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arXiv · 2308.11286

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

Lorenz Frühwirth·Manuel Hauke
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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.

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Lorenz Frühwirth, Manuel Hauke. 2023-08-22. On Birkhoff sums that satisfy no temporal distributional limit theorem for almost every irrational. https://arxiv.org/abs/2308.11286

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