Geodesic
The maximum pointwise rate of convergence in Birkhoff's ergodic theorem
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 498 (2020), pp. 18-25

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A criterion for the maximum possible pointwise convergence rate in Birkhoff's ergodic theorem for ergodic semiflows in a Lebesgue space is obtained. It is proved that higher rates of convergence in this theorem are impossible. 
@article{ZNSL_2020_498_a1,
     author = {A. G. Kachurovskii and I. V. Podvigin and A. A. Svishchev},
     title = {The maximum pointwise rate of convergence in {Birkhoff's} ergodic theorem},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {18--25},
     publisher = {mathdoc},
     volume = {498},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_498_a1/}
}
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A. G. Kachurovskii; I. V. Podvigin; A. A. Svishchev. The maximum pointwise rate of convergence in Birkhoff's ergodic theorem. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 498 (2020), pp. 18-25. http://geodesic.mathdoc.fr/item/ZNSL_2020_498_a1/