arXiv:2607.06907v2

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

Zhicheng Tong, Yong Li

math.DS37C5537A4637A4437C0537A10

Paper metadata

Original submission: July 8, 2026
Latest revision: August 13, 2026 · version 2
Claimed MSC: 37C55, 37A46, 37A44, 37C05, 37A10
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 43 pages, 2 figures. Any comments are welcome!

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.

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Audit summary

Audited against arXiv v2

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsContains wrong statements

The lower-bound existence assertions are false in part of their stated parameter range because Definition 2.3 makes the claimed observable class empty whenever its global Fourier-envelope sum diverges. The paper's explicit sparse Fourier constructions do converge absolutely, but that does not place them in the printed class. The upper-rate assertions of Theorem 3.2 are correct.

Theorem 3.1Incorrect

The claimed observable need not belong to the class in the theorem

Pages 8 and 12–13 · Definition 2.3 and Theorem 3.1 · arXiv:2607.06907v2

Definition 2.3 includes the class condition $\sum_{k\in\mathbb Z^d\setminus\{0\}}\widetilde\Delta(\lVert k\rVert_{\ell^\infty})^{-1}<\infty.$ Theorem 3.1 assumes only $x^\varepsilon\leq\widetilde\Delta(x)\leq\exp(x^\zeta)$ for some $\varepsilon>0$ and $0<\zeta<\beta^{-1}$ . Take $\widetilde\Delta(x)=x^{1/2}$ , which satisfies those hypotheses, but in the discrete one-dimensional assertion the defining sum is $2\sum_{n\geq1}n^{-1/2}=\infty$ . Consequently the printed class $\mathcal R_{\widetilde\Delta}^l(\mathbb T)$ is empty, contradicting the theorem's assertion that an observable in it exists. The same example also violates the two-dimensional sum in the continuous assertion and every Case (I)–(III).

Full paper, version 2 ↗

Corollary 3.1Incorrect

The higher-dimensional existence conclusion has the same counterexample

Page 13 · Corollary 3.1 · arXiv:2607.06907v2

The corollary retains the unrestricted hypothesis $\varepsilon>0$ while asserting existence in $\mathcal R_{\widetilde\Delta}^l(\mathbb T^d)$ . For $\widetilde\Delta(x)=x^{1/2}$ , Definition 2.3 again requires a divergent lattice sum. More generally, for $\widetilde\Delta(x)=x^L$ , that sum diverges whenever $L\leq d$ . Thus the asserted class member does not exist throughout a nonempty part of the corollary's stated range.

Full paper, version 2 ↗

Theorem 3.2, upper boundsCorrect

The universal upper-rate assertions

Pages 14–16 and 29–42 · Theorem 3.2 and its proof · arXiv:2607.06907v2

The finite-regularity upper regimes impose enough decay for the full Fourier lattice sum used in the proof, and the $C^\infty$ , logarithmic $C^\infty$ , and Gevrey envelopes do so as well. The Fourier and Poisson-summation estimates, the one-dimensional Denjoy–Koksma and Abel argument, the continuous small-divisor packing estimate, and the low/high-frequency decompositions give the displayed rates uniformly in the initial point.

Theorem 3.2, finite-regularity lower boundsIncorrect

The advertised all-regularity optimality range includes empty classes

Pages 14–15 and 28–29 · Case (I) and opening of the proof of Theorem 3.2 · arXiv:2607.06907v2

The finite-regularity clauses claim counterexamples for all polynomial exponents $L>0$ and explicitly derive them from Theorem 3.1 and Corollary 3.1. Under the printed Definition 2.3, however, $\mathcal R_{x^L}^l(\mathbb T^d)$ is empty when $L\leq d$ , since $\sum_{k\neq0}\lVert k\rVert_{\ell^\infty}^{-L}$ diverges. The existence-based statement that the listed rate is attained therefore fails in those regimes. This finding does not affect the separately proved upper bounds or the lower clauses whose envelope satisfies the defining global summability condition.

02Proofs3 reported findingsContains incorrect or incomplete proofs

The lower-bound construction checks absolute convergence only along its sparse frequency subsequence and incorrectly treats that as satisfying Definition 2.3's full-lattice condition. Restricting the hypotheses to globally summable envelopes, or explicitly stating the result for the constructed sparse-series class, repairs the affected lower bounds. The upper-bound proof chain is correct and complete.

Proof of Theorem 3.1 and Corollary 3.1Incorrect as written

Sparse absolute convergence does not verify membership in the printed class

Pages 21–29 · Sections 5.1–5.2, especially the constructed Fourier series · arXiv:2607.06907v2

The construction uses only frequencies $q_{m_j}$ and the exponential growth of this subsequence, so $\sum_j\widetilde\Delta(q_{m_j})^{-1}<\infty$ follows from $\widetilde\Delta(x)\geq x^\varepsilon$ . That proves the explicitly constructed sparse series is continuous and absolutely convergent. Definition 2.3 instead requires the envelope sum over every nonzero $k\in\mathbb Z^d$ , which the argument never proves and which is false for the concrete admissible choice $\widetilde\Delta(x)=x^{1/2}$ . Downstream dependency: every membership assertion in Theorem 3.1 and Corollary 3.1, and the lower-bound clauses of Theorem 3.2, use this step. Repair classification: Verified repair after changing the affected theorem scope. Either add the full-lattice summability condition to the hypotheses, or replace membership in $\mathcal R_{\widetilde\Delta}^l$ by membership in the explicitly defined absolutely convergent sparse-series class with the stated coefficient envelope; all subsequent lower-bound calculations then apply unchanged.

Full paper, version 2 ↗

Proof of Theorem 3.2, lower boundsIncorrect as written

The proof imports the defective class-membership conclusion

Pages 28–29 · opening paragraphs of Section 5.3 · arXiv:2607.06907v2

The proof says that all counterexample-type estimates follow directly from Theorem 3.1 and Corollary 3.1 and that convergence over the constructed subsequence justifies the weaker parameter restrictions. Subsequence convergence justifies the existence of those particular Fourier series, but it does not satisfy Definition 2.3's independent full-lattice envelope condition. Downstream dependency: the claimed finite-regularity optimality statements for all $L>0$ . Repair classification: Verified repair only with the same restriction or class redefinition described above.

Proof of Theorem 3.2, upper boundsCorrect and complete

Fourier, discrepancy, and frequency-splitting estimates

Pages 29–42 · Section 5.3 · arXiv:2607.06907v2

In every upper-bound regime, the proof first works in a range where the Fourier expansion is absolutely summable. The discrete estimates control the weighted exponential sums, the one-dimensional refinement uses Denjoy–Koksma followed by Abel summation, and the continuous argument separates the rare small divisors before summing the remaining modes. Repeated integration by parts and the stated endpoint-flatness of the weighting functions justify the logarithmic and Gevrey low/high-frequency splits. No missing substantive case was found.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.06907v2
Authors listed: Zhicheng Tong, Yong Li
Audit date: August 18, 2026

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P(delta)

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

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The claimed observable need not belong to the class in the theorem

The higher-dimensional existence conclusion has the same counterexample

The universal upper-rate assertions

The advertised all-regularity optimality range includes empty classes

Sparse absolute convergence does not verify membership in the printed class

The proof imports the defective class-membership conclusion

Fourier, discrepancy, and frequency-splitting estimates

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