Weak-type \((1, 1)\) inequality for discrete maximal functions and pointwise ergodic theorems along thin arithmetic sets (Q6573618)

In 1991 \textit{J. M. Rosenblatt} and \textit{M. Wierdl} [Lond. Math. Soc. Lect. Note Ser. 205, 3--151 (1995; Zbl 0848.28008)] conjectured that for any arithmetical set \(A\) with zero Banach density and aperiodic probability dynamical system \((X, \mathcal{B}, \mu, T)\), there exists a function \(f \in L_\mu^1(X)\), such that \N\[ M_{A, N} f=\frac{1}{|A \cap[1, N]|} \sum_{n \in A \cap[1, N]} f \circ T^n \quad \text { does not converge almost everywhere. } \] \NThis was disproved in 2006 by \textit{Z. Buczolich} [Acta Math. Hung. 117, No. 1--2, 91--140 (2007; Zbl 1164.37001)], where he provided a counterexample by constructing inductively an appropriate set \(A\) of zero Banach density for which one gets the pointwise convergence of the ergodic averages \(M_{A, N} f\) for all \(f \in L^1\). In [J. Anal. Math. 127, 247--281 (2015; Zbl 1327.42022)], \textit{M. Mirek} established the weak type \((1,1)\) bounds for \(\mathcal{M}_A\) and the corresponding pointwise ergodic theorem on \(L^1\) for sets of the form \(\left\{\left\lfloor n^c \ell(n)\right\rfloor: n \in \mathbb{N}\right\}\), where \N\[\mathcal{M}_A f(x)=\sup _{N \in \mathbb{N}} \frac{1}{|A \cap[1, N]|} \sum_{n \in A \cap[1, N]}|f(x-n)|,\] \N\(c\) is close to 1 and \(\ell\) is a certain kind of slowly varying function. In the article under review, the author considers the arithmetic sets \(B_{\pm}\) defined via \(\ell\), which were introduced by \textit{B. Krause} et al. [J. Funct. Anal. 271, No. 1, 164--181 (2016; Zbl 1338.30003)], and shows that the maximal function \N\[ \mathcal{M}_{B_{ \pm}} f(x)=\sup _{N \in \mathbb{N}} \frac{1}{\left|B_{ \pm} \cap[1, N]\right|} \sum_{n \in B_{ \pm} \cap[1, N]}|f(x-n)|\] \Nis of weak type (1,1) and bounded on \(L^p\) with \(p\in(1, \infty]\). As a corollary, he obtains the corresponding pointwise convergence result on \(L^1\). The authors also establish a multiparameter pointwise ergodic theorem by uniform oscillation estimates and certain vector-valued maximal estimates.