Solyanik estimates in ergodic theory

Let $U_1, \ldots, U_n$ be a collection of commuting measure preserving transformations on a probability space $(\Omega, \Sigma, \mu)$. Associated with these measure preserving transformations is the ergodic strong maximal operator $\mathsf M ^\ast _{\mathsf S}$ given by \[ \mathsf M ^\ast _{\mathsf S} f(\omega) := \sup_{0 \in R \subset \mathbb{R}^n}\frac{1}{\#(R \cap \mathbb{Z}^n)}\sum_{(j_1, \ldots, j_n) \in R\cap \mathbb{Z}^n}\big|f(U_1^{j_1}\cdots U_n^{j_n}\omega)\big|, \] where the supremum is taken over all open rectangles in $\mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. For $0 < \alpha < 1$ we define the sharp Tauberian constant of $\mathsf M ^\ast _{\mathsf S}$ with respect to $\alpha$ by \[ \mathsf C ^\ast _{\mathsf S} (\alpha) := \sup_{\substack{E \subset \Omega \\ \mu(E) > 0}}\frac{1}{\mu(E)}\mu(\{\omega \in \Omega : \mathsf M ^\ast _{\mathsf S} \chi_E (\omega) > \alpha\}). \] Motivated by previous work of A. A. Solyanik and the authors regarding Solyanik estimates for the geometric strong maximal operator in harmonic analysis, we show that the Solyanik estimate \[ \lim_{\alpha \rightarrow 1}\mathsf C ^\ast _{\mathsf S}(\alpha) = 1 \] holds, and that in particular we have \[\mathsf C ^\ast _{\mathsf S}(\alpha) - 1 \lesssim_n (1 - \frac{1}{\alpha})^{1/n}\] provided that $\alpha$ is sufficiently close to $1$. Solyanik estimates for centered and uncentered ergodic Hardy-Littlewood maximal operators associated with $U_1, \ldots, U_n$ are shown to hold as well. Further directions for research in the field of ergodic Solyanik estimates are also discussed.