Rates of convergence in the von Neumann ergodic theorem

It was proved in [Ka96] that a power rate of convergence in von Neumanns ergodic theorem is equivalent to the power (with the same exponent) singularity at zero point of a spectral measure of averaging function with respect to the dynamical system. I.e., it was shown that the estimates of convergence rates in this ergodic theorem are necessarily the spectral ones. In [KaPo16], asymptotically exact estimates of these rates were obtained for certain well-known billiards, and Anosov systems.
It turns out [KaKn18,KaPo18], that the Fejer sums for measures on the circle and the norms of the deviations from the limit in the von Neumann ergodic theorem both are calculating, in fact, with the same formulas (by integrating of the Fejer kernels) – and so, this ergodic theorem is a statement about the asymptotics of the growth of the Fejer sums at zero for the corresponding spectral measure. As a result, available in the harmonic analysis literature, numerous estimates for the deviations of Fejer sums at a point allowed to obtain new estimates for the rate of convergence in this ergodic theorem.
[Ka96] Alexander Kachurovskii. Rates of Convergence in Ergodic Theorems // Russian Math. Surveys. 1996. V. 51, No 4. P. 653–703.
[KaPo16] Alexander Kachurovskii, Ivan Podvigin. Estimates of the Rate of Convergence in the von Neumann and Birkhoff Ergodic Theorems // Trans. Moscow Math. Soc. 2016. V. 77. P. 1–53
[KaKn18] Alexander Kachurovskii, Kirill Knizhov. Deviations of Fejer Sums and Rates of Convergence in the von Neumann Ergodic Theorem // Dokl. Math. 2018. V. 97, No 3. P. 211–214.
[KaPo18] Alexander Kachurovskii, Ivan Podvigin. Fejer Sums for Periodic Measures and the von Neumann Ergodic Theorem // Dokl. Math. 2018. V. 98, No 1. P. 344–347.