Ergodic averages along sequences of slow growth

Abstract

We consider pointwise almost everywhere convergence of weighted ergodic averages along the sequence $\Omega \left(n\right)$, where $\Omega \left(n\right)$ denotes the number of prime factors of $n$ counted with multiplicities. It was previously shown that a pointwise ergodic theorem for $L^{\infty }$ functions does not hold along $\Omega \left(n\right)$. We classify the strength of this divergence by proving a double-logarithmic pointwise ergodic theorem for $L^1$ functions along $\Omega \left(n\right)$. This contrasts the behavior of Khintchine-type averages, for which, under any weaker form of averaging, there exists a bounded measurable function for which almost everywhere convergence fails. Moreover, we show that certain perturbations of increasing subpolynomial sequences fail to satisfy a pointwise ergodic theorem, yielding natural new examples of such sequences.