Around Krygin-Atkinson, Shneiberg theorems, the recurrence with zero integrals

Valery V. Ryzhikov

REVIEW 3 major objections 6 minor 23 references

Around Krygin-Atkinson, Shneiberg theorems, the recurrence with zero integrals

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For a zero-mean function on an ergodic flow, almost every point in any positive-measure set returns to that set at times when the accumulated integral is exactly zero.

desk verdict Plausible strengthening of Shneiberg's theorem, but the proof rests on an unproved recurrence assertion and a deferred preprint; the results are likely true but not established here. read the letter →

arxiv 2411.13486 v2 pith:3MBCZD3Y submitted 2024-11-20 math.DS

Valery V. Ryzhikov This is my paper · ORCID

classification math.DS MSC 37A1037A25

keywords ergodicflowzero-meanfunctionrecurrencecylindricalspecialzerointegralscocycleconservativecascade

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

The paper proposes a joint recurrence theorem for ergodic flows: if $T_t$ is an ergodic measure-preserving flow and $f$ has zero mean, then for every set $A$ of positive measure, almost every $x\in A$ with $f(x)\neq0$ has arbitrarily large times $t_k$ such that the accumulated integral $\int_0^{t_k}f(T_sx)\,ds$ is zero and the orbit point $T_{t_k}x$ returns to $A$. This matters because it fuses two separate recurrence phenomena—the vanishing of the integral along trajectories and the recurrence of orbits to a prescribed set—showing they can be witnessed at the same instants. The text proves the special-flow version in detail, strengthening it to returns arbitrarily close to the starting point, and reduces the general assertion to a complete proof contained in the author's preprint. It also records a conservativity theorem for infinite-measure cylindrical cascades and a criterion under which sublinear Birkhoff sums vanish infinitely often.

What carries the argument

The carrying mechanism is the cylindrical flow $F_t(x,r)=(T_tx,r+\sigma(t,x))$, together with the representation of a measurable flow as a special flow. In the special-flow model the phase space is the region under the graph of an integrable roof function, motion is vertical at unit speed, and the roof is identified with the base by an isomorphism. Recurrence of $F_t$ supplies, for a thin set $E$ built over points where $f(x)>0$, a time $t>N$ with $\bar\mu(F_tE\cap E)>0$. For a point in that intersection, the absolute continuity of the integral in $t$ yields an intermediate time $\Delta$ at which $\int_0^\Delta f(T_sT_tx)\,ds$ equals the required fiber displacement, so the total integral from $x$ up to $t+\Delta$ is zero while the spatial positions are close. This intermediate-value step is what converts recurrence of the extended flow into a zero-integral return.

What would settle it

Find an ergodic flow $T_t$ and a zero-mean integrable $f$ for which the cylindrical flow $F_t(x,r)=(T_tx,r+\sigma(t,x))$ is not recurrent; that would falsify the proof's load-bearing premise. Alternatively, exhibit a positive-measure set $A$ and a positive-measure subset of $\{x\in A:f(x)\neq0\}$ such that every sufficiently large $t$ with $T_tx\in A$ has $\int_0^t f(T_sx)\,ds\neq0$, which would contradict Theorem A directly.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the classical recurrence of the extended flow—the flow $F_t(x,r)=(T_tx,r+\sigma(t,x))$ that records the running integral—already contains enough information to force the integral's zero times to coincide with spatial returns. For a special ergodic flow (a suspension flow under a roof function), the paper proves Theorem B: for almost every $x$ with $f(x)\neq0$, there are times $t_k\to\infty$ with $\sigma(t_k,x)=0$ and $T_{t_k}x\to x$. The main assertion, Theorem A, claims the same synchronization with an arbitrary positive-measure set $A$: for almost every $x\in A$ with $f(x)\neq0$, some $t_k\to\infty$ satisfies $\sigma(t_k,x)=0$ and $T_{t_k}x\in A$. The proof of Theorem A is only outlined here and is completed in the cited preprint; what the text establishes directly is that the bad set of points without such returns has measure zero, once the cylindrical-flow recurrence is granted.

Load-bearing premise

The proof of the special-flow theorem assumes, without proof or citation, that the extended cylindrical flow is recurrent; if that recurrence ever fails for an ergodic flow with a zero-mean integrable function, the argument collapses, and the main theorem's complete proof is deferred to a preprint.

Editorial extensions

If this is right

For any positive-measure subset $A$ of a special ergodic flow's phase space, almost every $x\in A$ with $f(x)\neq0$ has arbitrarily large zero-integral return times to $A$.
For special ergodic flows and ergodic torus windings, the return can be made arbitrarily close to the starting point while the integral is zero.
Under the corollary's extra continuity and open-set assumptions, the same gives $T_{t_k}x\to x$ for almost every nonvanishing point of an ergodic flow on a metric space.
In the infinite-measure discrete setting, an ergodic automorphism with a zero-mean integer-valued function yields a conservative cylindrical cascade, so the cocycle returns to its starting level infinitely often.
When Birkhoff sums of an integer-valued function are sublinear in probability, the sums equal zero infinitely often with probability one.

Reading between the lines

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

Beyond the paper: since every measurable measure-preserving flow is isomorphic to a special flow, a complete proof of Theorem A would immediately cover all ergodic flows; the 'special' in the statement is a reduction, not a restriction.
Beyond the paper: the proof uses only positive-measure recurrence of the cylindrical flow, not mixing or any finer ergodic property, so the same synchronization should hold for any base flow for which the extended flow is recurrent, even without full ergodicity.
Beyond the paper: the lemma gives, for each $N$, a positive-measure set of starting points with a zero-integral return within a time horizon depending on $N$; estimating how the measure of that set decays with $N$ would yield quantitative return-rate information, a direction the paper leaves open.

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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

Two things you should know. First, the paper has a genuinely nice idea: strengthen Shneiberg's zero-integral theorem by adding the return of the trajectory to a positive-measure set A, and get rid of the compactness and continuity assumptions in Denisova's theorem. Second, the proof as written does not deliver. The supporting Theorem B assumes without proof that the cylindrical flow F_t is recurrent, and Theorem A's proof is deferred to the author's unpublished preprint [5]. The results may well be true, but this manuscript is not self-contained.

What is actually new: Theorem A (zero integral plus return to A for a special ergodic flow) and Theorem B (return to the point itself) are not stated in the cited literature. The infinite-measure version of Krygin-Atkinson (Theorem C) is a neat corollary via inducing, and Theorem D from Weiss is a useful remark. The exposition is mostly clear, apart from typos ('cilindrical', 'reccurence').

The soft spot is load-bearing. In the proof of Theorem B, after defining E, the text says 'Since the cilindrical flow Ft is recurrent, for any N there is t>N such that ¯µ(FtE∩E)>0.' No proof or reference. The cited Krygin-Atkinson theorem is for integer-valued cocycles over automorphisms; Atkinson's real-cocycle theorem gives only approximate returns |σ|<ε, not exact zeros. The rest of the proof—selecting (−h) and applying the intermediate value theorem—needs exactly the asserted exact recurrence. This is close to being the same statement as what the paper is trying to prove, so it cannot be assumed silently. Whether the cylindrical flow is recurrent in the required sense is nontrivial; it may be provable from known results, but it isn't shown here.

Theorem A's proof is only an outline ("using, for example, Theorem B and properties of absolutely continuous functions") and refers to the author's own preprint for the complete proof. For a main theorem, that is not enough.

Overall, the paper is short and the ideas are plausible. It deserves a serious referee, but the referee should push for a proof of the recurrence of F_t and for a complete proof of Theorem A. If those are supplied, the result is a solid, incremental contribution. As it stands, treat it as a research announcement.

Recommendation: send it to peer review with the caveat that the main proofs are incomplete.

Referee Report

3 major / 6 minor

Summary. The paper proposes Theorem A: for a special ergodic flow T_t on (X, μ), a zero-mean integrable function f, and a set A with μ(A) > 0, for almost every x ∈ A with f(x) ≠ 0 there is a sequence t_k → ∞ such that σ(t_k, x) = ∫_0^{t_k} f(T_s x) ds = 0 and T_{t_k} x ∈ A. The paper also proves Theorem B, a version with T_{t_k} x → x instead of return to A, and derives a corollary for metric spaces. Additional remarks concern an infinite-measure Krygin–Atkinson theorem (Theorem C), a result of Weiss (Theorem D), and applications to multiple mixing and the del Junco–Rudolph property. The proof of Theorem B is based on recurrence of the cylindrical flow F_t(x, r) = (T_t x, r + σ(t, x)), and the proof of Theorem A is only sketched, with the full proof deferred to the author's unpublished preprint [5].

Significance. If Theorem A is correct, it strengthens Shneiberg's classical zero-integral theorem by requiring the returning orbit points to lie in a prescribed positive-measure set A, and Theorem B gives a metric version for special flows. The paper is short and contains several attractive ideas, including a reduction of the infinite-measure Krygin–Atkinson theorem and connections to joinings and multiple mixing. However, the two load-bearing steps are not fully established in the manuscript: the recurrence of the real-valued cylindrical flow is asserted without proof or reference, and the proof of Theorem A is relegated to an unpublished preprint. In addition, the algebraic step in the proof of Theorem B appears to contain an error. The significance is therefore conditional on repairing these gaps.

major comments (3)

[Section 2, proof of Theorem B] After defining the set E, the proof states: 'Since the cilindrical flow Ft is recurrent, for any N there is t > N such that ¯μ(FtE ∩ E) > 0.' This recurrence is not proved or cited. The Krygin–Atkinson theorem cited in the introduction applies to integer-valued cocycles over automorphisms, and Atkinson's theorem for real-valued cocycles gives returns with |σ(t, x)| < ε rather than the exact recurrence of the skew product F_t needed here. Because the rest of the proof after this line depends entirely on the asserted positivity of ¯μ(F_t E ∩ E), this is a load-bearing gap. Please supply a proof or a precise reference for the recurrence of the cylindrical flow F_t for ergodic flows with zero-mean integrable f.
[Section 2, proof of Theorem B] The algebraic step following 'Let (T_t x, −h) ∈ F_t E ∩ E' does not yield σ(t + Δ, x) = 0. Since (T_t x, −h) ∈ F_t E, there exists (x, h') in E with h' + σ(t, x) = −h, so σ(t, x) = −h − h'. After choosing Δ with ∫_0^Δ f(T_s T_t x) ds = h, one obtains ∫_0^{t+Δ} f(T_s x) ds = σ(t, x) + h = −h' ≠ 0. The total integral is zero only if ∫_t^{t+Δ} f(T_s x) ds = h + h', not h. As written, the lemma and Theorem B do not follow from the displayed argument.
[Proof of Theorem A] The proof of Theorem A is not contained in the paper: it is deferred to the unpublished preprint [5], and the outline says only that 'using, for example, Theorem B and the properties of absolutely continuous functions, one can show that almost all points from A are good.' Since Theorem A is the central new claim, the manuscript should include a complete proof or a fully detailed derivation from Theorem B. In particular, the outline does not explain how the conclusion T_t x ∈ A is obtained from the metric closeness T_t x → x, nor how the condition t_1 ≥ 0 is handled. Reliance on an unpublished self-citation for the main theorem is not sufficient for a refereed journal.

minor comments (6)

[Section 2] There are several typos: 'cilindrical' should be 'cylindrical'; the Russian text contains 'bз теоремы' and 'расмотрим' which should be 'из теоремы' and 'рассмотрим'.
[Section 2, definition of E] The condition '0 > h > −δ(x)/4' is clearer when written as '−δ(x)/4 < h < 0'.
[Abstract and Theorem A] The abstract states the theorem for an 'ergodic flow', while Theorem A states 'special ergodic flow'; please make the hypotheses consistent.
[Theorem D] The Birkhoff sum notation in the statement of Theorem D is garbled; it should read ∑_{i=0}^{n-1} f(S^i x) or an equivalent, rather than the displayed '∑ n−1 0 f(T^i x)'.
[Section 3] The phrase 'The bibliography [4] lists many works' should read 'The bibliography of [4] lists many works'.
[References] Reference [5] is an unpublished preprint. If it is essential for the proof of Theorem A, please include the full proof in the paper or provide a publicly available version of the preprint.

Circularity Check

1 steps flagged · score 4.0 of 10

Theorem A's complete proof is deferred to the author's own unpublished preprint [5]; the in-paper outline depends on Theorem B, whose proof in turn assumes without proof or citation that the cylindrical flow is recurrent.

self citation load bearing [Section 2, Proof of Theorem A]
"Proof of Theorem A. The complete proof was given in [5], here we will briefly outline the main points."

The paper's headline assertion (Theorem A) is not proved in this manuscript: the complete proof is deferred to reference [5], an unpublished preprint by the same author. The in-text outline is explicitly only a sketch and invokes Theorem B, whose proof in this paper relies on an unproved recurrence assertion. Thus, as far as the text itself shows, the central theorem's justification reduces to a self-citation that is not independently verified. This is load-bearing self-citation rather than definitional circularity, so it contributes a moderate score.

full rationale

The main new claim, Theorem A, is not proven in the text: Section 2 states 'The complete proof was given in [5]', and [5] is V.V. Ryzhikov, 'Recurrence of integral zeros on trajectories of ergodic flow. Preprint (2024)' by the same author. This is a load-bearing self-citation for the headline result, and per the review rules an unpublished author-overlapping preprint is not independent support; it raises the circularity score to 4. The paper also contains a serious gap that is not, strictly speaking, circularity: the proof of Theorem B asserts 'Since the cilindrical flow Ft is recurrent...' with no proof or citation. For a real-valued zero-mean integrable f over an ergodic flow, conservativity/recurrence of the skew product is a nontrivial theorem (closely related to Atkinson's and Shneiberg's results) and cannot simply be assumed in a derivation of zero-integral recurrence. However, the assumed recurrence is not identical to the conclusion: the proof converts small integrals into exact zeros via the intermediate value theorem, so the step is an unproved input rather than a definitional equivalence. No parameter fitting, renaming, ansatz-by-citation, or imported uniqueness theorem occurs. The paper does contain independent content (the outline, Theorem C, Theorem D), so the circularity score is 4 rather than 6-8.

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

No free parameters or new entities. The main axiomatic load is the unproven recurrence of the cylindrical flow.

assumptions (4)

standard math Poincare recurrence and conservativeness of measure-preserving systems
Used to assert recurrence of the cylindrical flow and return of orbits to sets.
domain assumption Special flow representation theorem for measurable measure-preserving flows
The paper assumes every measurable flow is isomorphic to a special flow, used in Theorem B proof.
standard math Fubini's theorem and absolute continuity of indefinite integrals for integrable functions
Used to show Φ(t) is absolutely continuous and has derivative f(T_t x).
ad hoc to paper The cylindrical flow F_t is recurrent for zero-mean integrable f
Assumed without proof or citation in the proof of Theorem B; this is a gap.

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Cite this review

Pith. "Pith review of Around Krygin-Atkinson, Shneiberg theorems, the recurrence with zero integrals." pith.science (2026). https://pith.science/paper/3MBCZD3Y

@misc{pith2026241113486,
  author       = {Pith},
  title        = {Pith review of: Around Krygin-Atkinson, Shneiberg theorems, the recurrence with zero integrals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MBCZD3Y}},
  note         = {Machine review of arXiv:2411.13486}
}

abstract

We recall theorems by Krygin, Atkinson, Shneiberg and propose the following assertion. Let $T_t$ be an ergodic flow on $(X,\mu)$, let a function $f$ on $X$ have zero mean, and $\mu(A)>0$ for $A\subset X$. Then for almost all $x\in A$ with $f(x)\neq 0$ there exists a sequence $t_k\to\infty$ such that $\int_0^{t_k}f(T_sx)ds=0$ and $T_{t_k}x\in A$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 22 canonical work pages

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [5]

Ryzhikov, Recurrence of integral zeros on trajecto ries of ergodic ﬂow

V.V. Ryzhikov, Recurrence of integral zeros on trajecto ries of ergodic ﬂow. Preprint (2024)

work page 2024

[2] [1]

Krygin, An example of a cylindrical cascade with ano malous metric properties, Moscow State University Bulletin, ser

A.B. Krygin, An example of a cylindrical cascade with ano malous metric properties, Moscow State University Bulletin, ser. 1, 1975, no. 5, 26-32

work page 1975

[3] [3]

Shneiberg, Zeros of integrals along trajectories of ergodic systems, Funct

I.Ya. Shneiberg, Zeros of integrals along trajectories of ergodic systems, Funct. Anal. Appl., 19:2 (1985), 160-161

work page 1985

[4] [4]

N. V. Denisova, Recurrence of integrals of conditionall y periodic functions, Dokl. Math., 108:1 (2023), 316-319

work page 2023

[5] [6]

Moshchevitin, Recurrence of the integral of a smoot h three-frequency conditionally periodic function, Math

N.G. Moshchevitin, Recurrence of the integral of a smoot h three-frequency conditionally periodic function, Math. Notes, 58:5 (1995), 1187-1196 5

work page 1995

[6] [7]

Konyagin, Recurrence of the integral of an odd condi tionally periodic function, Math

S.V. Konyagin, Recurrence of the integral of an odd condi tionally periodic function, Math. Notes, 61:4 (1997), 473-479

work page 1997

[7] [8]

V. V. Kozlov, Methods of qualitative analysis in the dyna mics of a rigid body. Scientiﬁc Publishing Center Regular and Chaotic Dynamics, Izhevsk, 2000. 248 pp

work page 2000

[8] [9]

Kochergin, On the Growth of Birkhoﬀ Sums over a Rotat ion of the Circle, Math

A.V. Kochergin, On the Growth of Birkhoﬀ Sums over a Rotat ion of the Circle, Math. Notes, 113:6 (2023), 784-793

work page 2023