AI 中文总结
研究遍历理论中环面平移加权伯克霍夫平均收敛问题，发现即使给定非共振频率，也存在低正则性权重和可观测量族，使加权平均呈指数收敛，加深了对非共振与正则性相互作用的理解。
AI 中文摘要
自1978年克伦格尔的工作以来，遍历定理中不存在有效收敛速率已广为人知。对于环面平移，通过选择合适权重可将遍历平均的收敛加速到指数速率，但直观上这需要高度非共振频率和非常正则的可观测量。本文发现新现象：即使对于任何给定非共振频率，存在非平凡的低正则性权重和可观测量族，使得沿拟周期轨道的加权伯克霍夫平均以定量、均匀和指数速率收敛。这不仅加深了对遍历理论中非共振与正则性之间深层相互作用的理解，也是与约科兹型结果相对应的加权情形。
英文摘要
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].
CommentsTo appear in Bulletin of the London Mathematical Society. 14 pages, 2 figures. Any comments are welcome!
Journal refBulletin of the London Mathematical Society 58 (2026), no. 9, Paper No. e70474, 14 pp
