On the speed of convergence in the ergodic theorem for shift operators

Given a probability space (X, μ), a square integrable function f on such space and a (unilateral or bilateral) shift operator T, we prove under suitable assumptions that the ergodic means N−1 ∑N−1 n=0 Tn f converge pointwise almost everywhere to zero with a speed of convergence which, up to a small logarithmic transgression, is essentially of the order of N−1/2 . We also provide a few applications of our results, especially in the case of shifts associated with toral endomorphisms.