{"id":"0c70a5c4-7248-47cf-a15c-a94c5a64bd5a","arxiv_id":"2607.21950","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For every nonresonant torus frequency and any σ<1, there exist normalized weights and an A_B low-regularity observable making weighted Birkhoff averages converge faster than exp(-cN^σ).","lead":"This paper proves that weighted Birkhoff averages along any nonresonant quasi-periodic orbit can be accelerated to a stretched-exponential rate when the weights and a specially constructed low-regularity observable are chosen together. It provides a converse to classical slowness results and refines the tradeoff between frequency nonresonance and observable regularity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claimed sigma-range rests on the unproved derivative bound Lemma 2.1 from [TL25b]; if its beta exponent is not correct, Lemma 2.2 and the parameter balancing in Section 2.3 fail for sigma close to 1.","rationale":"I read the paper as a construction-based proof: choose Laskar-type bump weights, prove stretched-exponential decay for each Fourier mode via Poisson summation and a derivative-growth lemma, then balance the decay against a Fourier-coefficient/small-divisor decomposition. The d=1 non-Holder example is explicit and correct; the Poisson-summation step and the three-sum decomposition in Section 2.3 are coherent; the parameter constraints (2.13) are consistent once beta*sigma < 1 is assumed. The only real load-bearing uncertainty is Lemma 2.1, which is imported without proof from the author's own [TL25b] and fixes the exponent 1/beta in the final rate. The reader's conditional verdict is therefore appropriate; my stress-test does not move it. I do not regard the sketched d>1 construction as a separate threat to the central claim, since explicit summable choices are immediate and no d>1 non-Holder assertion is made. Minor typos such as the 'non-decreasing' g contradicting the given decreasing example are worth fixing but do not affect the argument, since (2.10) only needs summability of g(||k||).","tokens_in":12080,"tokens_out":36664,"duration_ms":342135,"concrete_test":"Independently prove or extract the proof of Lemma 4.1 in [TL25b] and check it applies to the exact ewpq with arbitrary real p,q > 0. Concretely, use Faa di Bruno and endpoint asymptotics to bound ||D^m ewpq||_{L^1(0,1)} for p=q=10 and p=20,q=30, m up to 200, and verify the fitted m-dependence has exponent at most 1+1/min{p,q}. If the true exponent is larger, recompute (2.7)-(2.13) for sigma close to 1; a positive result for beta <= 1+1/min{p,q} would remove the concern, while a failure would invalidate the claimed sigma-range.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 2.1 (quoted as Lemma 4.1 of [TL25b]) is not proved in this manuscript, yet the entire quantitative rate (2.14) is built on it. Lemma 2.2 converts the bound ||D^m ewpq||_{L^1} <= lambda^m m^(beta m) into the single-mode decay exp(-c(N dist)^(1/beta)). The parameter choices in Section 2.3 require beta*sigma < 1 to make the zeta-interval nonempty, and (2.13) then needs 1/beta, (1-zeta)/beta, kappa*zeta > sigma. If the true derivative growth were worse than m^((1+1/min{p,q})m) - for instance an extra factorial power or a larger exponent - the balancing would break for sigma close to 1, and no alternative weight function is analyzed in the paper. The d>1 construction of observables is also sketched, but this is secondary because for d>1 the theorem only needs f in A_B and explicit summable choices are easy; the low-regularity claim in d=1 is proved separately and is not the fragile part.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":12351,"tokens_out":15219,"duration_ms":138064,"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":[{"comment":"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.","section":"§2.1, Lemma 2.1 (used in §2.2, Eqs. (2.6)–(2.8), and §2.3, Eq. (2.13))"},{"comment":"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.","section":"§2.1, paragraph after Lemma 2.1"},{"comment":"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.","section":"§2.1 and Theorem 1.1 for d>1"}],"minor_comments":[{"comment":"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.","section":"§2.2, Eqs. (2.7)–(2.9)"},{"comment":"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.","section":"§2.3, Eq. (2.14)"},{"comment":"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.","section":"§1.1 and Remark 1.1"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is interesting and the main idea appears sound, but the referee report cannot fully certify Theorem 1.1 because the crucial derivative bound is imported from a previous paper without proof. This is a load-bearing point, especially for σ close to 1. The d>1 construction also needs a few lines of completion. I recommend major revision with a request to make the manuscript self-contained on these points. The unusual sentence about constructing other weighting functions should be addressed during revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, what to know: this paper does something new. For any nonresonant α∈R^d and any σ∈(0,1), it builds normalized weights and observables f∈A_B(T^d) such that the weighted Birkhoff averages converge like exp(-cN^σ), uniformly in x. That closes a real gap: earlier exponential results needed Diophantine frequencies and high regularity, while arbitrary nonresonance gave only polynomial rates. The d=1 statement that f can be non-Hölder is a concrete bonus.\n\nThe proof is coherent. The Poisson summation step, the local minimization over m, and the three-way split of Fourier modes (Λ≤, Λ>, Υ>) line up, and the parameter constraints in (2.13) are consistent. The construction of the non-Hölder example is explicit and correct, modulo a minor notation issue in the denominator (k log^2 k vs k log 2k).\n\nNow the soft spots. The largest one is Lemma 2.1, imported from the author's own [TL25b]. The bound ||D^m e_{p,q}||_{L^1} ≤ λ^m m^{βm} fixes the exponent 1/β in the final rate, so the whole balancing depends on it. The paper does not prove it or even sketch the argument. That is a real gap in self-containedness, but it is not a circularity or a hidden assumption—it is a concrete analytic estimate that can be checked independently. A referee should ask for a proof or a precise location.\n\nThe second issue is terminology. exp(-cN^σ) with σ<1 is stretched-exponential, not exponential. The title and abstract say 'exponential' repeatedly, which overstates the rate. Easy fix.\n\nThird, the d>1 construction is sketched rather than fully written out. Existence is clear from condition (2.1), but I would like to see an explicit family spelled out.\n\nOverall, the central claim appears correct and the result is a genuine advance. The flaws are not load-bearing: they are about presentation and verification. I would accept this for peer review and recommend the referee focus on Lemma 2.1 and the terminology. The paper deserves a serious referee, not a desk reject.","headline":"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.","tokens_in":12875,"tokens_out":5854,"would_cite":true,"duration_ms":50827,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37C55","37A25","37A30","37A44","37A46"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any nonresonant rotation, tailored weights give stretched-exponential Birkhoff averages.","keywords":["weighted Birkhoff averages","exponential convergence","nonresonant frequency","low regularity observables","toral translations","small divisors","stretched-exponential rate","Hölder regularity"],"falsifier":"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.","tokens_in":11857,"feed_emoji":"📉","tokens_out":12200,"duration_ms":96415,"temperature":0.7,"pith_summary":"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.","feed_headline":"Arbitrary nonresonance no longer blocks exponential averages","feed_subtitle":"This paper builds low-regularity observables whose weighted Birkhoff averages decay as exp(-c N^σ) on toral rotations.","key_machinery":"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 $\\sigma\\in(0,1)$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Lemma 2.1, the L^1 derivative bound for the bell weight that the exponential single-oscillator estimate depends on.","marker":"[TL25b]"},{"why":"Establishes exponential convergence of weighted Birkhoff averages under stronger hypotheses; the present construction extends its oscillation and truncation techniques.","marker":"[TL24]"},{"why":"First rigorous arbitrary polynomial acceleration for discrete quasi-periodic weighted averages, the baseline this paper pushes to exponential.","marker":"[DSSY17]"},{"why":"Extends the polynomial acceleration framework to related quasi-periodic settings and is cited as part of the same baseline.","marker":"[DY18]"},{"why":"The slow-convergence theorem for ergodic averages that motivates the whole question of whether acceleration is possible.","marker":"[Kre78]"},{"why":"Yoccoz's slow-convergence results for analytic observables under Liouvillean frequencies; the paper claims a weighted counterpart.","marker":"[Yoc80,Yoc95]"},{"why":"Establishes optimal slow lower bounds for generic almost-every rotation, used to contrast the accelerated non-generic family.","marker":"[TL26b]"}],"fun_headline_variants":["Exponential weighted averages for any nonresonant frequency","Low regularity still gives exponential Birkhoff averages","Arbitrary nonresonance: exponential convergence still possible","Weights beat nonresonance: exponential speed at low regularity","Any nonresonant frequency allows exponential weighted averages"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exponential weighted averages for any nonresonant frequency","Low regularity still gives exponential Birkhoff averages","Arbitrary nonresonance: exponential convergence still possible","Weights beat nonresonance: exponential speed at low regularity","Any nonresonant frequency allows exponential weighted averages"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001147,"raw_usage":{"total_tokens":4777,"prompt_tokens":988,"completion_tokens":3789,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":604,"completion_tokens_details":{"reasoning_tokens":3709}},"tokens_in":604,"tokens_out":3789,"duration_ms":23588,"temperature":1.0,"reasoning_tokens":3709,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:31:07.244098+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":2}