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

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.