{"id":"42b1802b-511c-4ab2-bdb2-66afc5080289","arxiv_id":"2411.13486","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For almost every point in a positive-measure set under an ergodic flow, there are arbitrarily late return times to the set with exactly zero integral of a zero-mean function.","lead":"This paper proposes a new recurrence theorem: in an ergodic flow, for almost every starting point in any positive-measure set, the flow returns to that set at times when the integrated function is exactly zero. The result extends classic theorems by Krygin, Atkinson, and Shneiberg to returns to arbitrary measurable sets.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem B's proof assumes without proof or citation that the cylindrical flow F_t is recurrent; this unproved recurrence is the load-bearing step, and Theorem A depends on Theorem B.","rationale":"The reader's weakest_assumption identifies precisely the most load-bearing point: the proof of Theorem B assumes cylindrical-flow recurrence without proof or citation. My reading of the manuscript confirms that this assumption is used essentially in the proof of the Lemma, and no reference is given for the real-valued flow version. The classical Krygin–Atkinson theorem cited in Section 1 covers integer-valued cocycles over automorphisms; for real-valued cocycles over flows, Atkinson-type results give recurrence up to ε, not exact local recurrence of the skew product. The proof's later argument depends only on the asserted positive-measure return to E, so if that assertion is unjustified the central proof is unsupported. I do not claim the theorem is false; the assumption may be a known consequence of Atkinson's theorem for real cocycles, and if so the paper merely needs a citation or a short proof. However, as presented, the gap is real and material. The additional reliance on the unpublished preprint [5] for the complete proof of Theorem A reinforces the conditional status: the main theorem is not fully proved in this paper. The reader's verdict of CONDITIONAL is therefore appropriate, and my stress-test does not change it.","tokens_in":8652,"tokens_out":12271,"duration_ms":156278,"concrete_test":"Check the unproved recurrence assertion against Atkinson's theorem for flows: state and prove, or locate in the literature, that for an ergodic flow T_t and zero-mean f∈L^1, for every positive-measure set U and every ε>0 there exists t>N with μ({x∈U: T_t x∈U, |σ(t,x)|<ε})>0. Then verify that this implies the needed return ¯μ(F_tE∩E)>0 for the specific set E in Section 2 by taking ε smaller than min_{x∈U} δ(x)/8 and applying the intermediate value theorem. If the derivation succeeds, the gap is a missing citation; if it fails, construct or identify a zero-mean integrable f on an ergodic special flow whose cylindrical flow is non-conservative, which would refute Theorem B.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 2, proof of Theorem B, after defining E = {(x,h): x∈U, f(x)>0, 0>h>−δ(x)/4}, the text states: 'Since the cilindrical flow Ft is recurrent, for any N there is t>N such that ¯μ(FtE∩E)>0.' No proof or reference is supplied for this recurrence in the setting of a real-valued integrable zero-mean function f over an ergodic flow. The Krygin–Atkinson theorem cited in Section 1 is for integer-valued cocycles over automorphisms, and Atkinson's theorem for real-valued cocycles gives only returns with |σ(t,x)|<ε, not the exact local recurrence of the skew product F_t needed here. This is not a cosmetic gap: everything after that line—selecting (Ttx,−h)∈FtE∩E, applying the intermediate value theorem to obtain Δ with ∫_0^Δ f(T_sT_tx)ds=h, and concluding σ(t+Δ,x)=0 with T_{t+Δ}x close to x—uses only the asserted positivity of ¯μ(FtE∩E). If F_t fails to be recurrent (or the specific set E fails to return), the Lemma and Theorem B collapse. Theorem A's proof then also collapses, since it invokes Theorem B, and its full proof is deferred to the unpublished preprint [5]. The recurrence assertion may be true and provable via Atkinson's real-cocycle theorem, but as written it is an unproved assumption that carries the central argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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].","tokens_in":9006,"tokens_out":6165,"duration_ms":65195,"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":[{"comment":"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":"Section 2, proof of Theorem B"},{"comment":"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.","section":"Section 2, proof of Theorem B"},{"comment":"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.","section":"Proof of Theorem A"}],"minor_comments":[{"comment":"There are several typos: 'cilindrical' should be 'cylindrical'; the Russian text contains 'bз теоремы' and 'расмотрим' which should be 'из теоремы' and 'рассмотрим'.","section":"Section 2"},{"comment":"The condition '0 > h > −δ(x)/4' is clearer when written as '−δ(x)/4 < h < 0'.","section":"Section 2, definition of E"},{"comment":"The abstract states the theorem for an 'ergodic flow', while Theorem A states 'special ergodic flow'; please make the hypotheses consistent.","section":"Abstract and Theorem A"},{"comment":"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":"Theorem D"},{"comment":"The phrase 'The bibliography [4] lists many works' should read 'The bibliography of [4] lists many works'.","section":"Section 3"},{"comment":"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.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central claim is plausible and the paper contains useful ideas, but the proof of Theorem B has an unproved recurrence assertion plus an algebraic error, and Theorem A is not proven in the manuscript. These are fixable in principle, but the revision needs substantial new content rather than local edits. The reliance on the author's own unpublished preprint [5] for the main theorem should also be addressed, either by including a full proof or by making the preprint available and clearly cited."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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').\n\nThe 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.\n\nTheorem 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.\n\nOverall, 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.\n\nRecommendation: send it to peer review with the caveat that the main proofs are incomplete.","headline":"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.","tokens_in":9455,"tokens_out":3904,"would_cite":false,"duration_ms":39877,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A10","37A25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["ergodic flow","zero-mean function","recurrence","cylindrical flow","special flow","zero integrals","cocycle","conservative cascade"],"falsifier":"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.","tokens_in":8494,"feed_emoji":"🔁","tokens_out":13552,"duration_ms":137729,"temperature":0.7,"pith_summary":"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.","feed_headline":"Return times with zero integral hit any positive-measure set","feed_subtitle":"An ergodic flow's trajectory returns to any set at times when the integral of f is zero","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original discrete-time cylindrical-cascade recurrence theorem that this paper extends to flows.","marker":"[1]"},{"why":"Independently establishes the same recurrence for ergodic automorphisms and zero-mean integer cocycles, forming the discrete base case.","marker":"[2]"},{"why":"Provides the prior flow theorem that integrals along almost every trajectory have infinitely many zero times, which Theorem B strengthens to simultaneous spatial recurrence.","marker":"[3]"},{"why":"Gives the compact-metric continuous-function predecessor whose hypotheses Theorem A relaxes to measurable functions and arbitrary positive-measure sets.","marker":"[4]"},{"why":"Contains the complete proof of Theorem A, which the present paper only outlines.","marker":"[5]"}],"fun_headline_variants":["Zero-integral returns hit any positive-measure set","Ergodic flow returns to sets at zero-integral times","Almost all points re-enter A with zero integral","Zero integral enforces recurrence in ergodic flows","Cylindrical flow forces zero-integral returns"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Zero-integral returns hit any positive-measure set","Ergodic flow returns to sets at zero-integral times","Almost all points re-enter A with zero integral","Zero integral enforces recurrence in ergodic flows","Cylindrical flow forces zero-integral returns"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000671,"raw_usage":{"total_tokens":3015,"prompt_tokens":863,"completion_tokens":2152,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":2075}},"tokens_in":479,"tokens_out":2152,"duration_ms":17561,"temperature":1.0,"reasoning_tokens":2075,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:22:05.007952+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Krygin, An example of a cylindrical cascade with ano malous metric properties, Moscow State University Bulletin, ser","cited_arxiv_id":null,"evidence_quote":"Supplies the original discrete-time cylindrical-cascade recurrence theorem that this paper extends to flows."},{"cited_title":"Shneiberg, Zeros of integrals along trajectories of ergodic systems, Funct","cited_arxiv_id":null,"evidence_quote":"Provides the prior flow theorem that integrals along almost every trajectory have infinitely many zero times, which Theorem B strengthens to simultaneous spatial recurrence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the compact-metric continuous-function predecessor whose hypotheses Theorem A relaxes to measurable functions and arbitrary positive-measure sets."},{"cited_title":"Ryzhikov, Recurrence of integral zeros on trajecto ries of ergodic flow","cited_arxiv_id":null,"evidence_quote":"Contains the complete proof of Theorem A, which the present paper only outlines."}],"review_version":1}