{"id":"a6eb1f31-deca-4387-a435-dd13cfaa073b","arxiv_id":"2507.16740","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Slow convergence of Birkhoff averages is realized for ergodic Z-actions by a Rokhlin tower construction and claimed, without proof, for Z^n-actions.","lead":"This note constructs integrable functions whose Birkhoff averages at selected times stay far from their mean, with the bad set occupying almost the whole space. It matters because it shows that no universal rate of convergence can be guaranteed for ergodic averages, even for actions of Z^n, though the Z^n proof is only sketched.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2 is not proved: the Z^n case is deferred to an exercise, and the Z-action induction leaves the quantitative tower estimates unverified, so the central claim rests on missing derivations.","rationale":"The reader's weakest assumption identifies exactly the load-bearing gap: the inductive tower construction requires simultaneous quantitative control that the manuscript never states or proves, and Theorem 2 is left as an exercise. My review confirms this. I do not claim the theorem is false; for Z-actions it is Krengel's known result, and a repaired tower construction may well go through. But under the stated reviewing rules, a central theorem whose proof is deferred cannot count as established. The printed Z-action proof also contains a concrete inconsistency at the first step if the tower height is literally N_1, and the final tail estimate is asserted without deriving the effect of later towers. These are gaps in the argument, not external disagreements. Therefore the reader's REJECT verdict is appropriate, and my stress-test does not move it.","tokens_in":5359,"tokens_out":17364,"duration_ms":192352,"concrete_test":"Complete the missing induction for Theorem 2 with explicit recursion rules: choose ε_k, δ_k, h_k, N_k and prove, for every k, (1) Birkhoff closeness m(|A(·,N_k,f_{k-1}) - ∫f_{k-1}| < a_k/10) > 1-δ_k; (2) the tower interior condition m({x∈E_k : the Q_{N_k}-cube at x lies in E_k}) ≥ (1 - nN_k/h_k)m(E_k); (3) the tail perturbation bound N_k∑_{i>k}ε_i ≤ δ_k; (4) the integral perturbation bound ∫_{∪_{i>k}E_i} f_0 dm ≤ a_k/4, with ∑ε_k < η and ∑δ_k < ∞. If all four can be satisfied simultaneously with m(X\\∪E_k) arbitrarily close to 1, the proof works; if any constraint cannot be met, the claimed transfer fails. As a sanity check for the Z-version, recompute the first stage with literal height N_1: for x in the top N_1 levels of E_1, the average A(x,N_1,f_1) is not zero, contradicting the displayed assertion unless the tower height is corrected to h_1 ≫ N_1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The manuscript's central claim, Theorem 2, is stated and then dismissed with \"We leave the details as an exercise.\" That alone leaves the main result without a derivation. The transfer from Z to Z^n is not automatic as written: the proof would require a quantitative Z^n Rokhlin-tower construction in which the cube averaging set Q_{N_k} stays inside the tower E_k = ⊔_{z∈Q_{h_k}} T^z B_k for most points of E_k. This needs a boundary estimate of the form m({x∈E_k : x + Q_{N_k} ⊂ E_k}) ≥ (1 - nN_k/h_k) m(E_k), and the paper does not state or prove it. Even in the Z-action proof of §1, the displayed first step takes E_1 = ⊔_{i=1}^{N_1} T^iB_1 and asserts A(x,N_1,f_1)=0 on most of E_1; this requires tower height h_1 ≫ N_1, not the printed height N_1. The final estimate m(|A(x,N_k,f)-∫f|>ε_k/2) > 1 - 2∑_{i≥k}δ_i is also asserted without deriving how the later towers affect A(x,N_k,f) and ∫f; one would need explicit bounds such as N_k∑_{i>k}ε_i ≤ δ_k and ∫_{∪_{i>k}E_i} f_0 dm ≤ δ_k. None of these inequalities appear, so the construction is not verified even for Z-actions, and for Z^n it is entirely omitted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":5648,"tokens_out":14391,"duration_ms":166191,"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":[{"comment":"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.","section":"§1, displayed definition of E1"},{"comment":"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.","section":"§1, estimate involving C1 and ∫f0 1_{C1}"},{"comment":"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.","section":"§1, final tail estimate"},{"comment":"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.","section":"§2, Theorem 2"}],"minor_comments":[{"comment":"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.","section":"§1, first display"},{"comment":"There is an extra closing parenthesis in the formula C = X \\ (∪_{k=1}^∞ E_k); the parentheses should be balanced.","section":"§1, definition of C"},{"comment":"The normalization ∥f0∥ = 2 is used nowhere; Theorem 1 only requires ∥f0∥ > 0, and the proof should say so explicitly.","section":"§1, normalization of f0"},{"comment":"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.","section":"§1 and §2"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The advertised main theorem is currently unproved, and the Z-action proof has load-bearing quantitative gaps. I would be willing to see a revised version in which the author supplies a complete proof of Theorem 2 and verifies the tower estimates in Theorem 1. If those details are not supplied, the paper should not be accepted. The presence of the full Russian duplicate suggests a preliminary draft rather than a final submission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nWhat 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.\n\nTheorem 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.\n\nWho 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.\n\nMy 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.","headline":"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.","tokens_in":6199,"tokens_out":4519,"would_cite":false,"duration_ms":47736,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A05","37A15","37A30","28D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every ergodic Z^n action, Birkhoff averages can be forced to converge as slowly as a prescribed schedule demands.","keywords":["Birkhoff ergodic averages","slow convergence","ergodic Z^n actions","Rokhlin-Halmos lemma","Rokhlin towers","L1 functions","cube averages"],"falsifier":"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.","tokens_in":5131,"feed_emoji":"📉","tokens_out":7182,"duration_ms":79611,"temperature":0.7,"pith_summary":"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.","feed_headline":"Birkhoff averages can be forced to converge as slowly as you like","feed_subtitle":"A tower construction zeros the function on small sets, making average errors exceed any prescribed threshold almost surely.","key_machinery":"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 $m(|A(x,N_k,f) - \\int f\\,dm| > \\varepsilon_k/2) > 1 - 2\\sum_{i\\ge k}\\delta_i$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the baseline result that for ergodic transformations slow convergence is possible via suitable functions, which the present paper reproves by an explicit tower method.","marker":"[1]"},{"why":"Provides the slow-convergence approach for ergodic averages that the paper modifies for Z-actions.","marker":"[3]"},{"why":"Extends the slow-convergence picture to weighted means and amenable group actions, the framework adapted here.","marker":"[4]"}],"fun_headline_variants":["Birkhoff averages can be made to converge arbitrarily slowly","Slow Birkhoff convergence is always possible for ergodic actions","Tower construction forces Birkhoff averages to lag any rate","Ergodic averages: arbitrary slow convergence via tower method","For any ergodic action, Birkhoff averages can be arbitrarily slow"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Birkhoff averages can be made to converge arbitrarily slowly","Slow Birkhoff convergence is always possible for ergodic actions","Tower construction forces Birkhoff averages to lag any rate","Ergodic averages: arbitrary slow convergence via tower method","For any ergodic action, Birkhoff averages can be arbitrarily slow"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1372,"prompt_tokens":923,"completion_tokens":449,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":366}},"tokens_in":539,"tokens_out":449,"duration_ms":5558,"temperature":1.0,"reasoning_tokens":366,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:04:24.444716+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Krengel, On the speed of convergence in the ergodic theorem Monatsh","cited_arxiv_id":null,"evidence_quote":"Supplies the baseline result that for ergodic transformations slow convergence is possible via suitable functions, which the present paper reproves by an explicit tower method."},{"cited_title":"Рыжиков, Медленные сходимости эргодических средних, Матем","cited_arxiv_id":null,"evidence_quote":"Provides the slow-convergence approach for ergodic averages that the paper modifies for Z-actions."},{"cited_title":"Рыжиков, Медленная сходимость взвешенных средних для потоков и дей- ствий счетных аменабельных групп","cited_arxiv_id":null,"evidence_quote":"Extends the slow-convergence picture to weighted means and amenable group actions, the framework adapted here."}],"review_version":1}