On the pointwise rate of convergence in the Birkhoff ergodic theorem: recent results

Let ${(\Omega,\mathcal{F},\mu)}$ be a probability measure space and ${T:\Omega\to\Omega}$ be a measure $\mu$ preserving ergodic transformation. For ${f\in L_1^0(\Omega,\mathcal{F},\mu)}$ consider the ergodic averages ${A_n^Tf(\omega)=\frac{1}{n}\sum\limits_{k=0}^{n-1}f(T^k\omega)}, \ n\in\mathbb{N}.$ The Birkhoff ergodic theorem states that a.e. ${A_n^Tf\to0}$ as ${n\to\infty}.$
It is well-known [1] that a.e. ${A_n^Tf(\omega)=o(1/n)}$ as ${n\to\infty}$ iff ${f\equiv0.}$ We discuss the existence of estimates ${\varphi\in c_0^{f,T}}$ for the rate of pointwise convergence of ergodic averages (i.e., ${A_n^Tf(\omega)=\mathcal{O}(\varphi(n))}$ as ${n\to\infty}$ a.e.) with the property ${\varphi(\ell_k)=o(1/{\ell_k})}$ for some increasing sequence ${\ell=\{\ell_k\}_{k\geq1}}$ of naturals.
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Theorem ([2]). Let ${T}$ be an ergodic automorphism, ${f\in L^0_1(\Omega, \mathcal{F}, \mu)}$, ${f\not\equiv0}$ and ${\ell=\{\ell_k\}_{k\geq1}}$ be monotone sequence of natural numbers. If ${\varphi\in c_0^{f,T}}$ and ${\varphi(\ell_k)=o(1/\ell_k)}$ as ${k\to\infty}$ then a.e. ${\frac{1}{N}\sum\limits_{k=1}^{N}f(T^{\ell_k}\omega)\to f(\omega)}$ as ${N\to\infty}.$
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Some corollaries of this statement will be considered. We also discuss the existence of the power-law estimates.
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Theorem ([3]). Let $T$ be an ergodic endomorphism, ${f\in L^0_1(\Omega, \mathcal{F}, \mu)}$ and ${f\not\equiv0}.$ Then the function ${\lambda_{f,T}(\omega):=\limsup\limits_{n\to\infty}\frac{\ln(1/n)}{\ln\left(\sup\limits_{k\geq n} |A^T_kf(\omega)|\right)}}$ is a constant a.e. Moreover, ${\lambda_{f,T}<\infty}$ iff there is a power-law pointwise estimate ${A_n^Tf(\omega)=\mathcal{O}(n^{-1/\delta})}$ for some ${\delta>0}.$
[1] Kachurovskii A. G. Rates of convergence in ergodic theorems // Russian Math. Surveys, 1996. V. 51. No. 4, P. 653–703.
[2] Podvigin I. V. On possible estimates of the rate of pointwise convergence in the Birkhoff ergodic theorem // Siberian Math. J., 2022. V. 63. No. 2, P. 316–325.
[3] Podvigin I. V. Exponent of Convergence of a Sequence of Ergodic Averages // Math. Notes, 2022. V. 112. No. 2, P. 271–280.