Exponential convergence of the weighted Birkhoff average (Q6573748)

While the ergodic theorem itself is known to not admit an effective rate result in general by work of \textit{U. Krengel} [Monatsh. Math. 86, 3--6 (1978; Zbl 0352.28008)], restricting the type of underlying dynamical system or the class of functions used as observables may permit rate results. Here irrational rotations of both finite and infinite dimensional tori are studied, and situations of both polynomial and exponential convergence rates are found. In particular the first universal exponential convergence in the quasi-periodic case and arbitrary polynomial convergence in the almost periodic case under analyticity are shown.