{"id":"b6fdaf91-7940-4120-9396-00d8c558b067","arxiv_id":"2608.02489","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Diophantine rotations on T^n, Hölder observables have Birkhoff-average errors T^-1, T^-1 log T, or T^-(k+α)/σ, with matching or near-matching counterexamples.","lead":"This paper derives sharp rates at which time-averages along quasi-periodic torus rotations converge to the spatial average, for observables with Hölder or Besov regularity. The rates improve prior bounds and are applied to homogenization of Hamilton–Jacobi equations and to stability of invariant measures.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The W^{1,1}-to-C^{0,1/2} composition step in Theorem 1.4 is false, leaving the claimed homogenization rate for W^{1,1} potentials unsupported.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: the unjustified step from f∈W^{1,1} to f_μ∈C^{0,1/2}. I confirmed the concern is mathematically substantive by constructing a concrete W^{1,1} function that is not C^{0,1/2}, and showing that composition with sqrt does not repair it. This affects the homogenization application (Theorem 1.4) but not the central Birkhoff-average estimates (Theorems 1.1–1.3). The paper's main results appear correct, but the Theorem 1.4 statement for W^{1,1} observables is unsupported as written. The appropriate verdict remains CONDITIONAL, requiring the authors to either correct the claim, replace W^{1,1} by W^{1,∞}, or provide a rigorous alternative derivation. Since this matches the reader's verdict, no change to the reader's assessment is needed.","tokens_in":39875,"tokens_out":12474,"duration_ms":116821,"concrete_test":"Explicitly construct the counterexample: let I_k be disjoint intervals of length ℓ_k = 2^{-k^2} on T^1, and set f(x) = Σ 2^{-k} φ((x-c_k)/ℓ_k) with φ a triangular bump of height 1. Extend to T^n by f_n(x)=f(x_1). This is in W^{1,1}(T^n) with uniformly bounded norm. Then for x_k at the center of I_k and y_k at distance ℓ_k/2, check that |(μ+f_n(x_k))^{1/2} - (μ+f_n(y_k))^{1/2}| / |x_k-y_k|^{1/2} ≥ c 2^{(k^2-k)/2}, which diverges as k→∞. If this computation is verified, the assertion in the proof is conclusively false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 4, proof of Theorem 1.4, the first bullet after (4.9), the authors assert: '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.' This is false. A function in W^{1,1}(T^n) (even when continuous) need not be Hölder of any positive exponent, and composing with the square root does not create uniform 1/2-Hölder regularity. For example, on T^1 define a sum of disjoint triangular spikes of height 2^{-k} on intervals of length 2^{-k^2}; this function lies in W^{1,1} but its C^{0,1/2} seminorm diverges. The Birkhoff estimate in Theorem 1.2 requires Hölder regularity of the observable, so the claimed ε^{1/(2σ)} rate for f∈W^{1,1} in (1.18)–(1.19) does not follow from the given argument. This is a genuine gap in a stated application, although Theorems 1.1–1.3 themselves are not affected.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":40208,"tokens_out":7055,"duration_ms":67836,"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":[{"comment":"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.","section":"Section 4, proof of Theorem 1.4, first bullet after (4.9)"},{"comment":"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.","section":"Theorem 1.2(ii) and Proposition 3.6"}],"minor_comments":[{"comment":"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.","section":"Section 4, proof of Theorem 1.4, upper bound, line after (4.18)"},{"comment":"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.","section":"Proposition 3.5"},{"comment":"The heading reads 'Corrspondence' – should be 'Correspondence'.","section":"Lemma 2.5 heading"},{"comment":"The phrase 'continuous fraction' should be 'continued fraction' (see Subsection 2.1 and elsewhere).","section":"Throughout"},{"comment":"Reference [20] is garbled: 'HAJERBAHOURI, RAPHAËLDANCHIN,ANDJEAN-YVESCHEMIN' should be 'BAHOURI, H., CHEMIN, J.-Y., AND DANCHIN, R.'","section":"References"},{"comment":"The sequence is denoted T_N in the lemma and T_j in Theorem 1.1; the notation should be unified.","section":"Lemma 3.4"}],"recommendation":"major_revision","confidential_remarks":"The core Besov/Hölder estimates are solid and publishable, but the W^{1,1} step in Theorem 1.4 is a genuine gap in a stated application, and the optimality claim in Theorem 1.2(ii) overreaches the proof. Both are fixable within the manuscript's scope: replace the W^{1,1} assumption or supply a correct regularity statement for the composed observable, and restrict/repair the lower-bound example. I recommend major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The core of the paper — the Besov-space Birkhoff estimates, the critical log T/T rate, and the lower-bound examples — is a genuine advance and looks correct. The homogenization application is not fully supported as written: the claim that f∈W^{1,1}(T^n) implies (2(μ+f))^{1/2}∈C^{0,1/2}(T^n) uniformly in μ is false, so the ε^{1/(2σ)} rate for W^{1,1} potentials in Theorem 1.4 does not follow from the given proof.\n\nWhat is actually new: Lemma 3.1 is a clean dyadic counting argument that captures the distribution of small divisors on each shell; it converts the Diophantine condition into a sharp ℓ^p bound. That gives optimal O(1/T) for supercritical Besov/Hölder regularity, the new O(log T/T) at the critical exponent, and an algebraic rate below it. The lower-bound construction in Lemma 3.4 for the critical case B^1_{∞,∞} is clever and convincing, and Proposition 3.9 gives a nearly matching subcritical lower bound for n=2. The transfer to discrete Denjoy–Koksma and the improvement over [25] are honest. The Besov tool is a real addition to the literature.\n\nSoft spots. The W^{1,1} claim is a load-bearing flaw in one branch of Theorem 1.4. W^{1,1} functions need not be Hölder of any positive order, and the square root does not fix that. The triangular-spikes example is enough to kill the claim. This means the ε^{1/(2σ)} rates for f∈W^{1,1} are unsupported. The C^2 nondegenerate-minimum branch is fine, and Theorems 1.1–1.3 are unaffected. A second, smaller issue: Proposition 3.6 is stated for C^{k,α} with arbitrary k, but the proof only constructs the k=0 case; the general case is likely trivial via finite Fourier series, but it is an overstatement as written. The n≥3 subcritical example is acknowledged to be essentially two-dimensional, which is fine but should be kept in mind.\n\nBottom line: the main estimates are solid, the central argument holds up, and the paper deserves serious refereeing. I would send it to a referee with clear instructions to focus on Theorem 1.4 and ask for a corrected or honestly weakened statement for the W^{1,1} case. If that gets fixed, I would cite it.","headline":"Main Besov estimates are a real advance and likely correct; the W^{1,1} branch of the homogenization claim rests on a false embedding.","tokens_in":40647,"tokens_out":4315,"would_cite":true,"duration_ms":46243,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A30","35B27","35B40","37C40","37J51","47A35","49L25"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Birkhoff ergodic theorem","Diophantine frequency","Besov space","Hölder continuous observable","Denjoy–Koksma inequality","Hamilton–Jacobi homogenization","statistical regularity","invariant measures"],"falsifier":"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.","tokens_in":39762,"feed_emoji":"⏱️","tokens_out":4474,"duration_ms":42668,"temperature":0.7,"pith_summary":"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.","feed_headline":"Birkhoff averages hit optimal rate on quasi-periodic tori","feed_subtitle":"A Besov-space method pins the threshold where the O(1/T) rate begins, improving homogenization and invariant-measure stability.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Besov method proves optimal T^-1 error for Birkhoff averages on tori","Optimal convergence rate for Hölder observables on quasi-periodic tori","Quasi-periodic Birkhoff averages: sharp rate via Besov regularity","Improved homogenization from optimal Birkhoff rates on quasi-periodic tori"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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).","fun_headline_variants_meta":{"raw":{"variants":["Besov method proves optimal T^-1 error for Birkhoff averages on tori","Optimal convergence rate for Hölder observables on quasi-periodic tori","Quasi-periodic Birkhoff averages: sharp rate via Besov regularity","Improved homogenization from optimal Birkhoff rates on quasi-periodic tori"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001257,"raw_usage":{"total_tokens":4962,"prompt_tokens":694,"completion_tokens":4268,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":4198}},"tokens_in":438,"tokens_out":4268,"duration_ms":57237,"temperature":1.0,"reasoning_tokens":4198,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:16:47.177657+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}