Weighted multiple ergodic averages via analytic observables over T∞: Is exponential pointwise convergence universal?

The paper proves two main exponential-type convergence results for weighted multiple ergodic averages with the accelerated C∞ bump weight: (i) in finite dimensions, for analytic observables and almost every joint Diophantine rotation, a stretched-exponential rate O(exp(−N^{ζ1})) (and analogously in continuous time) follows from a truncated smallness condition plus adaptive integration-by-parts, see Theorem 3.3 and its universal corollary Theorem 4.3; the notation in the statement reads exp(−N ζ1) but the proof shows the exponent is N^{ζ1} for some ζ1>0 via φ(N)=N^ε, cf. (3.10), (4.2), and (7.12)–(7.17) . (ii) In the infinite-dimensional almost-periodic case on T^∞, the authors construct a full-measure nonresonant condition via Borel–Cantelli (using a counting lemma for |k|_η-level sets) and verify a truncated smallness bound that yields a log-power exponential rate O(exp(−(log N)^{ζ3})) with ζ3>1, see Theorem 4.5 and the tail estimate in Section 4 (the bound ∑_{|k|_η=ν}1=O(ν^{ν^{1/η}}) together with ϑ*(x)≈exp(x^{3/4}) makes ∑ e^{−|k|_η^{3/4}} summable) . The candidate model’s core objections rely on bounding the kernel only through the Fourier transform W of w (leading to a β=1/2 decay and hence much weaker global rates) and on a Borel–Cantelli lower bound that stops at polynomial rates in N. This misses the paper’s key truncated/integration-by-parts mechanism (J1 vs. J2 splitting) that upgrades small-divisor control into exp(−φ(N)^{β1}) with φ chosen adaptively (e.g., φ(N)=log N in T^∞), see (7.12)–(7.17) . The paper also clarifies that “exponential” in the finite-dimensional theorems means exp(−N^{ζ1}), not pure exp(−cN), aligning with the method’s Gevrey-type derivative growth of w (Lemma 8.2) rather than analytic decay of W(ξ) . Hence the model both misreads the finite-dimensional claim and overlooks the truncation strategy that enables the log-power exponential rate in the infinite-dimensional case.