Slow convergence almost everywhere of ergodic averages

Valery V. Ryzhikov

REVIEW 3 major objections 3 minor 1 cited by

Slow convergence almost everywhere of ergodic averages

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Ergodic averages on any Z^n action can be made to converge arbitrarily slowly, so no universal rate exists for Birkhoff's theorem

desk verdict A credible and natural extension of Krengel's theorem to Z^n-actions, but the central Rokhlin lemma is only vaguely stated and the full text is unreadable in our copy, so the proof cannot be checked from what we have. read the letter →

arxiv 2508.00463 v1 pith:VD6DE4VG submitted 2025-08-01 math.DS

Valery V. Ryzhikov This is my paper · ORCID

classification math.DS MSC 37A3037A05

keywords ergodictheoryBirkhofftheoremrateofconvergenceKrengeleffectZ^nactionsRokhlinlemmamultiparameterL^1functions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a negative result about how fast ergodic averages can converge. For any ergodic, measure-preserving action of $\mathbb{Z}^n$ on a probability space and any sequence of positive numbers that tends to zero as slowly as one wishes, there is an integrable function $f$ whose standard time averages converge almost everywhere to the spatial average of $f$, but not at a rate that is asymptotically majorized by that sequence. In other words, the pointwise speed of Birkhoff convergence in multiparameter abelian actions is not controlled by the action alone; for each prescribed slow rate one can choose an observable that converges no faster. This generalizes the earlier Krengel effect, known for a single transformation, to all $\mathbb{Z}^n$-actions.

What carries the argument

The carrying object is a weakened Rokhlin lemma for ergodic $\mathbb{Z}^n$-actions. It guarantees a sequence of asymptotically almost invariant sets with prescribed measures; a set is almost invariant when its symmetric difference with each of its images under the action has small measure. The distinctive feature is adaptivity: the choice of each next set depends on the $L^1$ function $f$ and on the previously chosen sets, not just on the action. This lets the proof force the ergodic averages over large boxes to sit near a chosen positive value at selected times, which defeats any prescribed slow rate.

What would settle it

Find one ergodic measure-preserving $\mathbb{Z}^n$-action and one sequence $a_k\to 0$ such that every $f\in L^1$ has $\limsup_{k\to\infty} |A_k f-\int f|/a_k < \infty$ almost everywhere; this directly contradicts the theorem's assertion that some $f$ fails to be asymptotically majorized by that sequence.

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Extended reading notes

Core claim

The paper's central claim is that no universal rate estimate for Birkhoff's ergodic theorem exists for ergodic $\mathbb{Z}^n$-actions. Given an ergodic measure-preserving action $(T^v)_{v\in\mathbb{Z}^n}$ on a probability space $(X,\mu)$ and any sequence $a_k\to 0$ decreasing to zero arbitrarily slowly, there exists $f\in L^1(\mu)$ such that $A_k f = \frac{1}{|Q_k|}\sum_{v\in Q_k} f\circ T^v$ converges to $\int f\,d\mu$ almost everywhere, yet the error $|A_k f-\int f|$ is not asymptotically majorized by $a_k$. The construction produces infinitely many long time intervals where the deviation is uniformly close to a positive constant, so the slow convergence is geometrically visible rather than merely a tail effect. This is the multiparameter generalization of the Krengel effect.

Load-bearing premise

The construction rests on a weakened Rokhlin lemma for ergodic $\mathbb{Z}^n$-actions: after the function and the earlier almost invariant sets are fixed, the next set of prescribed measure must still be available; if this adaptive lemma fails for an action, so does the proof.

Editorial extensions

If this is right

Every ergodic $\mathbb{Z}^n$-action has observables with arbitrarily slow pointwise convergence of Birkhoff averages.
No sequence $a_k\to 0$ can serve as a universal rate certificate for the multiparameter ergodic theorem, even among $L^1$ functions.
The Krengel effect, previously a statement about a single transformation, holds in full generality for $\mathbb{Z}^n$-actions.
The proof's deviations persist uniformly over arbitrarily long time intervals, so the slow convergence is not a rare-tail phenomenon.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the same adaptive Rokhlin lemma exists for actions of other amenable groups, the no-universal-rate property would extend beyond $\mathbb{Z}^n$; the paper does not address that case.
One could ask whether the slow-converging functions form a residual set in $L^1$; the paper proves existence, not genericity, so the question is open.
The adaptive set-selection mechanism might also apply to weighted or subsequence ergodic averages, but its scope beyond the present setting is unexplored.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read on Ryzhikov's note. The result, if correct, is a natural and worthwhile extension of Krengel's no-universal-rate theorem from Z-actions to Z^n-actions. The theorem statement is clean: for any null sequence, an L^1 function exists whose cube averages deviate slowly. The adaptive Rokhlin construction is the obvious route, and the abstract's description is consistent with how such proofs work.

The soft spot is that the paper's central lemma is stated only in words. The weakened Rokhlin lemma must deliver not just sets that are almost invariant under the n generators, but almost-invariance on average over the entire cube defining the ergodic averages. One-step invariance alone gives an error that grows with the l1 norm of the shift, and averaging over a cube of side N costs another factor of N. A tall tower fixes this, but the abstract gives no quantitative condition linking tower height to N. The stress-test note is right to flag this as the hinge; it is exactly where the multiparameter proof has to work. I don't think this is a fatal objection—the standard multidimensional Rokhlin lemma with large height plus a careful adaptive choice should supply the needed control—but it is a real gap in what the abstract tells us, and since our copy of the full text is unreadable, I cannot verify that the proof actually delivers it.

Minor issues: the abstract has a typo ('wanishing'), and the hypotheses about the action (free vs. non-free) are not stated. The non-free case is probably fine by quotienting, so that is not the real risk.

Given the result is a natural question and the proof strategy is credible, I think this deserves refereeing. The referee should be asked to check the quantitative form of the Rokhlin lemma. If that works, it's a publishable short note. I would not block it.

Would I cite it? Only after checking the proof; right now I'd treat it as plausible but unverified. Bring it to reading group? Maybe, if we want to work through the construction.

Referee Report

3 major / 3 minor

Summary. The paper claims that for any ergodic measure-preserving action of Z^n on a probability space and any positive sequence decreasing to zero arbitrarily slowly, there exists f in L^1 such that the standard ergodic averages over cubes converge almost everywhere to the spatial average of f, but at a rate that is not asymptotically majorized by the given sequence. The proof is said to use a weakened version of Rokhlin's lemma for ergodic Z^n-actions, producing a sequence of asymptotically almost invariant sets with prescribed measures, chosen adaptively. This would generalize the Krengel effect on the absence of universal convergence rates for Birkhoff averages to multiparameter actions. The abstract is readable and states the main claim clearly, but the supplied full text is a corrupted, unreadable encoding, so no proof details, lemma statements, or references can be inspected.

Significance. If the theorem is correct, it is a substantial contribution: it rules out any universal rate estimate for ergodic averages in multiparameter actions, extending the classical one-parameter Krengel result. The proposed adaptive Rokhlin construction is a natural and potentially powerful mechanism, and the abstract's emphasis on the adaptive choice of almost invariant sets is a noteworthy idea. However, the significance can only be assessed fully once the proof is readable and the weakened Rokhlin lemma is stated in a quantitative form. The paper does not currently provide verifiable machine-checked proofs or reproducible code, so the assessment rests entirely on the abstract's assertions.

major comments (3)

[Full text] The supplied full text is not readable: it consists of corrupted characters and cannot be parsed as mathematical prose, definitions, or proofs. The central theorem, the weakened Rokhlin lemma, all intermediate statements, and the bibliography are therefore unavailable for verification. This is a load-bearing issue, not a presentation nicety: the abstract alone does not establish the result. The authors should provide a readable manuscript, preferably as a properly encoded PDF or TeX source, before the paper can be refereed.
[Abstract] The weakened Rokhlin lemma is asserted but not stated with any quantitative content, and that quantitative content is exactly what the proof needs. To realize a deviation of size c at cube side N on a set E, the construction must control the average occupancy of E over the full cube [0,N)^n, i.e. small values of (1/N^n) * sum_{g in [0,N)^n} mu(E Delta T^g E). Almost invariance under the n generators alone controls only one-step errors; the triangle inequality gives an error growing like |g|_1 times the one-step error, which can grow like N when averaged over the cube. The abstract does not state tower heights, floor placement, or any quantitative trade-off that would overcome this. Without a precise statement of this lemma, the main construction is unverified at its hinge.
[Abstract] The theorem is stated for 'an ergodic action of the group Z^n on a probability space' without specifying whether the action is free. Classical Rokhlin-type lemmas for amenable group actions typically require freeness or aperiodicity, and the weakened version announced here may or may not hold for non-free actions. If the theorem is intended for all ergodic actions, the reduction to the free case (by quotienting out the subgroup that acts trivially) should be stated explicitly. If the lemma is only for free actions, the scope of the theorem is unclear. This is a correctness-risk concern, not an assertion that the statement is false.

minor comments (3)

[Abstract] The word 'wanishing' appears twice in the abstract and should be 'vanishing'.
[Abstract] The sentence 'We not only can achieve the specified deviations arbitrarily far, but the sequence of such deviations from the average can be realized as a sequence wanishing arbitrarily slowly' is grammatically unclear and should be rewritten; the relation between 'arbitrarily far' and 'arbitrarily slowly' needs precise formulation.
[Full text] The readable portion does not include a formal theorem statement with hypotheses and conclusion; once the text is readable, the authors should ensure the main theorem is stated in a numbered environment with all assumptions (e.g., probability space, ergodicity, sequence properties) explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper proves an existence theorem via a Rokhlin-type lemma, with no fitted parameters, predicted constants, or self-citation chain.

full rationale

The central claim is an existence statement: for an ergodic Z^n-action and any sufficiently slowly decreasing null sequence, there is an L^1 function whose ergodic averages fail to be asymptotically majorized by that sequence. The proof is announced as a direct construction using a weakened Rokhlin lemma that supplies asymptotically almost invariant sets with prescribed measures, chosen adaptively. Nothing in the abstract or in the readable portions of the manuscript indicates that the conclusion is fed back into the construction. The Rokhlin lemma is a standard measure-theoretic tool, and the adaptive dependence on the earlier sets and on the function is a feature of the inductive construction, not a hidden assumption of the target result. There are no fitted parameters, no empirically calibrated constants, and no reliance on a self-citation to justify the load-bearing step. The generalization of the Krengel effect is used as motivation, not as an input. Consequently, no specific reduction of the theorem to its own inputs can be exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard ergodic theory plus the availability of a weakened Rokhlin lemma for Z^n-actions. No free parameters are fitted and no new entities are postulated. The main unverified input is the exact formulation and proof of the Rokhlin-type lemma and its adaptive variant.

assumptions (3)

domain assumption The action of Z^n is measure-preserving and ergodic on a probability space.
This is the theorem's setting; all ergodic averages are taken with respect to such an action.
standard math Birkhoff's pointwise ergodic theorem holds for Z^n-actions, guaranteeing that ergodic averages converge almost everywhere to the spatial average.
The theorem concerns the rate of this convergence and presumes its existence; Birkhoff's theorem for amenable group actions is the classical background result.
domain assumption A weakened Rokhlin lemma for ergodic Z^n-actions produces asymptotically almost invariant sets with any prescribed measures, and the construction remains valid when each new set is chosen adaptively after previous choices.
This is the main tool named in the abstract; the proof of the main theorem stands or falls on this lemma and its adaptive variant.

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Cite this review

Pith. "Pith review of Slow convergence almost everywhere of ergodic averages." pith.science (2026). https://pith.science/paper/VD6DE4VG

@misc{pith2026250800463,
  author       = {Pith},
  title        = {Pith review of: Slow convergence almost everywhere of ergodic averages},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VD6DE4VG}},
  note         = {Machine review of arXiv:2508.00463}
}

abstract

For an ergodic action of the group $Z^n$ on a probability space and a given arbitrarily slowly decreasing to zero sequence, there exists an integrable function such that the standard ergodic time averages for it converge almost everywhere to the spatial average of the function at a rate that is not asymptotically majorized by this sequence. This generalizes the Krengel effect about the absence of universal estimates for the rate of convergence in Birkhoff's ergodic theorem. The proof uses a weakened version of Rokhlin's lemma for ergodic $Z^n$-actions. It ensures the existence of the required sequence of asymptotically almost invariant sets with given measures. A feature of the construction of such a sequence is that the choice of the next almost invariant set depends on the original function and on the choice of the previous invariant sets. A significant deviation of the ergodic averages from the mean of a positive function can be uniformly realized over extremely large time intervals. The deviation can be greater than a positive constant and differ little from it on arbitrary long time intervals. We not only can achieve the specified deviations arbitrarily far, but the sequence of such deviations from the average can be realized as a sequence wanishing arbitrarily slowly.

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Forward citations

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