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

Frühwirth, Lorenz; Hauke, Manuel

Annales Henri Lebesgue, Tome 7 (2024), pp. 251-265

Dolgopyat and Sarig showed that for any piecewise smooth function $f:𝕋\to ℝ$ and almost every pair $(\alpha ,x_0)\in 𝕋\times 𝕋$, $S_N(f,\alpha ,x_0):={\sum }_{n=1}^{N}f(n\alpha +x_0)$ fails to fulfill a temporal distributional limit theorem. In this article, we show that the doubly metric statement can be sharpened to a single metric one: For almost every $\alpha \in 𝕋$ and all $x_0\in 𝕋$, $S_N(f,\alpha ,x_0)$ 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.

Dolgopyat et Sarig ont montré que pour toute fonction lisse par morceaux $f:𝕋\to ℝ$ et presque tout couple $(\alpha ,x_0)\in 𝕋\times 𝕋$, alors $S_N(f,\alpha ,x_0):={\sum }_{n=1}^{N}f(n\alpha +x_0)$ ne peut satisfaire un théorème limite distributionnel temporel. Dans cet article, nous montrons que l’énoncé sur le produit peut être raffiné en un énoncé sur une seule composante : pour presque tout $\alpha \in 𝕋$ et pour tout $x_0\in 𝕋$, $S_N(f,\alpha ,x_0)$ ne satisfait pas de théorème limite distributionnel temporel, quels que soient le centrage et la mise à l’échelle. Les résultats obtenus permettent en outre de progresser sur une question posée par Dolgopyat et Sarig.

Classification : 37E10, 11K60, 11K50, 11J83, 37A44
Keywords: Irrational circle rotation, metric Diophantine approximation, temporal limit theorems, ergodic sums

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