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-1n=0N-1 Tnf 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.