Summability and speed of convergence in an ergodic theorem

Abstract

Given an irrational vector $\alpha$ in ${\mathbb{R}}^d$, a continuous function $f(x)$ on the torus ${\mathbb{T}}^d$ and suitable weights $\Phi (N,n)$ such that ${\sum }_{n=-\infty }^{+\infty }\Phi (N,n)=1$, we estimate the speed of convergence to the integral ${\int }_{{\mathbb{T}}^d}f(y)dy$ of the weighted sum ${\sum }_{n=-\infty }^{+\infty }\Phi (N,n)f(x+n\alpha )$ as $N\to +\infty$. Whereas for the arithmetic means $N^{-1}{\sum }_{n=1}^{N}f(x+n\alpha )$ the speed of convergence is never faster than $c{N}^{-1}$, for other means such speed can be accelerated. We estimate the speed of convergence in two theorems with different flavor. The first result is a metric one, and it provides an estimate of the speed of convergence in terms of the Fourier transform of the weights $\Phi (N,n)$ and the smoothness of the function $f(x)$ which holds for almost every $\alpha$. The second result is a deterministic one, and the speed of convergence is estimated also in terms of the Diophantine properties of the given irrational vector $\alpha \in {\mathbb{R}}^d$.

28 pages