Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

Exponential convergence can happen in weighted Birkhoff averages via quasi-periodicity with arbitrary nonresonance and low regularity

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For any nonresonant rotation, tailored weights give stretched-exponential Birkhoff averages.

desk verdict A genuinely new existence result—stretched-exponential acceleration for arbitrary nonresonant frequencies and low-regularity observables—with a coherent proof that rests on one unproved derivative bound from the author's prior work. read the letter →

arxiv 2607.21950 v1 pith:EEAMHI37 submitted 2026-07-24 math.DS

Zhicheng Tong This is my paper · ORCID
classification math.DS MSC 37C5537A2537A3037A4437A46
keywords weightedBirkhoffaveragesexponentialconvergencenonresonantfrequencylowregularityobservablestoraltranslationssmalldivisorsstretched-exponentialrateHölder
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Since Krengel's 1978 result, ergodic averages are known to be able to converge with no effective rate, and for toral rotations fast weighted averages seemed to demand strongly nonresonant (Diophantine) frequencies and very regular observables. This paper proves that the demand is not necessary: for every nonresonant frequency vector $\alpha\in\mathbb{R}^d$ and every prescribed $\sigma\in(0,1)$, there exist normalized weights and observables in $A_B(\mathbb{T}^d)$---a space only slightly better than continuous---for which the weighted Birkhoff averages converge uniformly at a quantitative rate of order $\exp(-c N^{\sigma})$. A sympathetic reader should care because it isolates regularity and nonresonance as independent resources: low regularity can be compensated by weights tailored to the frequency, and even the mildest nonresonance admits this stretched-exponential acceleration. In dimension one the observables can be chosen to fail Hölder continuity of every order, so the regularity hypothesis is genuinely low.

What carries the argument

The argument is carried by a bell-shaped time weight $\tilde w_{p,q}(x)=\exp(-x^{-p}(1-x)^{-q})$ on $(0,1)$, with $w_{n,N}=Z_N^{-1}\tilde w_{p,q}(n/N)$, and by an imported estimate on its derivatives: $\|D^m \tilde w_{p,q}\|_{L^1}\le \lambda^m m^{\beta m}$ with $\beta=1+1/\min\{p,q\}$. Poisson summation converts each Fourier-mode sum $S_{N,k}(\alpha)=\sum w_{n,N} e^{2\pi i n\langle k,\alpha\rangle}$ into a series of integrals; integration by parts and a local minimization in $m$ (choosing $m\sim (N\operatorname{dist}(\langle k,\alpha\rangle,\mathbb{Z}))^{-1/\beta}$) turn that estimate into the key bound $|S_{N,k}(\alpha)|\le \exp(-c (N\operatorname{dist}(\langle k,\alpha\rangle,\mathbb{Z}))^{1/\beta})$. The remaining work splits the nonzero Fourier modes into three regions---near-resonant modes controlled by decay of the observables' Fourier coefficients, intermediate modes controlled by the same exponential bound, and nonresonant modes where dist is bounded below---and chooses the parameters $\kappa$, $\zeta$, $p$, $q$ so that all three contributions decay like $\exp(-c N^{\sigma})$. The balancing condition (2.13) is what produces the arbitrary exponent $\sigma\in(0,1)$.

What would settle it

Compute the $L^1$ norms of the $m$-th derivatives of $\tilde w_{p,q}(x)=\exp(-x^{-p}(1-x)^{-q})$ for increasing $m$ and check whether $\sup_m (\|D^m \tilde w_{p,q}\|_{L^1})^{1/m}/m^{\beta}$ stays bounded; if it grows without bound, Lemma 2.1 fails and the proof's rate cannot hold for that weight. A more direct check: for a fixed Liouvillean $\alpha$ and the constructed weights, evaluate $S_{N,k}(\alpha)$ for frequencies $k$ with $N\operatorname{dist}(\langle k,\alpha\rangle,\mathbb{Z})$ large and verify the exponent is at least $1/\beta$, not smaller.

Watch

Extended reading notes

Core claim

The paper's central discovery is Theorem 1.1: given any nonresonant frequency $\alpha\in\mathbb{R}^d$ and any $0<\sigma<1$, there is a nonzero family of normalized weights $w_{n,N}$ and observables $f\in A_B(\mathbb{T}^d)$ such that $\limsup_{N\to\infty} N^{-\sigma}\log \sup_{x\in\mathbb{T}^d}\bigl|\sum_{n=0}^{N-1} w_{n,N} f(T_\alpha^n x)-\int_{\mathbb{T}^d} f\,d\mu\bigr|_B<0$. Equivalently, the uniform error decays like $\exp(-c N^{\sigma})$ for some $c>0$, with a rate shared by the whole constructed family. In the case $d=1$, $B=\mathbb{R}$, the constructed observables can fail to belong to $C^a(\mathbb{T})$ for every $a\in(0,1)$, so the low regularity is not an artifact of the Banach-valued formulation. The paper presents this as the weighted counterpart to Yoccoz's slow-convergence results: there, analytic observables can be forced to converge arbitrarily slowly by badly nonresonant frequencies; here, any fixed nonresonant frequency admits observables of low regularity with quantitatively fast weighted convergence.

Load-bearing premise

The rate rests on the imported derivative bound $\|D^m \tilde w_{p,q}\|_{L^1}\le \lambda^m m^{\beta m}$ from [TL25b]; if that bound is false for the chosen bell weights, the exponential estimate for a single harmonic oscillator (Lemma 2.2) and hence the claimed $\sigma$-rate do not follow from this proof.

Editorial extensions

If this is right

  • For every fixed nonresonant rotation, there exist explicitly constructed weights and observables whose weighted Birkhoff averages converge uniformly with error at most $C\exp(-c N^{\sigma})$, for any desired $\sigma\in(0,1)$.
  • The same conclusion transfers to any dynamical system smoothly conjugate to a toral translation, so the constructed acceleration is available beyond the torus itself.
  • In one dimension the accelerating observables can be chosen outside every Hölder class $C^a$, so exponential-type acceleration does not require Hölder regularity of any positive order.
  • The proof identifies a three-region decomposition of Fourier modes and a balancing rule; the same template should yield quantitative rates for other weight functions that satisfy the derivative bound.
  • The result complements slow-convergence theorems: unweighted averages and generic low-regularity observables remain slow, while frequency-adapted weights and observables form a non-generic but explicit accelerated family.

Reading between the lines

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

  • Editorial reading of the balancing condition: $\sigma=1$ is the boundary, since reaching a true $\exp(-cN)$ rate would require $\beta=1$ in the derivative bound, corresponding to much smoother weights; the strictly sublinear exponents $\sigma<1$ appear intrinsic to this method.
  • The generic-observable contrast cited in the paper suggests a design principle: fast weighted averages are an engineered property, not a generic one, and the construction shows the engineering cost is low in regularity but high in adaptation to the specific $\alpha$.
  • The paper's hint that an almost-periodic analogue holds with slower exponential convergence suggests the same local-minimization-plus-truncation scheme may apply to systems with varying frequencies, with the rate reduced according to the complexity of the frequency set.
  • A numerical experiment could test the theorem directly: for a one-dimensional Liouvillean $\alpha$ and the constructed $\varphi$, plotting $-\log|\text{error}|$ against $\log N$ should show a slope near $\sigma$ for the chosen parameters; a slope below $\sigma$ would point at a gap in the derived estimate.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proves Theorem 1.1: for every nonresonant frequency vector α in R^d and every σ in (0,1), there exist N-dependent normalized weights {w_{n,N}} and an observable f in the Wiener algebra A_B(T^d) such that the weighted Birkhoff averages converge uniformly with limsup_{N→∞} N^{-σ} log(sup_x |Σ w_{n,N} f(T_α^n x) − ∫ f dμ|_B) < 0. In the case d=1, B=R, the observable can be chosen to fail the Hölder condition of every order a∈(0,1). The proof constructs the weight e_{p,q}(x)=exp(−x^{−p}(1−x)^{−q}), uses Poisson summation and integration by parts to obtain exponential decay of the single-mode sums S_{N,k}, then decomposes Fourier space into three regions (Λ^≤, Λ^>, Υ^>) and balances the resulting exponential rates through the parameters p,q, κ, ζ. The key quantitative input is an imported derivative bound, Lemma 2.1 from [TL25b], which fixes the exponent β_{p,q} in the rates and in the parameter constraints.

Significance. If the result is correct, it is a meaningful complement to the slow-convergence results of Krengel, del Junco–Rosenblatt, and Kachurovskii: it shows that, by choosing weights and observables jointly, exponential uniform acceleration is possible for every nonresonant toral translation, with no Diophantine condition at all. The paper is constructive: the weights are explicit, the non-Hölder example in d=1 is concrete, and the final exponential rate is obtained by direct parameter balancing rather than by an abstract existence argument. The main caveat is that the entire quantitative claim rests on the unproved derivative estimate in Lemma 2.1; if that estimate were false or had a different exponent, the balancing in Section 2.3 would fail for σ close to 1. As a paper, the result is novel in the context of Laskar-type weighted averages and deserves serious consideration, provided the imported lemma is made verifiable.

major comments (3)
  1. [§2.1, Lemma 2.1 (used in §2.2, Eqs. (2.6)–(2.8), and §2.3, Eq. (2.13))] The entire quantitative rate depends on the L^1 derivative bound ||D^m e_{p,q}||_{L^1} ≤ λ^m m^{β_{p,q} m} with β_{p,q}=1+1/min{p,q}, quoted as Lemma 4.1 of [TL25b] but not proved in this manuscript. The parameter choices in §2.3 require β_{p,q}σ<1 and the final rate uses exactly this β in all three summands of (2.14). Thus the exact value of β is load-bearing for the claim that every σ∈(0,1) is attainable. Please include a self-contained proof of Lemma 2.1, or at minimum give the precise dependence of λ on p,q and a complete reference to a proof that a reader can independently check. Without this, the σ-range in Theorem 1.1 cannot be verified from the manuscript alone.
  2. [§2.1, paragraph after Lemma 2.1] The sentence 'We point out that we can consider other weighting functions in C^∞_0([0,1]) satisfying the property stated in Lemma 2.1. Construction of such functions is a question.' is confusing and should be clarified. If the proof uses only the explicit function e_{p,q}, then the sentence about other functions is irrelevant and could be deleted or replaced by a precise description of the class actually used. If the theorem claims a whole family of admissible weights, the author should state what the family is and verify the derivative bound for every member.
  3. [§2.1 and Theorem 1.1 for d>1] The construction of the observable f for d>1 is only sketched: the Fourier coefficients are said to be 'free' on Υ^>_α, with no explicit choice guaranteeing f∈A_B(T^d). The explicit low-regularity example is given only for d=1. To make the theorem complete for all d≥1, the author should specify the coefficients on Υ^>_α (for instance, setting them to zero) and verify that the resulting f belongs to A_B(T^d). This is straightforward, but as written the d>1 existence part is not fully demonstrated.
minor comments (4)
  1. [§2.2, Eqs. (2.7)–(2.9)] The local-minimization step uses an asymptotic choice of the integer m and m'; the footnote about taking integer parts is helpful, but the argument would be more rigorous if the monotonicity analysis were written out explicitly, showing that the integer part of the minimizing m gives the same exponential bound up to a constant factor.
  2. [§2.3, Eq. (2.14)] The symbol 'ec' is unusual and can be misread as 'e' times 'c'. Please rename it as a single constant, e.g. c_3, to avoid confusion.
  3. [§1.1 and Remark 1.1] Remark 1.1 says exponential convergence 'typically demands extremely high regularity' for 'almost every frequency vector—even those satisfying a strong Diophantine condition'. The phrase 'almost every' is imprecise here: a strong Diophantine condition selects a measure-zero set, so the intended comparison should be stated as 'for a generic set of full measure' or rephrased to match the actual Diophantine class discussed.
  4. [Throughout] Minor typos and formatting issues should be corrected: for instance, the definition of A_B(T^d) says it is 'marginally better than mere continuity', which is arguable since absolute convergence of Fourier coefficients is a nontrivial smoothness condition; the text could say 'of Wiener type' instead. Also check the reference [TL25b] and the footnote numbering for consistency.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the existence proof is a direct construction with one independent self-cited derivative bound.

full rationale

The paper proves Theorem 1.1 by explicitly constructing normalized weights w_{n,N} = Z_N^{-1} \tilde w_{p,q}(n/N) and an observable f whose Fourier coefficients are required to satisfy the decay condition (2.1) relative to dist(⟨k,α⟩,Z). The final estimate (2.14) is obtained by bounding three separate sums through (2.10), (2.11), and (2.12); no parameter is fitted to data, no subset of the conclusion is reused as an input, and the target exponential convergence is never assumed in the construction. The only step imported from the author's earlier work is Lemma 2.1, quoted as Lemma 4.1 of [TL25b], which gives the derivative bound ||D^m \tilde w_{p,q}||_{L^1} ≤ λ^m m^{β_{p,q}m}. That bound is a concrete, parameter-free estimate whose stated assumptions do not include Theorem 1.1; it does not assert exponential convergence, it is not a uniqueness theorem, and it is not equivalent to the target statement. The fact that Lemma 2.1 is not proved in this manuscript is a legitimate correctness and completeness concern, but it is not circular reasoning. Similarly, the d>1 construction of observables is presented only through the coefficient condition (2.1) rather than an explicit closed form; this is an exposition gap, not a circular step. Overall, the central derivation is self-contained apart from one independent self-citation, so no circular step is identified.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The theorem is a construction proof, so its auxiliary parameters are chosen by hand to satisfy inequalities rather than inferred from data. The single non-standard input is the derivative bound Lemma 2.1 imported from [TL25b]. No new physical entities are introduced.

free parameters (4)
  • p,q = min{p,q} > (σ^{-1}-1)^{-1}
    Shape parameters of the compactly supported weight e_{p,q}; determine β=1+1/min{p,q} and must satisfy βσ<1 so the target rate N^σ is attainable.
  • κ = κ > (1-βσ)^{-1}σ
    Exponent in the Fourier-coefficient decay condition (2.1); needed for the near-resonant sum (2.10) to be exp(-N^{κζ}).
  • ζ = κ^{-1}σ < ζ < 1-βσ
    Threshold separating Λ^{≤} from Λ^{>} in Section 2.3; chosen so all three exponents in (2.13) exceed σ.
  • ϵ = 0<ϵ<1/8 in the d=1 example
    Defines the far-resonant set Υ^>_α; appears in the far-resonant rate (2.12) and in the non-Hölder example φ.
assumptions (3)
  • standard math Poisson summation formula and integration by parts for functions in the Schwartz class.
    Used in Section 2.2 to rewrite S_{N,k} as a sum of Fourier integrals.
  • standard math Weyl equidistribution for irrational rotations in dimension 1.
    Used in Section 2.1 to lower-bound the number of modes in Θ_h in the proof that φ is not Hölder.
  • ad hoc to paper Lemma 2.1 (= Lemma 4.1 of [TL25b]): the L^1 derivative bound for e_{p,q}.
    Imported without proof from the author's own earlier paper; it is the engine behind Lemma 2.2 and fixes the exponent 1/β.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Exponential convergence can happen in weighted Birkhoff averages via quasi-periodicity with arbitrary nonresonance and low regularity." pith.science (2026). https://pith.science/paper/EEAMHI37

@misc{pith2026260721950,
  author       = {Pith},
  title        = {Pith review of: Exponential convergence can happen in weighted Birkhoff averages via quasi-periodicity with arbitrary nonresonance and low regularity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EEAMHI37}},
  note         = {Machine review of arXiv:2607.21950}
}
read the original abstract

Since Krengel's work [Kre78] in 1978, it has been widely known that no effective rate of convergence exists in the ergodic theorem. For toral translations, however, by choosing appropriate weights one can accelerate the convergence of ergodic averages to an exponential rate, but this intuitively requires both highly nonresonant frequencies and very regular observables. In this paper, we uncover a new phenomenon: even for any given nonresonant frequency, there exists a non-trivial family of weights and observables of low regularity such that the weighted Birkhoff averages along quasi-periodic orbits converge at a quantitative, uniform, and exponential rate. This not only yields a finer understanding of the deep interaction between nonresonance and regularity in ergodic theory, but also stands as a weighted counterpart to a Yoccoz-type result [Yoc80,Yoc95].

Figures

Figures reproduced from arXiv: 2607.21950 by the authors.

Figure 1
Figure 1. Geometric properties and the commutative diagram of the conjugacy [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Laskar's frequency map analysis revisited

    math.DS 2026-08 conditional novelty 6.0 of 10

    Frequency map analysis converges with error O(e^{-cT^ζ}) for analytic quasi-periodic functions and with super-polynomial rates for Brjuno and almost-periodic cases.

Reference graph

Works this paper leans on

43 extracted references · 20 canonical work pages · cited by 1 Pith paper

  1. [1]

    Bellow, R

    A. Bellow, R. L. Jones, J. Rosenblatt, Almost everywhere convergence of weighted averages. Math. Ann. 293 (1992), pp. 399--426. https://doi.org/10.1007/BF01444724

    work page doi:10.1007/bf01444724 1992

  2. [2]

    Bellow, V

    A. Bellow, V. Losert, The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences. Trans. Amer. Math. Soc. 288 (1985), pp. 307--345. https://doi.org/10.2307/2000442

    work page doi:10.2307/2000442 1985

  3. [3]

    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

    doi:10.1016/j.jfa.2004.10.019 2005

  4. [4]

    A. D. Brjuno, Analytic form of differential equations. I, II. (Russian) Trudy Moskov. Mat. Ob s c . 25 (1971), pp. 119--262; ibid. 26 (1972), pp. 199--239

    work page 1971

  5. [5]

    Cheng, L

    C.-Q. Cheng, L. Wang, Destruction of Lagrangian torus for positive definite Hamiltonian systems. Geom. Funct. Anal. 23 (2013), pp. 848--866. https://doi.org/10.1007/s00039-013-0213-z

    doi:10.1007/s00039-013-0213-z 2013

  6. [6]

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

    doi:10.1088/1361-6544/aa84c2 2017

  7. [7]

    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

    doi:10.1088/1361-6544/aa99a0 2018

  8. [8]

    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

    doi:10.1007/bf01673506 1979

Show all 43 references
  1. [9]

    Duignan, J

    N. Duignan, J. D. Meiss, Distinguishing between regular and chaotic orbits of flows by the weighted Birkhoff average. Phys. D 449 (2023), Paper No. 133749, pp. 16. https://doi.org/10.1016/j.physd.2023.133749

    2023

  2. [10]

    Durand, D

    F. Durand, D. Schneider, Ergodic averages with deterministic weights. Ann. Inst. Fourier (Grenoble) 52 (2002), pp. 561--583. https://doi.org/10.5802/aif.1894

    2002 doi

  3. [11]

    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

  4. [12]

    A. Fan, J. Schmeling, Everywhere divergence of one-sided ergodic Hilbert transform. Ann. Inst. Fourier (Grenoble) 68 (2018), pp. 2477--2500. https://doi.org/10.5802/aif.3214

    2018 doi

  5. [13]

    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

  6. [14]

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

    1996 doi

  7. [15]

    A. G. Kachurovski , I. V. 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

  8. [16]

    A. G. Kachurovski , I. V. Podvigin, On measuring the rate of convergence in the Birkhoff ergodic theorem. (Russian) Mat. Zametki 106 (2019), pp. 40--52; translation in Math. Notes 106 (2019), pp. 52--62. https://doi.org/10.4213/mzm11817

    2019 doi

  9. [17]

    Kakutani, K

    S. Kakutani, K. Petersen, The speed of convergence in the ergodic theorem. Monatsh. Math. 91 (1981), pp. 11--18. https://doi.org/10.1007/BF01306954

    1981 doi

  10. [18]

    Katznelson, The action of diffeomorphism of the circle on the Lebesgue measure

    Y. Katznelson, The action of diffeomorphism of the circle on the Lebesgue measure. J. Analyse Math. 36 (1979), pp. 156--166 (1980). https://doi.org/10.1007/BF02798775

    1979 doi

  11. [19]

    Katznelson, An introduction to harmonic analysis

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

    2004 doi

  12. [20]

    Khanin, A

    K. Khanin, A. Teplinsky, Herman's theory revisited. Invent. Math. 178 (2009), pp. 333--344. https://doi.org/10.1007/s00222-009-0200-z

    2009 doi

  13. [21]

    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

  14. [22]

    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

  15. [23]

    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

    1993 doi

  16. [24]

    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

  17. [25]

    M. Lin, M. Weber, Weighted ergodic theorems and strong laws of large numbers. Ergodic Theory Dynam. Systems 27 (2007), pp. 511--543. https://doi.org/10.1017/S0143385706000769

    2007 doi

  18. [26]

    Montalto, M

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

    2021 doi

  19. [27]

    I. V. 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

  20. [28]

    I. V. 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

  21. [29]

    I. V. Podvigin, On the pointwise rate of convergence in the Birkhoff ergodic theorem: recent results. Ergodic theory and dynamical systems, pp. 117--125, De Gruyter Proc. Math., De Gruyter, Berlin, 2024. https://doi.org/10.1515/9783111435503-005

    2024 doi

  22. [30]

    V. V. Ryzhikov, Slow convergences of ergodic averages. (Russian) Mat. Zametki 113 (2023), pp. 742--746; translation in Math. Notes 113 (2023), pp. 704--707. https://doi.org/10.4213/sm9844

    2023 doi

  23. [31]

    V. 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

  24. [32]

    V. V. Ryzhikov, Slow convergence of Birkhoff ergodic averages. Preprint, 2025. https://doi.org/10.48550/arXiv.2507.16740

    2025 doi arXiv

  25. [33]

    V. V. Ryzhikov, Slow convergence almost everywhere of ergodic averages. Preprint, 2025. https://doi.org/10.48550/arXiv.2508.00463

    2025 doi arXiv

  26. [34]

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

    2004

  27. [35]

    Stein, R

    E. Stein, R. Shakarchi, Fourier analysis. An introduction. Princeton Lectures in Analysis, 1. Princeton University Press, Princeton, NJ, 2003. xvi+311 pp

    2003

  28. [36]

    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

  29. [37]

    Z. Tong, Y. Li, Towards continuity: Universal frequency-preserving KAM persistence and remaining regularity in nearly non-integrable Hamiltonian systems. Commun. Contemp. Math. 27 (2025), Paper No. 2450038, 42 pp. https://doi.org/10.1142/S021919972450038X

    2025 doi

  30. [38]

    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

  31. [39]

    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

  32. [40]

    Z. Tong, Y. Li, Optimal rates of uniform convergence for weighted Birkhoff averages via almost all rotations. Preprint, 2026. https://doi.org/10.48550/arXiv.2607.06907

    2026 doi arXiv

  33. [41]

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

    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

  34. [42]

    Yoccoz, Lin\' e arisation des germes de diff\' e omorphismes holomorphes de ( C , 0)

    J.-C. Yoccoz, Lin\' e arisation des germes de diff\' e omorphismes holomorphes de ( C , 0) . C. R. Acad. Sci. Paris S\' e r. I Math. 306 (1988), pp. 55--58

    1988

  35. [43]

    Yoccoz, Centralisateurs et conjugaison diff\' e rentiable des diff\' e omorphismes du cercle, Petits diviseurs en dimension 1 , Ast\' e risque No

    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

    1995

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.