{"id":"177f1746-0181-43d4-9f64-d2bbc4fee635","arxiv_id":"2607.06907","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Optimal uniform convergence rates for weighted Birkhoff averages on tori are established across finite-differentiability, C-infinity, logarithmic, and Gevrey regularity classes, with matching lower bounds showing no weighting function can do better in general.","lead":"This paper proves that weighted Birkhoff averages for toral translations converge at optimal polynomial or exponential rates determined by the regularity of the observable, for almost all rotations. It matters because it provides a sharp, nearly complete characterization of how fast these weighted averages can converge, replacing the slow O(N^{-1}) rate of classical ergodic theory.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. The inductive construction of {m_s} (conditions (a)-(c) in §5.1) is sketched but valid: exponential growth of {q_n} plus freedom to choose m_s far from m_{s-1} makes all three conditions simultaneously satisfiable.","rationale":"The reader identifies the right area — the inductive construction of {m_s} is the least-verified step and is crucial for the lower bound. However, the concern does not actually land: the construction is valid because (1) Borel-Bernstein provides infinitely many indices satisfying (a), (2) freedom in subsequence selection gives (b), and (3) the super-exponential decay of the weighting function's Fourier transform (from w ∈ W^β) dominates the at-most-exponential growth of ẽΔ, giving (c). The paper's one-sentence justification is brief but the underlying mathematics is sound. The reader's other concerns — gaps between upper and lower bounds in Cases (I) and (IV), and the d≥2 regime — are real limitations but are explicitly acknowledged by the authors (Remarks 3.4-3.5, Section 3.2 items (g), (s)-(u)). These gaps reflect limitations of the method, not errors in the argument. The paper makes a genuine contribution: first lower bounds for weighted Birkhoff averages, novel upper bound techniques (particularly the Salamon-inspired small divisor estimates in §5.3 and the Denjoy-Koksma application for d=1), and a transparent discussion of where optimality holds and where it is partial. The CONDITIONAL verdict is reasonable given the acknowledged gaps, but I would not lower it further, and the correctness risk is lower than 'unknown' — the key arguments I checked (Poisson summation application, S_1/S_2/S_3 decomposition, Denjoy-Koksma estimate in (5.22), continuous case adaptation) are all sound.","tokens_in":39327,"tokens_out":11715,"duration_ms":369710,"concrete_test":"Verify the inductive construction explicitly for the Gevrey case with α = 0.5, β = 2, d = 1: construct the first 5 terms of {m_s} for a specific ρ (e.g., the golden ratio conjugate, where partial quotients are all 1 and q_n are Fibonacci numbers), and check that conditions (a), (b), (c) are simultaneously satisfied. Note: for the golden ratio, q_{ν+1} > ν q_ν fails for all ν (since q_{ν+1}/q_ν → φ ≈ 1.618), so one should use a typical ρ with unbounded partial quotients. A concrete ρ with known partial quotients (e.g., ρ = [1,2,3,4,...]) would allow explicit verification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader correctly identifies the inductive construction of {m_s} in Section 5.1 as the least-detailed step in the lower bound proof. Conditions (a) q_{m_{s+1}} > m_s q_{m_s}, (b) |m_{s+1} - m_s| → ∞, and (c) a summability condition must hold simultaneously. The paper justifies this in one sentence: 'these requirements are indeed achievable, given that x^ε ≤ ẽΔ(x) ≤ exp(x^ζ) for some ε > 0 and 0 < ζ < β-1, and that q_n grows at least exponentially.' Upon verification, this claim holds. For condition (c), the dominant term is j = s-1, requiring exp(-C_1 (q_{m_s}/q_{m_{s-1}+1})^{β-1}) ≪ ẽΔ(q_{m_{s-1}})/ẽΔ(q_{m_s}). Since q_{m_s}/q_{m_{s-1}+1} ≥ 2^{(m_s - m_{s-1} - 1)/2} (exponential growth of {q_n}) and condition (b) lets us make this gap arbitrarily large, the left side decays super-exponentially in (m_s - m_{s-1}), while the right side decays at most as exp(q_{m_s}^ζ) with ζ < β-1. For almost every ρ, log q_n = O(n) (by the ergodic theorem on the Gauss map), so q_{m_s}^ζ = exp(O(ζ m_s)), which is dominated by the super-exponential decay when m_s - m_{s-1} is a positive fraction of m_s. The Borel-Bernstein theorem provides infinitely many valid indices, so such m_s can always be chosen. The sum over j < s-1 is controlled by even faster decay. The construction is valid; the sketch is brief but correct. The genuine limitations (gaps between upper and lower bounds in Cases (I) and (IV), the restriction α < β-1 for Gevrey lower bounds) are real but explicitly acknowledged in Remarks 3.4-3.5 and Section 3.2.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper studies weighted Birkhoff averages for toral translations, establishing both upper and lower bounds on uniform convergence rates for almost every rotation. The main results (Theorems 3.1, 3.2, and Corollary 3.1) address four regularity classes of observables—finitely differentiable, C^∞, logarithmic C^∞, and Gevrey—and demonstrate optimality of the rates in multiple aspects. The upper bounds are obtained via Fourier-analytic techniques (Poisson summation, integration by parts, the Denjoy-Koksma inequality for d=1, and a novel small-divisor estimate inspired by [Sal04] for d≥2). The lower bounds are established by constructing explicit counterexample observables using continued-fraction approximants of the rotation number. The paper also argues that no alternative weighting function can improve the rate in general.","tokens_in":40222,"tokens_out":1794,"duration_ms":209866,"significance":"The paper makes a substantial contribution to the quantitative theory of weighted ergodic averages. The simultaneous establishment of matching (or near-matching) upper and lower bounds across four regularity regimes, for almost every rotation and uniformly in the initial point, goes significantly beyond prior work (e.g., [DSSY17, DY18, TL24a]), which obtained only upper bounds under more restrictive conditions. The lower-bound construction (Theorem 3.1) is the first of its kind for weighted Birkhoff averages and reveals a clear connection between observable regularity and achievable convergence speed. The adaptation of the [Sal04] small-divisor counting technique to the weighted setting (§5.3, Case (I)-(ii)) is a noteworthy technical innovation. The results are parameter-free in the sense that the rates depend only on the regularity index of the observable and the Diophantine properties of a typical rotation.","major_comments":[{"comment":"§5.1, construction of the sequence {m_s}: The inductive selection of {m_s} satisfying conditions (a)–(c) simultaneously is the linchpin of the lower-bound proof. The justification given is a single sentence: 'these requirements are indeed achievable, given that x^ε ≤ ẽΔ(x) ≤ exp(x^ζ) for some ε > 0 and 0 < ζ < β−1, and that q_n grows at least exponentially.' While the argument appears correct upon verification (exponential growth of {q_n} plus the freedom to choose m_s far from m_{s−1} makes the super-exponential decay in condition (c) dominate the at-most-exponential growth of ẽΔ), this is a load-bearing step that deserves an explicit verification, not just a parenthetical assertion. The authors should add 2–3 sentences sketching the induction: how to choose m_s given m_{s−1}, why condition (c) is satisfiable for the dominant j = s−1 term, and why the sum over j < s−1 is controlled.","section":null},{"comment":"§5.3, Case (I)-(i), d=1 upper bound: The proof uses the Denjoy–Koksma inequality (Lemma 2.1) to obtain the key estimate (5.22): Σ_{k=1}^{q_n−1} ||⟨k,ρ⟩||_Z^{−ℓ'} = O(q_n^{ℓ'}). This step requires F to be of bounded variation, which is verified for the specific piecewise-defined F. However, the subsequent application of Abel's summation formula (5.23) to deduce convergence of the full series (5.21) requires controlling the partial sums Σ_{k=1}^{j} ||⟨k,ρ⟩||_Z^{−ℓ'} for all j, not just j = q_n − 1. The text addresses this by summing over dyadic blocks [q_{v−1}, q_v), but the estimate on each block uses (5.22) with q_n replaced by q_v, which is valid only when the block endpoint is an approximant denominator. For intermediate j (not of the form q_n − 1), the bound Σ_{k=1}^{j} ||⟨k,ρ⟩||_Z^{−ℓ'} = O(j^{ℓ'}) does not follow directly from (5.22). The authors should either justify this extension","section":null},{"comment":"§3.2, Item (h) and the concluding remark of §3.2: The paper claims that 'no alternative weighting function can yield a faster uniform rate' and that this is part of the optimality. However, the concluding paragraph of §3.2 explicitly states: 'we make no claim of optimality for arbitrary weighting functions, since the selection of alternative weighting functions may yield a further acceleration of convergence.' These two statements are in tension. The optimality in Item (h) is specifically about the regularity of w (i.e., using w ∈ C_0^∞ vs. w ∈ C_0^M with finite M), not about the choice of weighting function per se. The abstract and introduction should be revised to clarify that the unimprovability claim concerns the regularity of the observable dictating the rate (for a fixed class of weighting functions), not that no other weighting function can do better. As written, the abstract's ph","section":null}],"minor_comments":[{"comment":"The phrase 'twenty-one aspects' of optimality (§3.2) is unusual and slightly distracting. Consider replacing with a more standard description such as 'multiple aspects' or simply listing the categories.","section":null},{"comment":"§3.2, Item (h): The claim that the C_0^∞ condition is 'essential for rapid convergence' is supported by reference to [TL24b, TL25a] where counterexamples use Cesàro-weighted or multiple-weighted forms. A brief clarification that these are different averaging schemes would help the reader.","section":null},{"comment":"Remark 3.4 explains the asymmetry between upper and lower bound assumptions in Cases (I) and (IV), but the explanation is terse. A more explicit statement of why absolute convergence over the full lattice (upper bounds) requires stronger conditions than convergence over a subsequence (lower bounds) would help.","section":null},{"comment":"The notation ẽΔ (with a tilde over e and Delta) is unusual and can be hard to parse in display equations. Consider using a different letter entirely (e.g., Ψ or Φ for the approximation function governing regularity, reserving Δ for the nonresonance function).","section":null},{"comment":"In the continuous case of Theorem 3.1, the observable Φ is constructed on T^2 using a 2-dimensional rotation vector ρ = (ρ_1, ρ_2) with ρ_2/ρ_1 irrational. The connection between this 2-dimensional construction and the d-dimensional continuous case in Corollary 3.1 (which requires d ≥ 3) is not fully explained. A sentence clarifying why d ≥ 3 is needed would be helpful.","section":null},{"comment":"The paper states (§4) that some techniques 'potentially mak[e] their first appearance in weighted ergodic theory.' This is a reasonable claim but should be stated more precisely—specifically, the adaptation of [Sal04]'s near-resonance counting to the weighted Birkhoff setting is new in this context.","section":null},{"comment":"Several references are cited with future dates (e.g., [PR26], [TL26a], [TL26b], [BBC26], [Ryz25]). If these are accepted/in-press works, this should be noted; if preprints, the dates should reflect the actual availability.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is a strong contribution that would be appropriate for a serious dynamics or analysis journal. The main concern is the level of detail in the inductive construction (§5.1) and the Abel summation step (§5.3), both of which are correct but under-explained for load-bearing arguments. The tension between the abstract's unimprovability claim and the concluding remark of §3.2 should be resolved by careful rewording. I note that the authors have an extensive self-citation pattern ([TL24a, TL24b, TL25a, TL25b, TL26a, TL26b, Ton26]), which is understandable given that this paper synthesizes and extends a recent line of work, but the novelty relative to these prior papers should be clearly delineated (§4 attempts this but could be more explicit about which results are genuinely new vs. quantitative refinements)."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"This paper gives the first lower bound estimates for weighted Birkhoff averages on toral translations, then matches them with upper bounds across four regularity classes (finite differentiability, C∞, logarithmic C∞, Gevrey). The main structural result: the observable's regularity dictates the convergence rate, and no alternative weighting function can beat it in general. That's a clean, significant statement, and the lower bounds are genuinely new — prior work by Das-Yorke, Duignan-Meiss, and the authors themselves only established upper bounds. The proof structure is sound. Lower bounds come from explicit counterexamples built on continued fraction approximants; upper bounds use Fourier analysis, Poisson summation, integration by parts, and the Denjoy-Koksma inequality for d=1. The d=1 upper bound is finer than the higher-dimensional case and uses number-theoretic tools that hadn't been applied in this setting before. The adaptation of Salamon's KAM small-divisor estimates for the continuous d≥2 case is a nice technical move that relaxes the regularity requirement from ℓ>2d to ℓ>d. The stress-test note convinced me the inductive construction of {m_s} in Section 5.1 is valid, despite being sketched in one sentence. The exponential growth of {q_n} plus the freedom to choose m_s far from m_{s-1} makes conditions (a)–(c) simultaneously satisfiable. The reader's concern here is reasonable as a presentation issue but not a mathematical gap. The real soft spots are the acknowledged mismatches between upper and lower bound assumptions in Cases (I) and (IV) of Theorem 3.2. In Case (I)-(i) for d≥2, the upper bound needs ℓ>2d while the lower bound only needs ℓ>0, leaving an intermediate regime where optimality is unverified. In the Gevrey case, the restriction α<β−1 for lower bounds means the optimality claim doesn't cover all α. These are explicitly flagged in Remarks 3.4–3.5 and don't undermine the core contribution, but they do mean the 'optimal' label is sometimes weaker than it sounds. The self-citation pattern is heavy but legitimate — the upper bound machinery builds on the authors' prior work, and the lower bounds are new. This deserves a serious referee. The main things to check carefully are the small-divisor estimates in Section 5.3 (especially the cardinality bound (5.28) and the Abel summation in (5.29)–(5.32)) and whether the d=1 Denjoy-Koksma argument in (5.22) extends as claimed. Recommend full peer review.","headline":"First lower bounds for weighted Birkhoff averages, with matching upper bounds across four regularity classes. The core construction holds up; the gaps are real but acknowledged.","tokens_in":40275,"tokens_out":640,"would_cite":true,"duration_ms":86916,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Smoothness dictates convergence speed in weighted ergodic averages","keywords":[],"falsifier":"Construct a regularity class (an approximation function ẽΔ) for which the three inductive conditions (a), (b), (c) in Section 5.1 cannot be simultaneously satisfied, or exhibit a specific irrational rotation and observable in a stated regularity class for which the weighted average converges strictly faster than the claimed upper bound, contradicting the optimality assertion.","tokens_in":39513,"feed_emoji":"⚖️","tokens_out":1266,"duration_ms":189643,"temperature":0.7,"pith_summary":"When you average a function along the orbit of an irrational rotation on a torus, the standard Birkhoff average converges to the spatial mean at a painfully slow rate — at best O(N^{-1}), and often arbitrarily slower. The paper studies what happens when you replace the uniform weights 1/N by a smooth, compactly supported weighting function w (vanishing to high order at the endpoints of [0,1]). The central result is a complete, two-sided characterization: for almost every rotation vector, the convergence rate of the weighted average is dictated by the regularity of the observable — finitely differentiable functions get polynomial rates, C^∞ functions get arbitrary polynomial rates, logarithmic C^∞ functions get sub-exponential rates of the form exp(-c (log N)^λ), and Gevrey-class functions get stretched-exponential rates. Moreover, for each regularity class, the authors construct explicit counterexamples showing that no faster rate is achievable in general, and that no alternative weighting function can improve the rate either. The argument works by establishing sharp upper bounds through refined small-divisor estimates (using the Denjoy–Koksma inequality in dimension 1 and a near-resonance counting argument in higher dimensions) and matching lower bounds by constructing observables whose Fourier coefficients are concentrated at the denominators q_n of the rotation's continued-fraction approximants, exploiting the near-resonances that almost every rotation necessarily possesses.","feed_headline":"Smoothness sets the speed limit for weighted ergodic averages","feed_subtitle":"For almost every rotation, the regularity of the observable — not the choice of weighting function — determines the optimal convergence rate","key_machinery":"The proof rests on three pillars: (1) continued-fraction theory and the Borel–Bernstein theorem to identify, for almost every rotation ρ, infinitely many denominators q_n where the next approximant is anomalously large (q_{ν+1} > ν q_ν); (2) the Denjoy–Koksma inequality for bounded-variation functions on the circle, which yields a refined small-divisor sum estimate in dimension 1 that is unavailable in higher dimensions; and (3) a near-resonance counting argument (inspired by KAM-type techniques) that shows, in dimensions d ≥ 2, the set of lattice points k where ⟨k,ρ⟩ is genuinely small has controlled cardinality, so the sum over small divisors converges without requiring the full Diophantie","core_discovery":"The regularity of the observable is the sole determinant of the optimal convergence rate for weighted Birkhoff averages over almost all irrational rotations. The paper proves this by establishing, for four regularity classes (finite differentiability, C^∞, logarithmic C^∞, and Gevrey), that the upper bound on the convergence rate matches the lower bound up to sharp or nearly sharp constants, and that this bound cannot be improved by choosing a different weighting function. The mechanism is a duality: the upper bound uses the decay of the observable's Fourier coefficients against the small divisors inherent in almost every rotation, while the lower bound constructs an observable whose Fourier","pith_inferences":["The paper's claim that no alternative weighting function can yield a faster uniform rate in general leaves open the possibility that problem-specific (non-universal) weighting functions tailored to a particular observable's Fourier spectrum could achieve faster convergence — the authors announce a forthcoming paper addressing this question.","The restriction to 0 < α < β - 1 in the Gevrey lower bound (Theorem 3.1) means that for very smooth observables (large α) relative to the weighting function's decay parameter β, the lower bound may fail, suggesting that the interplay between observable regularity and weighting function regularity has a phase boundary that is not fully explored.","The near-resonance counting argument in dimensions d ≥ 2 could potentially be sharpened using deeper results from the geometry of numbers (e.g., Schmidt's subspace theorem), which might close the gap between upper and lower bounds in higher dimensions where the current results are optimal only up to constants."],"forward_implications":["The results provide a precise calibration tool for numerical simulations of quasiperiodic dynamical systems: given an observable of known regularity, one can predict the exact convergence rate of the weighted Birkhoff average and know that no better weighting function exists in general.","The dichotomy between dimension 1 (where the Denjoy–Koksma inequality gives sharper rates) and dimensions d ≥ 2 (where near-resonance counting is needed) suggests a fundamental difference in the arithmetic structure of small divisors across dimensions that may affect other cohomological-equation problems.","The optimality with respect to the weighting function implies that the popular Laskar weighting function is already near-optimal for general observables, and that further acceleration requires either higher regularity of the observable or problem-specific weighting tailored to a particular observable rather than universal weighting.","The extension to logarithmic C^∞ and Gevrey classes bridges the gap between the C^∞ and analytic regimes, providing a continuous spectrum of convergence rates that may inform KAM-type results where ultra-differentiable regularity plays a role."],"fun_headline_variants":["Regularity alone dictates convergence rates for weighted Birkhoff averages","Weighted ergodic averages converge exponentially for almost all rotations","Observable smoothness sets sharp convergence bounds over almost all rotations","No weighting function beats the regularity-bound rate for Birkhoff averages","Four regularity classes yield optimal convergence rates for weighted averages"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The lower-bound construction requires, for almost every rotation, the simultaneous satisfiability of three conditions on an inductively chosen subsequence of continued-fraction denominators: that the denominators grow sufficiently fast, that the gaps between selected indices diverge, and that a certain summability condition involving the approximation function and the weighting function's derivative decay holds. If these three conditions cannot be met simultaneously for some类","fun_headline_variants_meta":{"raw":{"variants":["Regularity alone dictates convergence rates for weighted Birkhoff averages","Weighted ergodic averages converge exponentially for almost all rotations","Observable smoothness sets sharp convergence bounds over almost all rotations","No weighting function beats the regularity-bound rate for Birkhoff averages","Four regularity classes yield optimal convergence rates for weighted averages"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":603,"prompt_tokens":535,"completion_tokens":68,"prompt_tokens_details":null},"tokens_in":535,"tokens_out":68,"duration_ms":17646,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T23:10:39.510583+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Construct a regularity class (an approximation function ẽΔ) for which the three inductive conditions (a), (b), (c) in Section 5.1 cannot be simultaneously satisfied, or exhibit a specific irrational rotation and observable in a stated regularity class for which the weighted average converges strictly faster than the claimed upper bound, contradicting the optimality assertion.","supporting_citations":[],"review_version":1}