Ergodic averages along sequences of slow growth

Abstract We consider pointwise almost everywhere convergence of weighted ergodic averages along the sequence , where denotes the number of prime factors of counted with multiplicities. It was previously shown that a pointwise ergodic theorem for functions does not hold along . We classify the strength of this divergence by proving a double‐logarithmic pointwise ergodic theorem for functions along . 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.