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arXiv · 2608.02489

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. 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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Son N. T. Tu, Jianlu Zhang, Siyao Zhu. 2026-08-03. Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications. https://arxiv.org/abs/2608.02489

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