arXiv:2608.02489v2

Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications

Son N. T. Tu, Jianlu Zhang, Siyao Zhu

math.APmath.DS37A3035B2735B4037C4037J5147A3549L25

Paper metadata

Original submission: August 3, 2026
Latest revision: August 13, 2026 · version 2
Claimed MSC: 37A30, 35B27, 35B40, 37C40, 37J51, 47A35, 49L25
arXiv categories: math.AP, math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 40 pages. Revised version incorporating structural changes to Theorem 1.1 and a new result in Theorem 1.2 establishing the optimal rate in all regimes. The author list has been updated to recognize Siyao Zhu's contributions during the early stages of the project, and an author contributions statement has been added

Abstract

In this paper, we establish quantitative Denjoy--Koksma type estimates for higher-dimensional quasi-periodic torus rotations. For Diophantine frequency vectors, we establish quantitative estimates on the discrepancy between Birkhoff averages and spatial averages for observables with various Besov-type regularities. By means of suitable Sobolev embeddings, these estimates yield, to the best of our knowledge, the sharpest currently available convergence rates for Hölder continuous observables. As applications, we obtain substantially improved quantitative homogenization results for Hamilton--Jacobi equations in spatially quasi-periodic settings, as well as nearly optimal statistical regularity estimates for invariant measures under perturbations.

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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.

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Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

The dyadic Besov estimates, their sharpness examples, the discrete analogue, and the statistical-stability theorem are correct. The Hamilton--Jacobi rates are also verified with the regularity actually used by their proof: the two occurrences of $W^{1,1}$ must be $W^{1,\infty}$ , and the constant must be allowed to depend on the relevant norm of the potential. This is a local hypothesis correction and does not change the rates or the paper's overall mathematical conclusion.

Theorem 1.1 and Corollary 1.3Correct

Continuous and discrete Birkhoff-average upper bounds

Pages 4–6 and 11–25 · Theorem 1.1, Corollary 1.3, and Sections 3.1 and 3.3 · arXiv:2608.02489v2

On each dyadic shell, the Diophantine condition separates the values $k\cdot\omega$ by $C_\omega 2^{-j\sigma}$ . Lemma 3.1 therefore gives the required $\ell^p$ small-divisor bound. Hausdorff--Young and the Besov shell norm then yield $T^{-1}$ above the threshold; splitting at $2^N\asymp T^{1/\sigma}$ gives $T^{-s/\sigma}$ below it and the critical factor $(\log T)^{1-1/q}$ . The suspension argument for the discrete rotation has the same shell estimates with the additional integer frequency and proves Corollary 1.3.

Theorem 1.2Correct

Sharpness and near-sharpness examples

Pages 4–5 and 15–23 · Theorem 1.2 and Section 3.2 · arXiv:2608.02489v2

A smooth one-mode coboundary already attains a nonzero multiple of $T^{-1}$ along a sequence. For badly approximable two-dimensional frequencies, the continued-fraction construction contributes a positive amount from each odd convergent and produces the critical $\log T/T$ lower bound while remaining in $B^1_{\infty,\infty}$ . A superlacunary subsequence of exact-index resonances yields the $T^{-s/\sigma}$ example. Finally, the large partial quotients supplied almost surely by Borel--Bernstein give the $T^{-\alpha}$ subsequential lower bound, while Theorem 1.1 supplies the printed upper bound. Two proof repairs for parts (i) and (ii) are recorded under Proofs.

Theorem 1.4Minor formal correction

Hamilton--Jacobi homogenization rates

Pages 6 and 25–32 · Theorem 1.4 and Section 4 · arXiv:2608.02489v2

The proof uses $\operatorname{Lip}(u_0)$ and asserts that $x\mapsto\sqrt{\mu+f(x)}$ is uniformly $1/2$ -Hölder. Those properties follow from $u_0\in W^{1,\infty}(\mathbb R)$ and $f\in W^{1,\infty}(\mathbb T^n)$ , not from the printed $W^{1,1}$ assumptions when $n\geq2$ . Replacing the two displayed $W^{1,1}$ classes by $W^{1,\infty}$ makes the proof valid and leaves every exponent unchanged. The constant in (1.18)–(1.19) must also depend on $\lVert f\rVert_{W^{1,\infty}}$ or the corresponding $C^2$ data; the proof already has this dependence. Under these locally corrected hypotheses, the Birkhoff bounds for the correctors and the convexity estimate for $\overline H$ give exactly the stated lower and upper rates.

Theorem 1.5Correct

Statistical regularity of invariant measures

Pages 7 and 32–35 · Theorem 1.5 and Section 5 · arXiv:2608.02489v2

Invariance permits time averaging under the perturbed flow. Its lift stays within $C|\delta|t$ of the unperturbed rotation, while Theorem 1.1 bounds the unperturbed Lipschitz average by $T^{-1/\sigma}$ , or by $T^{-1}\log T$ at $\sigma=1$ . Optimizing in $T$ gives (1.22). For the lower examples, rational approximants produce periodic-orbit measures whose Wasserstein distance from Lebesgue measure is comparable to $Q_j^{-1}$ ; the approximation exponent converts this to (1.23). The sequence is realized by the single Lipschitz family $V(x,\delta)=(1,\vartheta-\delta)$ after choosing approximants from one sign.

02Proofs6 reported findingsContains incorrect or incomplete proofs

The main upper-bound mechanism and both applications are correct after local repairs. The written proof of the smooth $T^{-1}$ sharpness statement does not establish its simultaneous-all-regularities and fixed-start formulation, and Proposition 3.7 uses an invalid common exponential model for continued-fraction denominators. Both defects have short verified repairs. The Hamilton--Jacobi section also contains the displayed Sobolev-index mismatch and two mechanically determined signs.

Lemma 3.1 and Propositions 3.2–3.3Correct and complete

Dyadic small-divisor estimates

Pages 11–15 · Section 3.1 · arXiv:2608.02489v2

The proof correctly uses separation of distinct shell frequencies, Hausdorff--Young for $1<p\leq2$ , and the high/low shell split. The critical logarithmic exponent is $1/q'=1-1/q$ , and the endpoint $q=1$ is covered by the same sequence estimates. The low-frequency block contains only finitely many modes and is harmless.

Theorem 1.2(i) and Proposition 3.4Incorrect as written

The printed construction does not prove the exact quantified claim

Pages 4 and 15–16 · Theorem 1.2(i), Lemma 3.6, and Proposition 3.4 · arXiv:2608.02489v2

The theorem asks for one function lying in $B^s_{\infty,\infty}$ for every $s>0$ and a lower bound along the fixed orbit in the statement. Proposition 3.4 instead constructs a different lacunary function for each selected $s=k+\alpha$ , and Lemma 3.6 proves a supremum over starting points. Repair classification: verified repair. Since nonresonance gives $\omega_1\neq0$ , take the single smooth mean-zero function $f(x)=\cos(2\pi x_1)$ . Then $\frac1T\int_0^T f(\omega t)\,dt=\frac{\sin(2\pi\omega_1T)}{2\pi\omega_1T}.$ Choosing $T_j$ so the sine has absolute value one proves the exact fixed-start $C/T_j$ lower bound, and this $f$ belongs to every positive Besov class.

Proposition 3.7, Equation (3.10)Incorrect as written

Badly approximable denominators need not follow one common exponential base

Pages 16–18 · proof of Proposition 3.7 · arXiv:2608.02489v2

The proof asserts $C_1\rho^k\leq Q_k\leq C_2\rho^k$ with one $\rho$ for the convergent denominators of every badly approximable number. Bounded partial quotients give exponential upper and lower bounds, but not generally with the same base: long blocks of different bounded partial quotients can change the local growth rate. Repair classification: verified repair. The only two uses need weaker standard facts. The recurrence gives $Q_{k+2\ell}\geq2^\ell Q_k$ , which uniformly bounds the number of odd convergents in a dyadic shell; bounded partial quotients give $Q_N\leq(A+1)^N$ , which yields $N\geq c\log Q_N$ . Substituting these two estimates leaves the $\log T_N/T_N$ conclusion unchanged.

Proof of Theorem 1.4Minor formal correction

The regularity used is Lipschitz, not $W^{1,1}$

Pages 26–29 · proof of Theorem 1.4 · arXiv:2608.02489v2

The estimates use $\operatorname{Lip}(u_0)$ explicitly and use $|f(x)-f(y)|\leq C|x-y|$ to place $\sqrt{\mu+f}$ uniformly in $C^{0,1/2}$ . The replacement $W^{1,1}\to W^{1,\infty}$ described under Statements is therefore required throughout this branch of the proof. With that replacement, the corrector estimates (4.10)–(4.18), the action comparison, and the final optimization are valid.

Equations (4.19)–(4.20) and the negative-energy comparisonTypo

The energy relation has two reversed signs

Pages 28–29 · Steps 1–2 of the proof of Theorem 1.4 · arXiv:2608.02489v2

Equation (4.2) gives $|\dot\eta|^2/2-V(\eta)=r$ , so (4.19)–(4.20) must read $\dot\eta=\pm\sqrt{2(r+V(\eta))}$ , not $\pm\sqrt{2(r-V(\eta))}$ . Likewise the displayed action of $\eta_0^+$ contains $\sqrt{-2V(x)}$ although $V\geq0$ ; it must be $\sqrt{2V(x)}$ . The surrounding equations (4.3), (4.5), (4.6), and (4.21) all use the plus sign, so the corrections are mechanical and no estimate changes.

Proof of Theorem 1.5Correct and complete

Time averaging, optimization, and rational-orbit lower example

Pages 32–35 · Section 5 · arXiv:2608.02489v2

The proof correctly separates orbit perturbation from the unperturbed Birkhoff discrepancy and optimizes the time horizon. The distance-to-orbit test function has Lipschitz norm one and integral $1/(4\sqrt{P_j^2+Q_j^2})$ , giving the claimed Wasserstein lower bound. The printed sequence of constant fields extends directly to a single parameter-Lipschitz family, so no additional assumption is needed.

03Novelty0 reported findingsNo non-novelty findings

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Paper: arXiv:2608.02489v2
Authors listed: Son N. T. Tu, Jianlu Zhang, Siyao Zhu
Audit date: August 18, 2026

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Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Continuous and discrete Birkhoff-average upper bounds

Sharpness and near-sharpness examples

Hamilton--Jacobi homogenization rates

Statistical regularity of invariant measures

Dyadic small-divisor estimates

The printed construction does not prove the exact quantified claim

Badly approximable denominators need not follow one common exponential base

The regularity used is Lipschitz, not W1,1W^{1,1}W1,1

The energy relation has two reversed signs

Time averaging, optimization, and rational-orbit lower example

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The regularity used is Lipschitz, not $W^{1,1}$