{"id":"f9f33d1e-611e-4af7-8a9f-e66e28ece553","arxiv_id":"2508.00463","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"For every ergodic Z^n-action and every sequence tending slowly to zero, some integrable function has ergodic averages that do not converge faster than that sequence almost everywhere.","lead":"This paper proves that for every ergodic action of the group Z^n on a probability space, one can find an integrable function whose ergodic averages converge almost everywhere to the spatial average with any prescribed slow rate, or even slower. It extends the classical Krengel effect, which says Birkhoff averages have no universal convergence speed, to multidimensional actions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unstated weakened Rokhlin lemma must supply cube-wide almost invariance, not just one-step generator invariance; the abstract gives no quantitative guarantee, so the main construction is unverified at its hinge.","rationale":"The abstract is the only legible part of the submission; the garbled full text contains no checkable proof. The central claim is a natural multiparameter analogue of Krengel's theorem and is plausible; no contradiction with known results suggests it is false. However, the proof rests on an unstated lemma whose quantitative strength is load-bearing. In particular, one-step almost invariance is not the same as control of ergodic averages over cubes of side N; the lemma must produce sets whose occupancy along every orbit over the whole cube is close to the prescribed measure. This is a concrete, verifiable condition, not a mere gap in exposition. The reader's weakest assumption pointed to the same proof ingredient, though framed around freeness and adaptivity; I regard the quantitative cube-wide condition as the sharper formulation. Since the full text cannot be inspected, the appropriate disposition remains UNVERDICTED; my concern does not move the verdict, so I mark UNCHANGED. A successful read of the actual lemma statement, plus a check that it supplies the N-uniform occupancy control, would settle the objection.","tokens_in":2781,"tokens_out":23010,"duration_ms":257545,"concrete_test":"Obtain an uncorrupted copy of the paper and extract the exact statement of the weakened Rokhlin lemma. Check whether, for the side length N_k used at step k and a prescribed measure delta, the lemma guarantees a set E satisfying (1/N_k^n) * sum_{g in [0,N_k)^n} mu(E Delta T^g E) < epsilon_k, equivalently that E is built from a tower of height H much larger than N_k/epsilon_k with a low-discrepancy choice of floors. If the lemma only gives mu(E Delta T_i E) < epsilon for the n generators, run the construction with E a half-tower of height H approximately N_k: then the cube average of 1_E over side N_k is near 1/2 rather than near 1, so the claimed uniform deviation over E fails. If the lemma as stated has the cube-wide form, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that for any null sequence there exists f in L^1 whose cube averages deviate from the spatial mean at a rate not majorized by the sequence. The abstract reduces the whole proof to a 'weakened version of Rokhlin's lemma' producing asymptotically almost invariant sets with prescribed measures, chosen adaptively. The precise quantitative form of this lemma is not stated, and that is exactly where the proof must carry the multiparameter weight. To realize a deviation of size c at cube side N on a set E, the construction needs control of 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 only controls one-step moves; by the triangle inequality the error for a single shift g can be |g|_1 times the one-step error, and the average over the cube can grow like N. A tower of height H much larger than N repairs this, but that is a quantitative condition on H and on the placement of floors, precisely the 'adaptive' feature. Neither this condition nor the lemma's hypotheses are visible in the abstract. The reader's worry about non-free actions is secondary: a non-free ergodic Z^n action can be reduced by quotienting out the subgroup that acts trivially, so freeness is not the real risk. The real risk is that the announced lemma is not strong enough to control average occupancy over the cubes defining the ergodic averages. Because the full text is unreadable in the provided file, this hinge cannot be checked.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":3097,"tokens_out":2666,"duration_ms":27793,"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":[{"comment":"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.","section":"Full text"},{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Abstract"}],"minor_comments":[{"comment":"The word 'wanishing' appears twice in the abstract and should be 'vanishing'.","section":"Abstract"},{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Full text"}],"recommendation":"major_revision","confidential_remarks":"The submission file appears to be corrupted at the source: the full text is mojibake and no proof is inspectable. Before sending the paper to another referee, the editor should ask the authors to resubmit a readable PDF or source file. The abstract's claim is interesting and plausible, but the missing quantitative statement of the weakened Rokhlin lemma is a serious gap that must be addressed in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nMinor 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.\n\nGiven 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.\n\nWould 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.","headline":"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.","tokens_in":3593,"tokens_out":2631,"would_cite":false,"duration_ms":26482,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A30","37A05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Ergodic averages on any Z^n action can be made to converge arbitrarily slowly, so no universal rate exists for Birkhoff's theorem","keywords":["ergodic theory","Birkhoff ergodic theorem","rate of convergence","Krengel effect","Z^n actions","Rokhlin lemma","multiparameter ergodic theorem","L^1 functions"],"falsifier":"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.","tokens_in":2606,"feed_emoji":"⏳","tokens_out":11284,"duration_ms":104376,"temperature":0.7,"pith_summary":"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.","feed_headline":"Ergodic averages can converge arbitrarily slowly","feed_subtitle":"For any ergodic Z^n action and any prescribed slow rate, some integrable function converges no faster than it.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[],"fun_headline_variants":["No universal rate bound for ergodic averages","Multiparameter ergodic averages can converge arbitrarily slowly","Birkhoff averages in Z^n: no worst-case convergence speed","For any slow rate, some ergodic function converges no faster","Convergence of ergodic averages has no worst-case rate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["No universal rate bound for ergodic averages","Multiparameter ergodic averages can converge arbitrarily slowly","Birkhoff averages in Z^n: no worst-case convergence speed","For any slow rate, some ergodic function converges no faster","Convergence of ergodic averages has no worst-case rate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001224,"raw_usage":{"total_tokens":5029,"prompt_tokens":942,"completion_tokens":4087,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":4005}},"tokens_in":558,"tokens_out":4087,"duration_ms":28757,"temperature":1.0,"reasoning_tokens":4005,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:07:09.516892+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}