arXiv:2607.21950v1
Exponential convergence can happen in weighted Birkhoff averages via quasi-periodicity with arbitrary nonresonance and low regularity
math.DS37C5537A2537A3037A4437A46
Abstract
Since Krengel's work [Kre78] in 1978, it has been widely known that no effective rate of convergence exists in the ergodic theorem. For toral translations, however, by choosing appropriate weights one can accelerate the convergence of ergodic averages to an exponential rate, but this intuitively requires both highly nonresonant frequencies and very regular observables. In this paper, we uncover a new phenomenon: even for any given nonresonant frequency, there exists a non-trivial family of weights and observables of low regularity such that the weighted Birkhoff averages along quasi-periodic orbits converge at a quantitative, uniform, and exponential rate. This not only yields a finer understanding of the deep interaction between nonresonance and regularity in ergodic theory, but also stands as a weighted counterpart to a Yoccoz-type result [Yoc80,Yoc95].
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01Statements3 reported findingsCorrect
Exponential convergence of weighted Birkhoff averages. Theorem 1.1 treats an arbitrary toral translation and every frequency vector satisfying the paper's nonresonance condition. For each admissible auxiliary decay function, it constructs the stated low-regularity observable and normalized weight so that the weighted Birkhoff averages converge at the announced exponential rate. The Fourier coefficients meet the exact modulus-of-continuity or Sobolev bound printed in the theorem, and the construction does not assume a Diophantine lower bound stronger than nonresonance. The two local notation corrections below do not alter this conclusion.
Theorem 1.1Correct
Exponential convergence of weighted Birkhoff averages
Pages 4–6 · Theorem 1.1 · arXiv:2607.21950v1
Theorem 1.1 treats an arbitrary toral translation and every frequency vector satisfying the paper's nonresonance condition. For each admissible auxiliary decay function, it constructs the stated low-regularity observable and normalized weight so that the weighted Birkhoff averages converge at the announced exponential rate. The Fourier coefficients meet the exact modulus-of-continuity or Sobolev bound printed in the theorem, and the construction does not assume a Diophantine lower bound stronger than nonresonance. The two local notation corrections below do not alter this conclusion.
Theorem 1.1, regularity clauseCorrect
The constructed observable has the claimed low regularity
Pages 4–6 and 11–14 · Theorem 1.1 and coefficient construction · arXiv:2607.21950v1
The Fourier support is lacunary enough that the modulus of continuity is bounded by the sum of the low-frequency coefficient tail and the high-frequency amplitude tail. The monotonicity and integrability assumptions on g make both terms no larger than the function printed in Theorem 1.1. At the selected resonant frequencies the matching lower estimate prevents an accidental improvement of regularity when the theorem claims sharpness. The correction from ‘non-decreasing’ to ‘non-increasing’ below is uniquely forced by this calculation.
Theorem 1.1, convergence clauseCorrect
Exponential weighted-average error
Pages 4–6 and 14–18 · Theorem 1.1 and final estimate · arXiv:2607.21950v1
Poisson summation expresses each Fourier mode's weighted average through the rapidly decaying transform of the smooth weight. The near-integer scalar products selected from nonresonance place the desired term at the appropriate dual frequency, and lacunarity keeps all other modes outside its window. Summing their transform tails yields the displayed exponential error uniformly in the averaging parameter. No Diophantine exponent beyond nonresonance is used.
02Proofs4 reported findingsCorrect
Fourier localization, Poisson summation, and truncation. A sequence of integer frequencies is chosen so that its scalar products with the translation vector approach integers at the prescribed small-divisor scale while remaining nonzero. The observable is supported on these isolated Fourier modes; the monotonicity and summability assumptions on the auxiliary function give the stated regularity estimate term by term. The compactly supported smooth weight is analyzed by Poisson summation, and repeated integration by parts makes every nonzero dual term exponentially small relative to the chosen truncation. The distinguished near-resonant mode supplies the target main term, while separation of the remaining modes prevents cancellation. The estimates are uniform for all averaging lengths after the finite initial range is absorbed into the constant.
Small-divisor constructionCorrect and complete
Fourier localization, Poisson summation, and truncation
Pages 7–18 · Section 2 · arXiv:2607.21950v1
A sequence of integer frequencies is chosen so that its scalar products with the translation vector approach integers at the prescribed small-divisor scale while remaining nonzero. The observable is supported on these isolated Fourier modes; the monotonicity and summability assumptions on the auxiliary function give the stated regularity estimate term by term. The compactly supported smooth weight is analyzed by Poisson summation, and repeated integration by parts makes every nonzero dual term exponentially small relative to the chosen truncation. The distinguished near-resonant mode supplies the target main term, while separation of the remaining modes prevents cancellation. The estimates are uniform for all averaging lengths after the finite initial range is absorbed into the constant.
Sections 2.2–2.4Correct and complete
Fourier coefficient bounds and Poisson-summation estimate
Pages 10–18 · Sections 2.2–2.4 · arXiv:2607.21950v1
The small-divisor subsequence is selected inductively so that every new frequency is separated from the finitely many earlier resonances. Coefficients are chosen at the largest size allowed by the target modulus, and lacunarity makes the resulting Fourier series converge in the stated function space. For finite partial sums Poisson summation is immediate; rapid decay gives a summable majorant independent of the truncation, so passage to the full observable is justified. The main and error terms then have the exact exponential scales used in Theorem 1.1.
Equation (2.2)Typo · no status impact
The monotonicity direction of the auxiliary function is reversed
Page 7 · Section 2.1, Equation (2.2) · arXiv:2607.21950v1
The text calls g non-decreasing, while the integrability condition and the displayed example require g to be non-increasing. Reversing this word is forced by every subsequent use and does not change the construction.
Lemma 2.2Minor formal correction · no status impact
The nearest-integer interval excludes the tie case
Page 9 · proof of Lemma 2.2 · arXiv:2607.21950v1
A nearest integer is selected with distance strictly between 0 and 1/2. A half-integer can occur even under nonresonance. Replacing the upper strict inequality by a non-strict one and choosing either nearest integer in the tie case leaves all subsequent bounds unchanged.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.