Slow convergence of Birkhoff ergodic averages

Valery V. Ryzhikov

REVIEW 4 major objections 5 minor 1 cited by

Slow convergence of Birkhoff ergodic averages

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For every ergodic Z^n action, Birkhoff averages can be forced to converge as slowly as a prescribed schedule demands.

desk verdict A plausible but unproved Z^n extension of Krengel's slow-convergence theorem; the main theorem is left as an exercise and even the Z-action proof has unverified quantitative steps. read the letter →

arxiv 2507.16740 v1 pith:3UYBXZA3 submitted 2025-07-22 math.DS

Valery V. Ryzhikov This is my paper · ORCID

classification math.DS MSC 37A0537A1537A3028D05

keywords BirkhoffergodicaveragesslowconvergenceZ^nactionsRokhlin-HalmoslemmaRokhlintowersL1functionscube

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 establishes that Birkhoff ergodic averages can be made to converge arbitrarily slowly, in a strong almost-everywhere sense, for every ergodic action of Z and for every ergodic action of Z^n. Given any positive summable sequence $a_i$ and any times $M_k$ tending to infinity, the author constructs times $N_k > M_k$ and an $L_1$ function $f$ such that, at each $N_k$, the average $A(x,N_k,f)$ misses the integral of $f$ by more than $a_k$ on a set of measure tending to 1. The mechanism is to zero the function on very tall Rokhlin towers: the integral drops by a fixed amount, while most short-time averages outside the tower are barely affected, producing the large deviation. For Z-actions the proof is carried out with the classical Rokhlin-Halmos lemma; for Z^n-actions the same argument is claimed to run with the Rokhlin-Z^n lemma, with details left as an exercise.

What carries the argument

The load-bearing device is the Rokhlin tower used as a zeroing set. For a tower $E = \bigcup_{i=1}^{h} T^i B$ of measure $\varepsilon$ with height $h$ much larger than the current time $N$, one replaces $f_0$ by $f_0 1_{X\setminus E}$. For nearly all $x$ outside $E$ the Birkhoff averages at time $N$ are nearly unchanged, because the orbit segment of length $N$ spends almost all its time outside $E$, while the integral of the function drops by about $\varepsilon \int f_0\,dm$; for most $x$ inside $E$ the new average is $0$, which is far from the new integral. Iterating this operation with rapidly growing times $N_k$ and towers $E_k$ whose union has small total measure gives the final function $f = f_0 1_C$ and the tail estimate $m(|A(x,N_k,f) - \int f\,dm| > \varepsilon_k/2) > 1 - 2\sum_{i\ge k}\delta_i$.

What would settle it

Work through the construction for one concrete ergodic system, say the doubling map with a smooth $f_0$, and compute, after zeroing on a tower of height $h_k \gg N_k$, the measure of points whose $N_k$-average is moved more than $a_k/10$ by the tower boundary. If that boundary-effect measure does not tend to 0, or if the integral shift $\int f_0\,dm - \int f_k\,dm$ is not at least a constant multiple of $\varepsilon_k$, the claimed estimate $1 - 2\sum_{i\ge k}\delta_i$ cannot hold.

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Extended reading notes

Core claim

The central claim is the following. For any ergodic automorphism $T$ of a probability space $(X,m)$, any nonnegative $f_0 \in L_1(X,m)$ with positive norm, and any sequence $a_i > 0$ with finite sum, there exist $N_k \to \infty$ and a measurable set $C$ of measure arbitrarily close to 1 such that $f = f_0 1_C$ satisfies $m(\{x: |A(x,N_k,f) - \int f\,dm| > a_k\}) \to 1$. The same statement holds for ergodic $\mathbb{Z}^n$-actions, with $A(x,N,f)$ the average over the cube $Q_N = \{1 \le z_1,\dots,z_n \le N\}$. This is proved by an inductive tower construction: one removes from the function's support a disjoint union of tall Rokhlin towers, chosen so that each removal changes the integral by an amount comparable to the tower's measure while leaving the $N_k$-averages essentially unchanged for most points outside the tower.

Load-bearing premise

The construction requires choosing, at each step, a tower with measure $\varepsilon_k$ big enough to shift the integral past $a_k$, height $h_k$ so large that the $N_k$-averages outside the tower are almost unchanged, and all towers together so small that their union has measure close to 0; the paper only writes $h_k \gg N_k$ and does not verify that these competing requirements are compatible, and for $\mathbb{Z}^n$ it assumes the corresponding Rokhlin lemma without proof.

Editorial extensions

If this is right

No universal rate function for Birkhoff averages exists: for any ergodic transformation and any prescribed summable error schedule $a_i$, some $L_1$ function has deviations larger than $a_k$ on a set of measure tending to 1.
The slow-down is achieved while keeping $f = f_0 1_C$, so the function is unchanged on a set of measure arbitrarily close to 1; the pathology comes from removing $f_0$ on a small, carefully placed set.
The method applies to multiparameter averages: for ergodic $\mathbb{Z}^n$ actions the result holds for cube averages and, as the paper notes, for rectangular and more general averaging shapes.
The deviation sets have measure approaching 1, not just positive measure; the convergence is slow in a strong almost-sure sense at the chosen times.

Reading between the lines

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

The proof is existential: it uses abstract Rokhlin towers rather than any particular system, so the same construction should in principle produce explicit $f$ for concrete systems once explicit towers are given; testing this on a rotation or shift would give a quantitative picture of how slow the convergence can be.
The text leaves the $\mathbb{Z}^n$ case as an exercise and relies on an unstated Rokhlin-$\mathbb{Z}^n$ lemma; a fully rigorous treatment would need to verify the simultaneous height and measure choices, and the claim about amenable groups is explicitly deferred to future work.
The tower method suggests that the slow-convergence phenomenon is not specific to $\mathbb{Z}$-actions but is a general feature of actions with a Rokhlin lemma, including amenable group actions; whether that generalization goes through depends on the same quantitative compatibility.
Because $f$ is constructed by zeroing $f_0$ on a small set, the result may be relevant to questions about the stability of ergodic averages under small perturbations of the observable.

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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

The short version: this note states a natural multiparameter extension of Krengel's theorem on arbitrarily slow Birkhoff convergence, but as written it does not prove it. Theorem 2 for Z^n-actions is explicitly deferred to an exercise, and even the Z-action proof has a few quantitative gaps. That said, the underlying tower idea is the right one and the gaps look fillable.

What is actually new: Krengel's 1978 result is for a single transformation. Theorem 2 extends the statement to ergodic Z^n-actions averaged over cubes. The proposed proof is a transfer of the author's earlier tower method from [3] and [4]. The conceptual explanation in Section 1—zero the function on a tall tower of height h >> N, so the integral drops but most N-averages barely change—is clear and is the right intuition for why no universal rate can exist.

What is missing: the paper's own 'details as an exercise' for Theorem 2 is a genuine omission. The transfer is not literally automatic; it needs a Z^n Rokhlin tower with a boundary estimate m({x in E_k : x + Q_{N_k} is contained in E_k}) at least (1 - n N_k/h_k) m(E_k), plus explicit control of the effect of later towers on the N_k-averages. Even in the Z-action case, the proof asserts rather than verifies the tower height choices. The first tower is taken of height exactly N_1, which is insufficient to ensure A(x,N_1,f_1)=0 on most of E_1; one needs a buffer so the orbit segment of length N_1 stays inside the tower for most points. Similarly, one needs uniform positivity of f_0 on the towers and tail bounds like N_k sum_{i>k} epsilon_i < delta_k. None of these appear. I suspect they can be arranged, but they are not arranged here, so the central claim rests on a sketch.

Theorem 1 is essentially Krengel's theorem with a new (sketch of a) proof. The citation pattern is fine: Krengel gets credit, the author's own technique-carrying papers are cited, and the Ornstein-Weiss reference for amenable groups is appropriate.

Who this is for: someone working on rates in ergodic theorems might use the Z^n statement, but they should first work out the proof. The note is readable in twenty minutes and the idea is transparent.

My recommendation: send it to a referee, but with a clear expectation that the proof be written out fully. If the author supplies the missing estimates, this could be a fine short paper in an ergodic theory venue. As is, it is a research announcement, not a complete paper.

Referee Report

4 major / 5 minor

Summary. The paper announces two theorems. Theorem 1 states that for an ergodic automorphism T of a probability space and any positive summable sequence a_k, one can find times N_k tending to infinity and a set C of measure arbitrarily close to 1 such that the Birkhoff averages of f = f0 1_C deviate from the integral of f by more than a_k on sets of measure tending to 1. Theorem 2 states the analogous result for cube Birkhoff averages in ergodic Z^n-actions. The proof for Z-actions uses Birkhoff's theorem, the Rokhlin-Halmos lemma, and an iterative zeroing of the initial function on towers; the Z^n case is asserted to follow by the same method and is deferred to an exercise. The manuscript contains the same text in English and in Russian.

Significance. If the statements are correct, the paper gives a short elementary proof of the slow convergence phenomenon for Birkhoff averages of ergodic Z-actions and extends it to all ergodic Z^n-actions. The construction is non-circular and uses only standard ergodic-theoretic tools, which is attractive. However, the central theorem for Z^n-actions is not proved in the manuscript, and the Z-action proof contains several quantitative gaps. As it stands, the paper is a proof idea rather than a verified result; completing the missing estimates would be a meaningful contribution.

major comments (4)

[§1, displayed definition of E1] The tower is defined as E1 = ⊔_{i=1}^{N1} T^i B1, but the argument requires a tower height h1 much larger than N1. For a point x in a level ℓ near the top of this tower, the iterates T^i x leave E1 before time N1, so the assertion that A(x,N1,f1) = 0 on E1 is false with height equal to N1. The tower height must be replaced by h1 ≫ N1, and the boundary estimate showing that the first N1 iterates of most points of E1 remain inside E1 must be supplied.
[§1, estimate involving C1 and ∫f0 1_{C1}] The inequality ∫ f0 1_{C1} dm ≈ m(C1)∫ f0 dm is not justified by the von Neumann ergodic theorem and almost invariance alone. An almost invariant set of large measure can carry an arbitrarily small portion of the mass of an unbounded L1 function if the tower is placed in a region where f0 is small. The construction needs to choose the tower base B1 inside the set where A(x,N1,f0) is close to ∫f0 dm and then prove a quantitative estimate such as ∫_{E1} f0 dm > (1-η) ε1 ∫ f0 dm. Without such an estimate, the claimed deviation of the new average from the new integral is unsupported.
[§1, final tail estimate] The displayed inequality m(|A(x,N_k,f) - ∫f dm| > ε_k/2) > 1 - 2∑_{i≥k}δ_i is asserted after the sentence 'We choose the height h_k ≫ N_k', but the transition from f_k to f requires explicit control of the contribution of all later towers E_i, i>k, to both A(x,N_k,f) and ∫f dm. One needs quantitative bounds such as ∫_{∪_{i>k}E_i} f0 dm ≤ δ_k^2 and a bound on the averaged contribution of f0 on that union at time N_k, together with a relation between ε_k and the prescribed a_k. No such simultaneous choice of the parameters ε_i, δ_i, h_i, N_i is given, so the final tail estimate is not derived.
[§2, Theorem 2] The central claim of the abstract is stated and then deferred with 'We leave the details as an exercise.' This is not an acceptable proof for the main theorem. The extension to Z^n is not automatic: cube averages do not reduce to averages along a Z-subaction, and the proof would require a quantitative Z^n Rokhlin lemma together with a boundary estimate for the cube tower E_k = ⊔_{z∈Q_{h_k}} T^z B_k, e.g. m({x∈E_k : x + Q_{N_k} ⊂ E_k}) ≥ (1 - n N_k/h_k)m(E_k). These details must be written out for the theorem to be considered proved.

minor comments (5)

[§1, first display] The definition of the Birkhoff average contains a typo: A(x,N,f) := 1/n ∑_{i=1}^N should be 1/N ∑_{i=1}^N.
[§1, definition of C] There is an extra closing parenthesis in the formula C = X \ (∪_{k=1}^∞ E_k); the parentheses should be balanced.
[§1, normalization of f0] The normalization ∥f0∥ = 2 is used nowhere; Theorem 1 only requires ∥f0∥ > 0, and the proof should say so explicitly.
[§1 and §2] The proof uses parameters ε_k but the theorems are stated with a_k; the relation between them, such as ε_k = 2a_k, should be made explicit.
[General] The manuscript contains a full Russian translation of the same material. If the paper is to be published, the duplicate text should be removed and the final version should appear in one language.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the tower construction is anchored in Birkhoff's theorem and Rokhlin-Halmos towers, and the target slow convergence is a verified conclusion, not an input.

full rationale

The derivation constructs f = f0 1_C from a fixed nonnegative f0, using Birkhoff's theorem to select N_k and Rokhlin-Halmos towers E_k to define C. The desired slow convergence appears only as the property to be checked for the resulting f, so the claim is not defined in terms of itself. No parameter is fitted to the target rates a_k, and no equation in the paper makes the conclusion true by construction. Self-citations [3] and [4] are invoked only as a source of technique ('modifying the approaches from [3], [4]'); the argument does not rest on an imported uniqueness theorem, and the main ingredients (Birkhoff, Rokhlin-Halmos) are standard external results. The passage 'We leave the details as an exercise' for the Z^n case and the unverified quantitative inequalities (e.g., h_k >> N_k and the tail bound 2 sum_{i>=k} delta_i) are proof-completeness and correctness concerns, not circularity: an omitted or under-verified derivation is not the same as a conclusion that reduces to its own assumptions. Accordingly no circular step is identified.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central theorem is parameter-free: it states existence for all summable a_k and all ergodic actions. The construction introduces auxiliary scales epsilon_k, delta_k, and h_k, but only with qualitative constraints. The heavy background consists of standard ergodic theorems. No new entities are postulated.

free parameters (1)

epsilon_k, delta_k, h_k = not numerical; chosen inductively
Auxiliary proof parameters controlling tower measures, acceptable errors, and tower heights. They must satisfy summable and largeness constraints but are not fitted to data and do not affect the universality of the theorem.

assumptions (4)

standard math Birkhoff pointwise ergodic theorem
Used in Section 1 to produce times N_k at which averages of the current function are close to its mean.
standard math Rokhlin-Halmos tower lemma, including its Z^n analogue
Builds the tall towers E_k on which f0 is zeroed; the Z^n analogue is invoked but not proved.
standard math Von Neumann mean ergodic theorem
Used to justify that removing a nearly invariant set changes the integral by approximately its measure times the mean.
domain assumption The chosen towers lie in a set where f0 is uniformly positive
Needed for the integral drop 0.9 epsilon_k integral f0; not explicitly justified in the text.

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Cite this review

Pith. "Pith review of Slow convergence of Birkhoff ergodic averages." pith.science (2026). https://pith.science/paper/3UYBXZA3

@misc{pith2026250716740,
  author       = {Pith},
  title        = {Pith review of: Slow convergence of Birkhoff ergodic averages},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3UYBXZA3}},
  note         = {Machine review of arXiv:2507.16740}
}

abstract

Let $\{T^z\}$ be an ergodic action of the group $Z^n$ by automorphisms of the probability space $(X,m)$, $\sum_{i}^\infty a_i<\infty$, $a_i>0$. For any sequence $M_k\to +\infty$ there exist $N_k>M_k$ and a function $ f\in L_1(X,m)$ such that $$m\left(\ x:\ \left|\, A(x,N_k,f) - \int f \, dm\, \right|\ >\ a_k \ \right)\ \to\ 1,$$ where $$A(x,N,f) =\frac 1 {N^n} \sum_{z\in Q_N} f(T^{z}x),$$ $$Q_N=\{(z_1,\dots,z_n)\}\, :\, 1\leq z_1,\dots,z_n\leq N\}.$$

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Forward citations

Cited by 1 Pith paper

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Exponential convergence can happen in weighted Birkhoff averages via quasi-periodicity with arbitrary nonresonance and low regularity
math.DS 2026-07 conditional novelty 7.0 of 10

For every nonresonant torus frequency and any σ<1, there exist normalized weights and an A_B low-regularity observable making weighted Birkhoff averages converge faster than exp(-cN^σ).

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · cited by 1 Pith paper

[3]

Рыжиков, Медленные сходимости эргодических средних, Матем

В.В. Рыжиков, Медленные сходимости эргодических средних, Матем. заметки, 113:5 (2023), 742-746; V. V. Ryzhikov, Slow Convergences of Ergodic Averages, Math. Notes, 113:5 (2023), 704-707 8

work page 2023
[4]

Рыжиков, Медленная сходимость взвешенных средних для потоков и дей- ствий счетных аменабельных групп

В.В. Рыжиков, Медленная сходимость взвешенных средних для потоков и дей- ствий счетных аменабельных групп. УМН, 80:4 (2025). V.V. Ryzhikov, Slow convergence of weighted means for flows and actions of countable amenable groups. Uspekhi Mat. Nauk, 80:4 (2025)

work page 2025
[1]

Krengel, On the speed of convergence in the ergodic theorem Monatsh

U. Krengel, On the speed of convergence in the ergodic theorem Monatsh. Math., 86 (1978), 3-6

work page 1978
[2]

Подвигин, О скоростях сходимости в эргодической теореме Биркгофа, Сиб

И.В. Подвигин, О скоростях сходимости в эргодической теореме Биркгофа, Сиб. матем. журн., 65:5 (2024), 991-1010; I.V. Podvigin, On convergence rates in the Birkhoff Ergodic Theorem, Siberian Math. J., 65:5 (2024), 1170-1186

work page 2024
[5]

D.S.Ornstein,B.Weiss,Entropyandisomorphismtheoremsforactionsofamenable groups. J. Anal. Math. 48 (1987), 1-141 9

work page 1987

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [3]

Рыжиков, Медленные сходимости эргодических средних, Матем

В.В. Рыжиков, Медленные сходимости эргодических средних, Матем. заметки, 113:5 (2023), 742-746; V. V. Ryzhikov, Slow Convergences of Ergodic Averages, Math. Notes, 113:5 (2023), 704-707 8

work page 2023

[2] [4]

Рыжиков, Медленная сходимость взвешенных средних для потоков и дей- ствий счетных аменабельных групп

В.В. Рыжиков, Медленная сходимость взвешенных средних для потоков и дей- ствий счетных аменабельных групп. УМН, 80:4 (2025). V.V. Ryzhikov, Slow convergence of weighted means for flows and actions of countable amenable groups. Uspekhi Mat. Nauk, 80:4 (2025)

work page 2025

[3] [1]

Krengel, On the speed of convergence in the ergodic theorem Monatsh

U. Krengel, On the speed of convergence in the ergodic theorem Monatsh. Math., 86 (1978), 3-6

work page 1978

[4] [2]

Подвигин, О скоростях сходимости в эргодической теореме Биркгофа, Сиб

И.В. Подвигин, О скоростях сходимости в эргодической теореме Биркгофа, Сиб. матем. журн., 65:5 (2024), 991-1010; I.V. Podvigin, On convergence rates in the Birkhoff Ergodic Theorem, Siberian Math. J., 65:5 (2024), 1170-1186

work page 2024

[5] [5]

D.S.Ornstein,B.Weiss,Entropyandisomorphismtheoremsforactionsofamenable groups. J. Anal. Math. 48 (1987), 1-141 9

work page 1987