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PreprintarXiv licencearXiv

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

  • Son N. T. Tu
  • Jianlu Zhang
  • Siyao Zhu
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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. The Besov framework is naturally adapted to the small-divisor structure of the problem, allowing a sharp analysis of resonant frequencies and yielding optimal rates throughout the entire parameter range. 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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Cite

BibTeXRIS
BibTeX
@misc{tu2026convergence,
  title = {{Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications}},
  author = {Son N. T. Tu and Jianlu Zhang and Siyao Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.02489},
  eprint = {2608.02489},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.02489},
}
RIS
TY  - UNPB
TI  - Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications
AU  - Son N. T. Tu
AU  - Jianlu Zhang
AU  - Siyao Zhu
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2608.02489
UR  - https://arxiv.org/abs/2608.02489
AB  - 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. The Besov framework is naturally adapted to the small-divisor structure of the problem, allowing a sharp analysis of resonant frequencies and yielding optimal rates throughout the entire parameter range. 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.
ER  -