  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.
