( ( 1 ) ) f n ( ω ) = 1 n ∑ i = 1 n f ( z n i ω ) , ω ∈ Ω , \begin{equation}\tag {$(1)$}{{f_n}(\omega ) = \frac {1} {n}\sum \limits _{i = 1}^n {f(z_n^i\omega )} ,} \qquad \omega \in \Omega ,\end{equation} where z n {z_n} is a primitive nth root of unity. The present paper generalizes this result of Jessen [1934]: if n ( k ) n(k) is an increasing sequence of positive integers with n ( k ) n(k) dividing n ( k ′ ) n(k’) whenever k > k ′ k > k’ , then f n ( k ) {f_{n(k)}} converges almost surely as k → ∞ k \to \infty .