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

TL;DR

This paper proves that for almost every irrational number, Birkhoff sums of certain functions do not satisfy any temporal distributional limit theorem, regardless of centering or scaling, advancing understanding in dynamical systems.

Contribution

It shows that the failure of temporal distributional limit theorems for Birkhoff sums holds for individual irrationals, not just on average, improving previous results.

Findings

Birkhoff sums do not satisfy temporal distributional limit theorems for almost every irrational.

The result holds uniformly for all initial points without averaging over pairs.

Progress is made on a question posed by Dolgopyat and Sarig.

Abstract

Dolgpoyat and Sarig showed that for any piecewise smooth function $f: \mathbb{T} \to \mathbb{R}$ and almost every pair $(\alpha,x_0) \in \mathbb{T} \times \mathbb{T}$, $S_N(f,\alpha,x_0) := \sum_{n =1}^{N} f(n\alpha + x_0)$ 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 $\alpha \in \mathbb{T}$ and all $x_0 \in \mathbb{T}$, $S_N(f,\alpha,x_0)$ 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.