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

Given a probability space $(X,\mu )$, 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 $N^{-1}{\sum }_{n=0}^{N-1}{T}^{n}f$ converge pointwise almost everywhere to zero with a speed of convergence which, up to a small logarithmic transgression, is essentially of the order of $N^{-1\mathrm{/}2}$. We also provide a few applications of our results, especially in the case of shifts associated with toral endomorphisms.