On Birkhoff sums that satisfy no temporal distributional limit theorem for almost every irrational

Lorenz Frühwirth; Manuel Hauke

paper

On Birkhoff sums that satisfy no temporal distributional limit theorem for almost every irrational

arXiv:2308.11286

Abstract

Dolgpoyat and Sarig showed that for any piecewise smooth function $f : T \to R$ and almost every pair $(α, x_{0}) \in T \times T$ , $S_{N} (f, α, x_{0}) := \sum_{n = 1}^{N} f (n α + x_{0})$ fails to fulfill a temporal distributional limit theorem. In this article, we show that the two-dimensional average is in fact not needed: For almost every $α \in T$ and all $x_{0} \in T$ , $S_{N} (f, α, x_{0})$ does not satisfy a temporal distributional limit theorem, regardless of centering and scaling. The obtained results additionally lead to progress in a question posed by Dolgopyat and Sarig.

14 pages, 1 figure, all comments are appreciated