Volume 7 (2024)
Frühwirth, Lorenz; Hauke, Manuel
On Birkhoff sums that satisfy no temporal distributional limit theorem for almost every irrational
Annales Henri Lebesgue, Volume 7 (2024), pp. 251-265.

Metadata

DOI10.5802/ahl.200
Keywords Irrational circle rotation, metric Diophantine approximation, temporal limit theorems, ergodic sums

Abstract

Dolgopyat and Sarig showed that for any piecewise smooth function f:𝕋→ℝ and almost every pair (α,x 0 )∈𝕋×𝕋, S N (f,α,x 0 ):=∑ n=1 N f(nα+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 α∈𝕋 and all x 0 ∈𝕋, S N (f,α,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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