Absolute and Unconditional Convergence of Series of Ergodic Averages and Lebesgue Derivatives

Bryan Johnson, Joseph Rosenblatt

correcthigh confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves the equivalence precisely and for all ergodic measure-preserving transformations (not assumed invertible), via the Conze weak-approximation method and Fourier analysis on rotations, culminating in Theorem 3.1 with a detailed (a)⇒(b)⇒(a) argument . The model’s (ii)⇒(i) direction is fine (spectral calculus yields unconditional convergence), but its (i)⇒(ii) uses forward Rokhlin towers and an appeal to “natural unitary extension/compression” that does not justify transferring the uniform operator bounds to the extension; this step fails for non-invertible systems as written. A correct fix would require either (a) the paper’s Conze approximation route or (b) operator-theoretic tools (Wold decomposition/von Neumann inequality). As stated, the model’s proof has a real gap in the non-invertible case.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript delivers a sharp and natural equivalence (Theorem 3.1) for unconditional convergence of series of ergodic-average differences, with a robust proof that avoids delicate invertibility issues by using the Conze approximation method and Fourier analysis on rotations. The result clarifies when differences of ergodic averages behave like orthogonal martingale differences and complements illuminating examples (e.g., n\_k=2\^k is good; n\_k=k fails). The exposition is largely clear, but a few steps in the Conze approximation and the invocation of Theorem 2.4 could benefit from slightly more explicit signposting for readers not already familiar with these techniques.