Summability and speed of convergence in an ergodic theorem

Given an irrational vector α in Rd, a continuous function f(x) on the torus Td and suitable weights Φ(N,n) such that ∑n=−∞+∞Φ(N,n)=1, we estimate the speed of convergence to the integral ∫Tf(y)dy of the weighted sum ∑n=−∞+∞Φ(N,n)f(x+nα) as N→+∞. Whereas for the arithmetic means N−1∑n=1Nf(x+nα) the speed of convergence is never faster than cN−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 Φ(N,n) and the smoothness of the function f(x) which holds for almost every α. 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 α∈Rd.