Ergodic averages along sequences of slow growth

TL;DR

This paper investigates the convergence properties of weighted ergodic averages along the sequence of the number of prime factors, revealing nuanced behavior and conditions under which convergence or non-convergence occurs.

Contribution

It classifies the non-convergence strength of the sequence (n) and shows the persistence of strong sweeping out under certain perturbations.

Findings

(n) satisfies a double-logarithmic pointwise ergodic theorem.

(n) does not exhibit strong sweeping out despite non-convergence.

Strong sweeping out persists under specific sequence perturbations.

Abstract

We consider pointwise convergence of weighted ergodic averages along the sequence $\Omega(n)$, where $\Omega(n)$ denotes the number of prime factors of $n$ counted with multiplicities. It was previously shown that $\Omega(n)$ satisfies the strong sweeping out property, implying that a pointwise ergodic theorem does not hold for $\Omega(n)$. We further classify the strength of non-convergence exhibited by $\Omega(n)$ by verifying a double-logarithmic pointwise ergodic theorem along $\Omega(n)$. In particular, this demonstrates that $\Omega(n)$ is not inherently strong sweeping out. We also show that the strong sweeping out property for slow growing sequences persists under certain perturbations, yielding natural new examples of sequences with the strong sweeping out property.