Solyanik estimates in ergodic theory (Q2833639)

\textit{A. A. Solyanik}'s estimates [Math. Notes 54, No. 6, 1241--1245 (1993; Zbl 0836.42014); translation from Mat. Zametki 54, No. 6, 82--89 (1993)] were introduced in harmonic analysis to study fine properties of the Hardy-Littlewood and strong maximal functions. Here estimates of this sort are introduced to the setting of ergodic theory, specifically for commuting measure-preserving transformations \(U_1,\dots,U_n\) on a probability space \((\Omega,\mu)\) and the associated ergodic strong maximal operator \(f\mapsto M_S^*f\) where \(M_S^*f(\omega)\) is the supremum over all open rectangles \(R\) in \(\mathbb{R}^n\) with sides parallel to the axes containing the origin of the average NEWLINE\[NEWLINE\frac{1}{| R\cap\mathbb{Z}^n|} \sum_{(j_1,\dots,j_n)\in R\cap\mathbb{Z}^n} | f(U_1^{j_1}\cdots U_n^{j_n}\omega)|.NEWLINE\]NEWLINE For \(\alpha\in(0,1)\) a sharp Tauberian constant \(C_S^*(\alpha)\) is defined as the supremum over all measurable sets \(E\) of positive measure of \(\frac{1}{\mu(A)}\mu(\{\omega\in\Omega\mid M_{S}^*\chi_E(\omega)>\alpha\})\) and it is shown that \(C_S^*(\alpha)\to1\) as \(\alpha\to 1\) (along with more refined estimates). Several further research questions are discussed, including the question of which of the bounds obtained here are sharp.