Geodesic
Measuring the Rate of Convergence in the Birkhoff Ergodic Theorem
Matematičeskie zametki, Tome 106 (2019) no. 1, pp. 40-52

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Estimates of the rate of convergence in the Birkhoff ergodic theorem which hold almost everywhere are considered. For the action of an ergodic automorphism, the existence of such estimates is proved, their structure is studied, and unimprovability questions are considered.
Keywords: individual ergodic theorem, rate of convergence in ergodic theorems, unimprovability of estimates, return time, lattice. 
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A. G. Kachurovskii; I. V. Podvigin. Measuring the Rate of Convergence in the Birkhoff Ergodic Theorem. Matematičeskie zametki, Tome 106 (2019) no. 1, pp. 40-52. http://geodesic.mathdoc.fr/item/MZM_2019_106_1_a3/