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

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.