Fruhwirth 2024 birkhoff sums that satisfy no temporal

../

---

Lorenz Frühwirth, Manuel Hauke, On Birkhoff sums that satisfy no temporal distributional limit theorem for almost every irrational. Annales Henri Lebesgue 7 (2024), 251-265. doi:10.5802/ahl.200.

Theorem 1.2 proves that for every piecewise smooth zero-mean function f (Definition 1.1: differentiable off finitely many points, derivative of bounded variation, at least one genuine jump), for Haar-almost every alpha in the torus and for every starting point x_0, the Birkhoff sums S_N(f,alpha,x_0) of the irrational rotation satisfy no temporal distributional limit theorem, whatever centering A_M and scaling B_M are chosen. This sharpens the doubly metric theorem of Dolgopyat and Sarig (Theorem A in the paper, from their 2018 paper, over almost all pairs (alpha,x_0)) to a single metric statement, showing the full-measure set where no such theorem holds can be taken as A x T with A of full measure, and it strengthens their sawtooth corollary (Corollary 2.3 of their 2020 paper); it also advances a question they posed about general piecewise smooth f. The method avoids Fourier analysis: f is decomposed into the sawtooth plus indicator functions (Proposition 3.1), the metric theory of continued fractions supplies almost surely infinitely many unusually large partial quotients with well-behaved convergent denominators (Lemma 3.4), Lemma 3.7 then identifies limit distributions along suitable subsequences, and two of them are shown to differ. For problem 1002 this is the quenched-temporal negative result, a different averaging regime from Erdos's spatial question (n fixed, alpha random); the introduction surveys Kesten's spatial and Dolgopyat-Sarig's annealed temporal Cauchy limits, so #1002 itself is not settled here.

Source: https://doi.org/10.5802/ahl.200. The file prints "Published under license CC BY 4.0." and "This journal is a member of Centre Mersenne." on p. 265: the Creative Commons Attribution 4.0 license.

Bears on. #1002

Results to transcribe.

• Theorem 1.2: For any piecewise smooth f, almost every alpha and every x_0, S_N(f,alpha,x_0) with N uniform on {1,...,M} satisfies no distributional limit theorem for any centering and scaling.
• Definition 1.1: Piecewise smooth functions: zero mean, differentiable off finitely many points, derivative extending to bounded variation, with at least one nonzero jump.
• Remark after Theorem 1.2: The full-measure set E in Dolgopyat-Sarig's Theorem A can be taken as A x T with A of full one-dimensional Haar measure; the authors do not know whether the Fourier-analytic method of Dolgopyat and Sarig could be adapted to give this.
• Lemma 3.4 (and Remark 3.5): For almost every alpha there are infinitely many exceptionally large partial quotients, and the corresponding convergent denominators can be taken to have further properties; the authors suggest this may be of interest in itself.
• Lemma 3.7 / equation (3.2): Limit distributions of S_N(f,alpha,x_0) along certain subsequences, in a family comparable to the one Dolgopyat and Sarig found for the sawtooth.