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Optimal rates of uniform convergence for weighted Birkhoff averages via almost all rotations

T0 review · 3 major / 7 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Smoothness dictates convergence speed in weighted ergodic averages

desk verdict 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. read the letter →

arxiv 2607.06907 v1 pith:VFQFHSVY submitted 2026-07-08 math.DS

Zhicheng Tong , Yong Li This is my paper · ORCID
classification math.DS
keywords convergenceaveragesweightedbirkhoffoptimalrateratesuniform
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The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

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.

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 (3)
  1. §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.
  2. §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
  3. §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
minor comments (7)
  1. 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.
  2. §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.
  3. 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.
  4. 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).
  5. 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.
  6. 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.
  7. 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained with parameter-free constructions and independent upper/lower bound arguments.

full rationale

This is a pure mathematics paper establishing optimal convergence rates for weighted Birkhoff averages. The derivation chain has two independent branches that meet to establish optimality, neither of which reduces to the other by construction. (1) The upper bound estimates (Theorem 3.2, proved in Section 5.3) derive convergence rates from the nonresonance condition (2.5)/(2.6), the Fourier coefficient decay encoded in the Banach space R^l_{ẽΔ}(T^d) (Definition 2.3), and the derivative bounds on the weighting function w ∈ W^β (Definition 2.4). These inputs are stated independently of the target result. The key technical steps—Poisson summation (5.7), the Denjoy-Koksma inequality (5.22), Abel summation (5.23), and the near-resonance counting argument adapted from [Sal04] (5.28)—are standard tools applied to the paper's own framework, not fitted parameters renamed as predictions. (2) The lower bound estimates (Theorem 3.1, proved in Section 5.1) construct explicit counterexample observables Ψ(θ) = Re Σ exp(2πi q_{m_j} θ)/ẽΔ(q_{m_j}) using continued fraction approximants of ρ. The convergence rate of the weighted average for these observables is then computed directly via decomposition into S_1, S_2, S_3 terms (5.1), with S_2 shown to dominate (5.11). The lower bound is not defined in terms of the upper bound; it is an independent computation on an explicitly constructed observable. While the paper heavily cites the authors' own prior work [TL24a, TL25b, TL26a, TL26b], these citations provide technical tools (quantitative derivative bounds, integrability frameworks) rather than uniqueness theorems that would forbid alternatives. The optimality claim—that no alternative weighting function can yield a faster rate—rests on the lower bound construction in Theorem 3.1, which works for any w ∈ W^β (and even w ∈ C^M_0([0,1]) as shown in the proof of Case (I)-(i)), not on a self-cited uniqueness result. The gap between upper and lower bounds in certain cases (e.g., Gevrey Case (IV) where ξ < α(αβ+d)^{-1} vs. lower bound O(exp(-N^α))) is explicitly acknowledged in Remark 3.4, confirming the bounds are not tautologically matched. No step in the derivation chain reduces to its inputs by definition, fitting, or self-citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

This is a pure mathematics paper with no free parameters, no invented entities, and no ad-hoc axioms. All background results are standard (Borel-Bernstein, Denjoy-Koksma, Poisson summation) or verified properties of specific functions (Laskar's weighting function). The Banach space R^l_{ẽΔ}(T^d) is a standard construction in harmonic analysis.

assumptions (6)
  • standard math Borel-Bernstein theorem: for a.e. ρ, infinitely many ν satisfy q_{ν+1} > ν q_ν
    Used in Section 5.1 to establish the existence of the sequence {m_s}. Standard result in metric Diophantine approximation.
  • standard math Denjoy-Koksma inequality for functions of bounded variation on T
    Lemma 2.1, used critically in the d=1 upper bound proof (Section 5.3, equation 5.22) to bound sums of small divisors.
  • standard math Poisson summation formula for compactly supported C∞ functions
    Used in Section 5.1 to convert discrete sums to integrals in the estimation of S1.
  • standard math For a.e. ρ∈R^d, the nonresonance condition holds with Δ(x)~x^d(log x)^2 (discrete) or x^{d-1}(log x)^2 (continuous)
    Remark 2.1, used throughout to specify the nonresonance condition for universal rotations.
  • domain assumption The weighting function w∈W^β satisfies ∥D^m w∥_{L^1} ≤ C̄ λ^m m^{mβ}
    Definition 2.4, Condition (2). This is a quantitative property of Laskar's function verified in [TL25b] and is load-bearing for all upper bound estimates.
  • domain assumption The observable class R^l_{ẽΔ}(T^d) is well-defined (Fourier coefficients absolutely summable)
    Definition 2.3. Required for the Fourier analysis framework; imposes conditions like ℓ>d for eΔ(x)~x^ℓ.

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Pith. "Pith review of Optimal rates of uniform convergence for weighted Birkhoff averages via almost all rotations." pith.science (2026). https://pith.science/paper/VFQFHSVY

@misc{pith2026260706907,
  author       = {Pith},
  title        = {Pith review of: Optimal rates of uniform convergence for weighted Birkhoff averages via almost all rotations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VFQFHSVY}},
  note         = {Machine review of arXiv:2607.06907}
}
abstract

In this paper, we investigate weighted Birkhoff averages for toral translations associated with compactly supported weighting functions. By introducing several new analytical techniques, we establish optimal uniform convergence rates for almost all rotations and specific (or even all) initial points. Unlike the $\mathcal{O}(N^{-1})$ rate best achieved in classical ergodic theory, we show that these weighted averages exhibit polynomial or even exponential convergence. We establish the optimality of these convergence rates in multiple aspects, particularly concerning regularity indices across four distinct cases: finite differentiability, the $C^\infty$ class, logarithmic $C^\infty$ classes, and Gevrey classes. Our results demonstrate that the regularity of the observable essentially dictates the convergence rate; furthermore, we prove that in general settings, no alternative weighting function can yield a faster uniform rate. In contrast to the generically slow convergence of standard time averages, this work provides an optimal and nearly complete characterization of rapid convergence for weighted Birkhoff averages.

Figures

Figures reproduced from arXiv: 2607.06907 by the authors.

Figure 1
Figure 1. Geometric properties (for d = 1) and the commutative diagram. 3.2 On the optimality of Theorem 3.2: Multiple aspects The present section is devoted to a detailed elaboration on the optimality of Theorem 3.2. The optimality manifests itself in twenty-one aspects, including rotations, initial points, regularity as￾sumptions, and the comparison of indices for the upper and lower bounds of the convergence rates, among o… view at source ↗

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Works this paper leans on

77 extracted references · 77 canonical work pages · cited by 1 Pith paper

  1. [1]

    Aaronson, M

    J. Aaronson, M. Lema\' n czyk, C. Mauduit, H. Nakada, Koksma's inequality and group extensions of Kronecker transformations, Algorithms, fractals, and dynamics (Okayama/Kyoto, 1992). pp. 27--50, Plenum, New York, 1995. https://doi.org/10.1007/978-1-4613-0321-3_2

    work page doi:10.1007/978-1-4613-0321-3_2 1992

  2. [2]

    Algom, Z

    A. Algom, Z. Wang, Arbitrarily slow decay in the M \" o bius disjointness conjecture. Ergodic Theory Dynam. Systems 43 (2023), pp. 2863--2880. https://doi.org/10.1017/etds.2022.61

    work page doi:10.1017/etds.2022.61 2023

  3. [3]

    Baroni, R

    R. Baroni, R. de Carvalho, J. Junior, I. Caldas, Shearless barriers in the conservative Ikeda map. Phys. Rev. E 112 (2025), Paper No. 054212, pp. 11. https://doi.org/10.1103/q3hf-jm99

    work page doi:10.1103/q3hf-jm99 2025

  4. [4]

    Bergelson, J

    V. Bergelson, J. Moreira, F. Richter, Single and multiple recurrence along non-polynomial sequences. Adv. Math. 368 (2020), 107146, pp. 69. https://doi.org/10.1016/j.aim.2020.107146

    work page doi:10.1016/j.aim.2020.107146 2020

  5. [5]

    Biasco, J

    L. Biasco, J. Massetti, M. Procesi, An abstract Birkhoff normal form theorem and exponential type stability of the 1d NLS. Comm. Math. Phys. 375 (2020), pp. 2089--2153. https://doi.org/10.1007/s00220-019-03618-x

    work page doi:10.1007/s00220-019-03618-x 2020

  6. [6]

    Blessing, J

    D. Blessing, J. Mireles James, Weighted Birkhoff averages and the parameterization method. SIAM J. Appl. Dyn. Syst. 23 (2024), pp. 1766--1804. https://doi.org/10.1137/23M1579546

    work page doi:10.1137/23m1579546 2024

  7. [7]

    Bogachev, Non-uniform Kozlov-Treschev averagings in the ergodic theorem

    V. Bogachev, Non-uniform Kozlov-Treschev averagings in the ergodic theorem. (Russian) Uspekhi Mat. Nauk 75 (2020), pp. 3--36; translation in Russian Math. Surveys 75 (2020), pp. 393--425. https://doi.org/10.1070/RM9940

    work page doi:10.1070/rm9940 2020

  8. [8]

    Bounemoura, J

    A. Bounemoura, J. F\' e joz, K AM , - G evrey regularity and the - B runo- R \" u ssmann condition. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), pp. 1225--1279. https://doi.org/10.2422/2036-2145.201707_009

    work page doi:10.2422/2036-2145.201707_009 2019

Show all 77 references
  1. [9]

    Bounemoura, J

    A. Bounemoura, J. F\' e joz, Hamiltonian perturbation theory for ultra-differentiable functions. Mem. Amer. Math. Soc. 270 (2021), pp. v+89. https://doi.org/10.1090/memo/1319

    2021 doi

  2. [10]

    Bourgain, On invariant tori of full dimension for 1D periodic NLS

    J. Bourgain, On invariant tori of full dimension for 1D periodic NLS. J. Funct. Anal. 229 (2005), pp. 62--94. https://doi.org/10.1016/j.jfa.2004.10.019

    2005 doi

  3. [11]

    Bou-Sakr-El-Tayar, J

    M. Bou-Sakr-El-Tayar, J. Bramburger, M. Colbrook, Weighted Birkhoff averages accelerate data-driven methods. Proc. A 482 (2026), Paper No. 20250979, pp. 19. https://doi.org/10.1098/rspa.2025.0979

    2026 doi

  4. [12]

    Bourgain, On the correlation of the Moebius function with rank-one systems

    J. Bourgain, On the correlation of the Moebius function with rank-one systems. J. Anal. Math. 120 (2013), pp. 105--130. https://doi.org/10.1007/s11854-013-0016-z

    2013 doi

  5. [13]

    Bourgain, P

    J. Bourgain, P. Sarnak, T. Ziegler, Disjointness of Moebius from horocycle flows, From Fourier analysis and number theory to Radon transforms and geometry. pp. 67--83, Dev. Math., 28, Springer, New York, 2013. https://doi.org/10.1007/978-1-4614-4075-8_5

    2013 doi

  6. [14]

    Calleja, A

    R. Calleja, A. Celletti, J. Gimeno, R. de la Llave, Accurate computations up to breakdown of quasi-periodic attractors in the dissipative spin-orbit problem. J. Nonlinear Sci. 34 (2024), Paper No. 12, pp. 38. https://doi.org/10.1007/s00332-023-09988-w

    2024 doi

  7. [15]

    S. Das, Y. Saiki, E. Sander, J. Yorke, Quantitative quasiperiodicity. Nonlinearity 30 (2017), pp. 4111--4140. https://doi.org/10.1088/1361-6544/aa84c2

    2017 doi

  8. [16]

    S. Das, J. Yorke, Super convergence of ergodic averages for quasiperiodic orbits. Nonlinearity 31 (2018), pp. 491--501. https://doi.org/10.1088/1361-6544/aa99a0

    2018 doi

  9. [17]

    de Faveri, M\" o bius disjointness for C^ 1+ skew products

    A. de Faveri, M\" o bius disjointness for C^ 1+ skew products. Int. Math. Res. Not. IMRN 2022, pp. 2513--2531. https://doi.org/10.1093/imrn/rnaa185

    2022 doi

  10. [18]

    del Junco, J

    A. del Junco, J. Rosenblatt, Counterexamples in ergodic theory and number theory. Math. Ann. 245 (1979), pp. 185--197. https://doi.org/10.1007/BF01673506

    1979 doi

  11. [19]

    Dolgopyat, B

    D. Dolgopyat, B. Fayad, Deviations of ergodic sums for toral translations I. Convex bodies. Geom. Funct. Anal. 24 (2014), pp. 85--115. https://doi.org/10.1007/s00039-014-0254-y

    2014 doi

  12. [20]

    Dolgopyat, B

    D. Dolgopyat, B. Fayad, Limit theorems for toral translations, Hyperbolic dynamics, fluctuations and large deviations. pp. 227--277, Proc. Sympos. Pure Math., 89, Amer. Math. Soc., Providence, RI, 2015. https://doi.org/10.1090/pspum/089/01492

    2015 doi

  13. [21]

    Dolgopyat, B

    D. Dolgopyat, B. Fayad, Deviations of ergodic sums for toral translations II. Boxes. Publ. Math. Inst. Hautes \' E tudes Sci. 132 (2020), pp. 293--352. https://doi.org/10.1007/s10240-020-00120-2

    2020 doi

  14. [22]

    Duignan, J

    N. Duignan, J. Meiss, Distinguishing between regular and chaotic orbits of flows by the weighted Birkhoff average. Phys. D 449 (2023), pp. 16. https://doi.org/10.1016/j.physd.2023.133749

    2023 doi

  15. [23]

    Eliasson, Discrete one-dimensional quasi-periodic Schr\"odinger operators with pure point spectrum

    L. Eliasson, Discrete one-dimensional quasi-periodic Schr\"odinger operators with pure point spectrum. Acta Math. 179 (1997), pp. 153--196. https://doi.org/10.1007/BF02392742

    1997 doi

  16. [24]

    Fan, Weighted Birkhoff ergodic theorem with oscillating weights

    A. Fan, Weighted Birkhoff ergodic theorem with oscillating weights. Ergodic Theory Dynam. Systems 39 (2019), pp. 1275--1289. https://doi.org/10.1017/etds.2017.81

    2019 doi

  17. [25]

    Fan, Multifractal analysis of weighted ergodic averages

    A. Fan, Multifractal analysis of weighted ergodic averages. Adv. Math. 377 (2021), pp. 34. https://doi.org/10.1016/j.aim.2020.107488

    2021 doi

  18. [26]

    A. Fan, Y. Jiang, Oscillating sequences, MMA and MMLS flows and Sarnak's conjecture. Ergodic Theory Dynam. Systems 38 (2018), pp. 1709--1744. https://doi.org/10.1017/etds.2016.121

    2018 doi

  19. [27]

    Frantzikinakis, A multidimensional Szemer\' e di theorem for Hardy sequences of different growth

    N. Frantzikinakis, A multidimensional Szemer\' e di theorem for Hardy sequences of different growth. Trans. Amer. Math. Soc. 367 (2015), pp. 5653--5692. https://doi.org/10.1090/S0002-9947-2014-06275-2

    2015 doi

  20. [28]

    Frantzikinakis, Furstenberg systems of Hardy field sequences and applications

    N. Frantzikinakis, Furstenberg systems of Hardy field sequences and applications. J. Anal. Math. 147 (2022), pp. 333--372. https://doi.org/10.1007/s11854-022-0221-8

    2022 doi

  21. [29]

    Frantzikinakis, B

    N. Frantzikinakis, B. Host, The logarithmic Sarnak conjecture for ergodic weights. Ann. of Math. (2) 187 (2018), pp. 869--931. https://doi.org/10.4007/annals.2018.187.3.6

    2018 doi

  22. [30]

    Frantzikinakis, B

    N. Frantzikinakis, B. Host, Weighted multiple ergodic averages and correlation sequences. Ergodic Theory Dynam. Systems 38 (2018), pp. 81--142. https://doi.org/10.1017/etds.2016.19

    2018 doi

  23. [31]

    Grafakos, Classical Fourier analysis

    L. Grafakos, Classical Fourier analysis. Third edition. Graduate Texts in Mathematics, 249. Springer, New York, 2014. pp. xviii+638. https://doi.org/10.1007/978-1-4939-1194-3

    2014 doi

  24. [32]

    Green, T

    B. Green, T. Tao, The M \" o bius function is strongly orthogonal to nilsequences. Ann. of Math. (2) 175 (2012), pp. 541--566. https://doi.org/10.4007/annals.2012.175.2.3

    2012 doi

  25. [33]

    a rcalc\

    G. Hardy, Orders of infinity. T he I nfinit\" a rcalc\" u l of P aul du B ois- R eymond, Hafner, New York, 1971. pp. v+62

    1971

  26. [34]

    Herman, Sur la conjugaison diff\' e rentiable des diff\' e omorphismes du cercle \`a des rotations

    M. Herman, Sur la conjugaison diff\' e rentiable des diff\' e omorphismes du cercle \`a des rotations. Inst. Hautes \' E tudes Sci. Publ. Math. No. 49 (1979), pp. 5--233. http://www.numdam.org/item?id=PMIHES_1979__49__5_0

    1979

  27. [35]

    Huang, Z

    W. Huang, Z. Wang, X. Ye, Measure complexity and M\"obius disjointness. Adv. Math. 347 (2019), pp. 827--858. https://doi.org/10.1016/j.aim.2019.03.007

    2019 doi

  28. [36]

    Kachurovski , Rates of convergence in ergodic theorems

    A. Kachurovski , Rates of convergence in ergodic theorems. Uspekhi Mat. Nauk 51 (1996), (310), pp. 73--124; translation in Russian Math. Surveys 51 (1996), pp. 653--703. https://doi.org/10.1070/RM1996v051n04ABEH002964

    1996 doi

  29. [37]

    Kachurovski , I

    A. Kachurovski , I. Podvigin, Estimates of the rate of convergence in the von Neumann and Birkhoff ergodic theorems. Trans. Moscow Math. Soc. 2016, pp. 1--53. https://doi.org/10.1090/mosc/256

    2016 doi

  30. [38]

    Katok, Combinatorial constructions in ergodic theory and dynamics

    A. Katok, Combinatorial constructions in ergodic theory and dynamics. University Lecture Series, 30. American Mathematical Society, Providence, RI, 2003. pp. iv+121. https://doi.org/10.1090/ulect/030

    2003 doi

  31. [39]

    Katok, B

    A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995. pp. xviii+802. https://doi.org/10.10...

    1995 doi

  32. [40]

    Katznelson, An introduction to harmonic analysis

    Y. Katznelson, An introduction to harmonic analysis. Third edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2004, pp. xviii+314. https://doi.org/10.1017/CBO9781139165372

    2004 doi

  33. [41]

    Klein, X

    S. Klein, X. Liu, A. Melo, Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps. Ergodic Theory Dynam. Systems 41 (2021), pp. 3363--3388. https://doi.org/10.1017/etds.2020.99

    2021 doi

  34. [42]

    Kozlov, D

    V. Kozlov, D. Treschev, On new forms of the ergodic theorem. J. Dynam. Control Systems 9 (2003), pp. 449--453. https://doi.org/10.1023/A:1024677322911

    2003 doi

  35. [43]

    Krengel, On the speed of convergence in the ergodic theorem

    U. Krengel, On the speed of convergence in the ergodic theorem. Monatsh. Math. 86 (1978/79), pp. 3--6. https://doi.org/10.1007/BF01300052

    1978 doi

  36. [44]

    Laskar, Frequency analysis for multi-dimensional systems

    J. Laskar, Frequency analysis for multi-dimensional systems. Global dynamics and diffusion. Phys. D 67 (1993), pp. 257--281. https://doi.org/10.1016/0167-2789(93)90210-R

    1993 doi

  37. [45]

    Laskar, Frequency analysis of a dynamical system

    J. Laskar, Frequency analysis of a dynamical system. Qualitative and quantitative behaviour of planetary systems (Ramsau, 1992). Celestial Mech. Dynam. Astronom. 56 (1993), pp. 191--196. https://doi.org/10.1007/BF00699731

    1992 doi

  38. [46]

    Laskar, Introduction to frequency map analysis

    J. Laskar, Introduction to frequency map analysis. Hamiltonian systems with three or more degrees of freedom ( S ' A gar\' o , 1995), pp. 134--150, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, 1999. http://doi.org/10.1007/978-94-011-4673-9_13

    1995 doi

  39. [47]

    Lema\' n czyk, C

    M. Lema\' n czyk, C. Mauduit, Ergodicity of a class of cocycles over irrational rotations. J. London Math. Soc. (2) 49 (1994), pp. 124--132. https://doi.org/10.1112/jlms/49.1.124

    1994 doi

  40. [48]

    J. Liu, P. Sarnak, The M \" o bius function and distal flows. Duke Math. J. 164 (2015), pp. 1353--1399. https://doi.org/10.1215/00127094-2916213

    2015 doi

  41. [49]

    Meiss, E

    J. Meiss, E. Sander, Birkhoff averages and the breakdown of invariant tori in volume-preserving maps. Phys. D 428 (2021), Paper No. 133048, pp. 20. https://doi.org/10.1016/j.physd.2021.133048

    2021 doi

  42. [50]

    Meiss, E

    J. Meiss, E. Sander, Resonance and weak chaos in quasiperiodically-forced circle maps. Commun. Nonlinear Sci. Numer. Simul. 142 (2025), Paper No. 108562, pp 20. https://doi.org/10.1016/j.cnsns.2024.108562

    2025 doi

  43. [51]

    Montalto, M

    R. Montalto, M. Procesi, Linear S chr\" o dinger equation with an almost periodic potential. SIAM J. Math. Anal. 53 (2021), pp. 386--434. https://doi.org/10.1137/20M1320742

    2021 doi

  44. [52]

    Podvigin, On possible estimates of the rate of pointwise convergence in the Birkhoff ergodic theorem

    I. Podvigin, On possible estimates of the rate of pointwise convergence in the Birkhoff ergodic theorem. Translation of Sibirsk. Mat. Zh. 63 (2022), pp. 379--391. Sib. Math. J. 63 (2022), pp. 316--325. https://doi.org/10.1134/s0037446622020094

    2022 doi

  45. [53]

    Podvigin, On convergence rates in the Birkhoff ergodic theorem

    I. Podvigin, On convergence rates in the Birkhoff ergodic theorem. Translation of Sibirsk. Mat. Zh. 65 (2024), pp. 991--1010. Sib. Math. J. 65 (2024), pp. 1170--1186. https://doi.org/10.1134/S0037446624050161

    2024 doi

  46. [54]

    Podvigin, V

    I. Podvigin, V. Ryzhikov, The full diapason of convergence rates of Birkhoff averages for ergodic flows. Preprint, 2026. https://doi.org/10.48550/arXiv.2601.21404

    2026 doi

  47. [55]

    Popov, KAM theorem for Gevrey Hamiltonians

    G. Popov, KAM theorem for Gevrey Hamiltonians. Ergodic Theory Dynam. Systems 24 (2004), pp. 1753--1786. https://doi.org/10.1017/S0143385704000458

    2004 doi

  48. [56]

    Queff\' e lec, M

    H. Queff\' e lec, M. Queff\' e lec, Diophantine approximation and Dirichlet series. Harish-Chandra Research Institute Lecture Notes, 2. Hindustan Book Agency, New Delhi, 2013. pp. xii+232

    2013

  49. [57]

    Richter, Uniform distribution in nilmanifolds along functions from a Hardy field

    F. Richter, Uniform distribution in nilmanifolds along functions from a Hardy field. J. Anal. Math. 149 (2023), pp. 421--483. https://doi.org/10.1007/s11854-022-0253-0

    2023 doi

  50. [58]

    M. Ruth, D. Bindel, Finding Birkhoff averages via adaptive filtering. Chaos 34 (2024), Paper No. 123109, pp. 20. https://doi.org/10.1063/5.0215396

    2024 doi

  51. [59]

    M. Ruth, D. Bindel, Level set learning for Poincar\'e plots of symplectic maps. SIAM J. Appl. Dyn. Syst. 24 (2025), pp. 611--632. https://doi.org/10.1137/23M1622179

    2025 doi

  52. [60]

    R\"ussmann, Note on sums containing small divisors

    H. R\"ussmann, Note on sums containing small divisors. Comm. Pure Appl. Math. 29 (1976), pp. 755--758. https://doi.org/10.1002/cpa.3160290615

    1976 doi

  53. [61]

    Ryzhikov, Slow convergences of ergodic averages

    V. Ryzhikov, Slow convergences of ergodic averages. Mat. Zametki 113 (2023), pp. 742--746; translation in Math. Notes 113 (2023), pp. 704--707. https://doi.org/10.1134/S0001434623050103

    2023 doi

  54. [62]

    Ryzhikov, Slow convergence of weighted averages for flows and actions of countable amenable groups

    V. Ryzhikov, Slow convergence of weighted averages for flows and actions of countable amenable groups. (Russian) Uspekhi Mat. Nauk 80 (2025), pp. 179--180; translation in Russian Math. Surveys 80 (2025), pp. 915--918. https://doi.org/10.4213/rm10251e

    2025 doi

  55. [63]

    Salamon, The Kolmogorov-Arnold-Moser theorem

    D. Salamon, The Kolmogorov-Arnold-Moser theorem. Math. Phys. Electron. J. 10 (2004), Paper 3, 37 pp

    2004

  56. [64]

    Sander, J

    E. Sander, J. Meiss, Birkhoff averages and rotational invariant circles for area-preserving maps. Phys. D 411 (2020), 132569, pp. 19. https://doi.org/10.1016/j.physd.2020.132569

    2020 doi

  57. [65]

    Sander, J

    E. Sander, J. Meiss, Computing Lyapunov exponents using weighted Birkhoff averages. J. Phys. A 58 (2025), Paper No. 355701, pp. 17. https://dx.doi.org/10.1088/1751-8121/adfac1

    2025 doi

  58. [66]

    Sander, J

    E. Sander, J. Meiss, Proportions of incommensurate, resonant, and chaotic orbits for torus maps. Chaos 35 (2025), Paper No. 013147, pp. 15. https://doi.org/10.1063/5.0226617

    2025 doi

  59. [67]

    Tong, Exponential convergence can happen in weighted Birkhoff averages via quasi-periodicity with arbitrarily weak nonresonance and low regularity

    Z. Tong, Exponential convergence can happen in weighted Birkhoff averages via quasi-periodicity with arbitrarily weak nonresonance and low regularity. Revised, 2026

    2026

  60. [68]

    Z. Tong, Y. Li, Exponential convergence of the weighted Birkhoff average. J. Math. Pures Appl. (9) 188 (2024), pp. 470--492. https://doi.org/10.1016/j.matpur.2024.06.003

    2024 doi

  61. [69]

    Z. Tong, Y. Li, Weighted multiple ergodic averages via analytic observables over T ^ : Is exponential pointwise convergence universal? Preprint, 2024. https://doi.org/10.48550/arXiv.2405.02866

    2024 doi

  62. [70]

    Z. Tong, Y. Li, A note on exponential convergence of Ces\` a ro weighted Birkhoff average and multimodal weighted approach. Acta Math. Sin. (Engl. Ser.) 41 (2025), pp. 2301--2323. https://doi.org/10.1007/s10114-025-3365-5

    2025 doi

  63. [71]

    Z. Tong, Y. Li, Quantitative uniform exponential acceleration of averages along decaying waves. Izv. Ross. Akad. Nauk Ser. Mat. 89 (2025), pp. 131--161; reprinted in Izv. Math. 89 (2025), pp. 1208--1238. https://doi.org/10.4213/im9666e

    2025 doi

  64. [72]

    Z. Tong, Y. Li, Weighted Birkhoff averages: Deterministic and probabilistic perspectives. Math. Ann. 394 (2026), Paper No. 18, 46 pp. https://doi.org/10.1007/s00208-026-03311-0

    2026 doi

  65. [73]

    Z. Tong, Y. Li, Towards sharp regularity: Full-dimensional tori in C^ vector fields over T ^ . Revised, 2026. https://doi.org/10.48550/arXiv.2306.08211

    2026 doi arXiv

  66. [74]

    Tsinas, Joint ergodicity of Hardy field sequences

    K. Tsinas, Joint ergodicity of Hardy field sequences. Trans. Amer. Math. Soc. 376 (2023), pp. 3191--3263. https://doi.org/10.1090/tran/8752

    2023 doi

  67. [75]

    Wang, M\" o bius disjointness for analytic skew products

    Z. Wang, M\" o bius disjointness for analytic skew products. Invent. Math. 209 (2017), pp. 175--196. https://doi.org/10.1007/s00222-016-0707-z

    2017 doi

  68. [76]

    Yoccoz, Sur la disparition de propri\' e t\' e s de type D enjoy- K oksma en dimension 2

    J.-C. Yoccoz, Sur la disparition de propri\' e t\' e s de type D enjoy- K oksma en dimension 2 . C. R. Acad. Sci. Paris S\' e r. A-B 291 (1980), pp. A655--A658

    1980

  69. [77]

    Yoccoz, Centralisateurs et conjugaison diff\' e rentiable des diff\' e omorphismes du cercle, Petits diviseurs en dimension 1

    J.-C. Yoccoz, Centralisateurs et conjugaison diff\' e rentiable des diff\' e omorphismes du cercle, Petits diviseurs en dimension 1 . Ast\' e risque No. 231 (1995), pp. 89--242. http://www.numdam.org/item?id=AST_1995__231__89_0

    1995

Pith tools

Reviewed July 9, 2026 · model on record in the stance chip above.