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For quasi-periodic rotations on the n-torus, Birkhoff averages of Hölder observables converge at rates that are optimal up to a logarithm, with the threshold set by the Diophantine index.

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2026-08-04 06:16 UTC pith:HRIFOLW3

load-bearing objection Main Besov estimates are a real advance and likely correct; the W^{1,1} branch of the homogenization claim rests on a false embedding. the 2 major comments →

arxiv 2608.02489 v1 pith:HRIFOLW3 submitted 2026-08-03 math.AP math.DS

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

Son N.T. Tu , Jianlu Zhang This is my paper
classification math.AP math.DS MSC 37A3035B2735B4037C4037J5147A3549L25
keywords Birkhoff ergodic theoremDiophantine frequencyBesov spaceHölder continuous observableDenjoy–Koksma inequalityHamilton–Jacobi homogenizationstatistical regularityinvariant measures
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes sharp quantitative Denjoy–Koksma type estimates: for Diophantine frequency vectors of index σ, the Birkhoff average of a C^{k,α} observable over time T differs from its spatial average by at most C T^{-1} when the regularity k+α exceeds σ, by C T^{-1} log T in the critical case, and by C T^{-(k+α)/σ} below. It proves the supercritical rate is optimal and that the subcritical rate is nearly optimal in dimension two for Hölder observables. These rates are derived through a Besov-space / Littlewood–Paley framework that treats the small divisors on each dyadic frequency shell, and they improve the previously known rates for Hölder observables. The rates carry over to a discrete analogue and are then used to obtain nearly optimal homogenization rates for one-dimensional Hamilton–Jacobi equations with quasi-periodic potentials and statistical regularity bounds for invariant measures under perturbations.

Core claim

Theorem 1.2 is the paper's core claim: for n≥2, ω∈D(σ,C_ω,n), and f∈C^{k,α}(T^n), the Birkhoff-average error is bounded by C∥f∥ T^{-1} if k+α>σ, by C∥f∥ T^{-1} log T if k+α=σ, and by C∥f∥ T^{-(k+α)/σ} if k+α<σ, with explicit constants. The supercritical rate is optimal, demonstrated by a co-boundary construction that achieves order T^{-1}. For n=2 and k=0 the subcritical rate is nearly optimal: a constructed example gives a lower bound of order T^{-α} against an upper bound T^{-α/(n-1+ε)}. The paper also shows the critical logarithmic loss is optimal for B^{1}_{∞,∞}(T^2).

What carries the argument

The Besov-space decomposition f=Σ Δ_j f with a dyadic partition of unity, combined with Lemma 3.1, which controls the distribution of small divisors |ξ·ω| on each dyadic annulus: the Diophantine condition forces the values ξ·ω to be separated by at least δ_j ~ C_ω 2^{-jσ}, so the ℓ^p sum of 1/|ξ·ω| over the annulus is bounded by (constant)/δ_j. This turns the usual Fourier small-divisor sum into a dyadic estimate that is sharp at the critical regularity.

Load-bearing premise

The paper's homogenization application for W^{1,1} potentials assumes that f_μ=(2(μ+f))^{1/2} is uniformly 1/2-Hölder when f∈W^{1,1}; a general W^{1,1} function need not be Hölder, so the ε^{1/(2σ)} rate for that case rests on a step that is not valid as written—this appears in the proof of Theorem 1.4, right after equation (4.9).

What would settle it

Take a test function on T^1 whose derivative is in L^1 but unbounded, such as a lacunary series with narrow triangular spikes, and compute the C^{0,1/2} seminorm of (2(μ+f))^{1/2} as μ→0. If the seminorm is unbounded, the claimed uniform Hölder regularity in the proof of Theorem 1.4 fails; a concrete counterexample to the statement 'W^{1,1}⊂C^{0,1/2}' would directly invalidate that step.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For any Diophantine frequency, the optimal O(1/T) convergence rate holds for observables with regularity above σ, and the sharp threshold is exactly the Diophantine index.
  • The critical regularity σ carries a mandatory logarithmic loss; the paper's example shows that log T / T is the true rate for B^{1}_{∞,∞} in two dimensions.
  • The Hölder subcritical rate T^{-(k+α)/σ} is nearly optimal in two dimensions, essentially settling the correct exponent for the Denjoy–Koksma phenomenon on the torus.
  • The discrete analogue gives the sharpest known Denjoy–Koksma type inequality for toral rotations in dimension n+1.
  • The Birkhoff-average rates directly improve the homogenization rate for quasi-periodic Hamilton–Jacobi equations and the statistical regularity of invariant measures, with exponents that are nearly optimal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the paper leaves open the critical case for general n—whether C^{n-1} observables (regularity exactly n-1) admit the optimal T^{-1} rate without the logarithmic loss; a higher-dimensional analogue of the Lemma 3.4 construction would settle it.
  • Editorial extension: the Besov/Littlewood–Paley method should extend to compact group extensions or skew products where similar small-divisor separation holds, potentially yielding rates for observables on homogeneous spaces.
  • Editorial extension: the W^{1,1} branch of Theorem 1.4 hinges on a regularity assertion for the lifted observable that is not justified as written; if that step fails, the ε^{1/(2σ)} rate for W^{1,1} potentials needs a different argument, while the C^2-with-nondegenerate-minima branch appears unaffected.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper develops Besov-space estimates for the convergence rate of Birkhoff averages along Diophantine quasi-periodic rotations on T^n, n≥2. Theorem 1.1 gives, for observables in B^s_{p,q}, rates O(T^{-1}) when s>σ, O(T^{-1}\log T) when s=σ, and O(T^{-s/σ}) when s<σ, with an optimality example for the critical log loss. Theorem 1.2 translates these to Hölder and C^{k,α} observables, with sharp or nearly sharp lower-bound examples. Theorem 1.3 gives the discrete analogue. These estimates are applied in Theorem 1.4 to quantitative homogenization of quasi-periodic Hamilton–Jacobi equations, yielding ε^{1/(2σ)} and ε^{1/σ}|log ε| rates for W^{1,1} and C^2 nondegenerate-minimum potentials respectively, and in Theorem 1.5 to statistical regularity of invariant measures under perturbations, with an almost-optimal Wasserstein bound. The central Besov/Hölder upper-bound arguments are elementary, explicit, and built on a Littlewood–Paley frequency decomposition; the lower-bound constructions use continued-fraction Diophantine approximation.

Significance. If the results stand, they represent a substantial improvement on existing quantitative Denjoy–Koksma estimates for higher-dimensional rotations, identifying the sharp regularity threshold σ and providing explicit rates with no fitted parameters. The applications to Hamilton–Jacobi homogenization and statistical regularity are of independent interest and improve earlier rates in [25] and related literature. The main Besov-space machinery (Lemmas 3.1–3.3 and their discrete analogues) appears correct and is a genuine contribution. However, the homogenization application in Theorem 1.4 contains a false regularity assertion for W^{1,1} observables, and the optimality statement of Theorem 1.2(ii) is overstated relative to its proof. These issues do not affect Theorems 1.1–1.3, but they do affect stated applications and should be corrected before publication.

major comments (2)
  1. [Section 4, proof of Theorem 1.4, first bullet after (4.9)] The assertion 'If f∈W^{1,1}(T^n), then f_μ := (2(μ+f))^{1/2} ∈ C^{0,1/2}(T^n) uniformly in 0≤μ≤μ_0' is false. A W^{1,1} function need not be Hölder of any positive exponent, and composition with the square root does not create uniform 1/2-Hölder regularity. For example, on T^1 a sum of disjoint triangular spikes of height 2^{-k} on intervals of length 2^{-k^2} is in W^{1,1} but has divergent C^{0,1/2} seminorm. Consequently the estimate (4.10), and the resulting ε^{1/(2σ)} rates in (1.18)–(1.19), are unsupported as stated. The authors should either replace the W^{1,1} assumption by a Hölder-type assumption (e.g., C^{0,1/2}), or prove a suitable Besov/Bessel-potential regularity estimate for f_μ that yields the claimed rate.
  2. [Theorem 1.2(ii) and Proposition 3.6] The statement claims that for almost every ω and every k∈N, α∈(0,1], there exists f∈C^{k,α}(T^n) whose Birkhoff averages decay no faster than O(T^{-1}). The proof constructs f(x)=∇_ω ψ(x) with ψ(x)=∑ 2^{-j(1+α)} cos(2π 2^j x_1). This gives f∈C^{0,α}(T^n), but not f∈C^{k,α} for k≥1: the k-th derivative has Fourier coefficients of size 2^{j(k-α)}, which diverge unless k=0. Thus the optimality claim is overbroad. The construction only proves the case k=0; if the authors wish to claim optimality for C^{k,α} with k≥1, a different construction is needed, or the theorem should be restricted accordingly.
minor comments (6)
  1. [Section 4, proof of Theorem 1.4, upper bound, line after (4.18)] In the C^2 non-degenerate case, the estimate (4.10) gives |v_p(s)/s| ≤ C log s / s^{1/σ}, so the additive term in the line before (4.18) should be C log t / t^{1/σ}, not C/t^{1/(2σ)}. The final rate appears unaffected, but the displayed intermediate bound is inconsistent.
  2. [Proposition 3.5] For α=1, the proof states ∥f∥_{B^{k+1}_{∞,∞}} ≤ C∥f∥_{C^{k,1}} with 'C:=C(n,f)'. The constant should be independent of f for the norm inequality to be meaningful; as written it suggests a non-uniform estimate.
  3. [Lemma 2.5 heading] The heading reads 'Corrspondence' – should be 'Correspondence'.
  4. [Throughout] The phrase 'continuous fraction' should be 'continued fraction' (see Subsection 2.1 and elsewhere).
  5. [References] Reference [20] is garbled: 'HAJERBAHOURI, RAPHAËLDANCHIN,ANDJEAN-YVESCHEMIN' should be 'BAHOURI, H., CHEMIN, J.-Y., AND DANCHIN, R.'
  6. [Lemma 3.4] The sequence is denoted T_N in the lemma and T_j in Theorem 1.1; the notation should be unified.

Circularity Check

0 steps flagged

No circular derivation: the Birkhoff rates are proved from first principles; self-citations are contextual, not load-bearing.

full rationale

The main results Theorems 1.1–1.3 are self-contained. Lemma 3.1 derives the dyadic small-divisor bound directly from the Diophantine condition (1.5); Lemmas 3.2 and 3.3 combine it with Hölder, Hausdorff–Young, and the Littlewood–Paley decomposition; Theorem 1.2 follows through the standard Besov–Hölder identification cited to [39,49]. There is no fitted parameter, no observable defined in terms of the predicted rate, and no uniqueness theorem invoked from the authors' prior work. The self-citations to [25] in Section 4 (e.g., Lemma 4.9, Propositions 4.10/4.11) provide the Hamilton–Jacobi variational framework, not the convergence rate itself; the rate improvement comes from substituting the independently proven Birkhoff estimates, so these citations are context, not circular input. Likewise, [47] in Remark 3 is only a remark. Per the reviewing rule, I flag one mathematical gap that is not circular: Section 4, first bullet after (4.9), asserts 'If f∈W^{1,1}(T^n), then f_μ := ... ∈ C^{0,1/2}(T^n) uniformly in μ'; W^{1,1} does not imply Hölder continuity, so the ε^{1/(2σ)} homogenization rate for W^{1,1} potentials is unsupported by the given argument. This affects the correctness of Theorem 1.4, not the circularity of the derivation chain, and Theorems 1.1–1.3 are unaffected. Overall circularity score 1 (minor, non-load-bearing self-citation).

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The central estimates rely on standard harmonic analysis and Diophantine approximation; no parameters are fitted to data. The only ad hoc assumption is the false W^{1,1}→C^{0,1/2} embedding used in the homogenization application.

axioms (5)
  • standard math Littlewood–Paley decomposition and Bernstein inequalities on T^n
    Used throughout §3 in the dyadic decomposition f=∑Δ_j f and in bounding ∥Δ_j f∥_{L^r}; standard harmonic analysis (see [20,39,49]).
  • standard math Diophantine approximation facts: σ≥n−1; full-measure sets D(n−1+δ,C_ω,n); continued fraction properties and Borel–Bernstein theorem
    Used in Lemma 3.1, Lemma 3.10, Proposition 3.9, and the a.e. statements; standard number theory (Cassels, Schmidt, Bugeaud).
  • domain assumption Existence of effective Hamiltonian and correctors for quasi-periodic H-J equations
    Theorem 1.4 relies on qualitative homogenization and cell-problem structure from [26,50]; not re-derived in this paper.
  • ad hoc to paper f∈W^{1,1}(T^n) implies (2(μ+f))^{1/2}∈C^{0,1/2}(T^n)
    Asserted without proof in the proof of Theorem 1.4; false in general for n≥1. This ad hoc assumption undercuts the W^{1,1} homogenization rate.
  • domain assumption Perturbation bound ∥V(·,δ)−ω∥_{L^∞}≤|δ| (assumption (5.2))
    Hypothesis of Theorem 1.5 controlling the closeness of the vector field to the rotation.

pith-pipeline@v1.3.0-daily-deepseek · 39518 in / 30059 out tokens · 277853 ms · 2026-08-04T06:16:47.177657+00:00 · methodology

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read the original 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 H\"older 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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