Ergodic Averages Along Sequences of Slow Growth

The paper proves a.e. convergence in L1 for the double-logarithmic averages along Ω(n) by regrouping into k-almost-prime weights ηN(k), controlling their support via Hardy–Ramanujan, establishing an L1 maximal inequality (via transference), and a lacunary tail estimate, thereby delivering Theorem 1.2 rigorously . The candidate solution’s spectral/Dirichlet-series argument correctly yields L2 convergence (multipliers KN(e^{iθ}) → 0 off θ=0), but the final upgrade to µ-a.e. convergence for all f∈L1 via a “Banach principle” is not justified as stated: L2 convergence alone on a dense class does not imply a.e. convergence for all L1 without an appropriate maximal inequality or a valid pointwise convergence principle. The paper’s proof explicitly provides this missing ingredient; the model’s does not.