Ergodic averages along sequences of slow growth

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.