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.


References

[ADDS15] Avila, Artur; Dolgopyat, Dmitry; Duryev, Eduard; Sarig, Omri M. The visits to zero of a random walk driven by an irrational rotation, Isr. J. Math., Volume 207 (2015), pp. 653-717 | DOI | Zbl

[AK82] Aaronson, Jon; Keane, Michael The visits to zero of some deterministic random walks, Proc. Lond. Math. Soc., Volume 44 (1982), pp. 535-553 | DOI | Zbl

[AS03] Allouche, Jean-Paul; Shallit, Jeffrey Automatic Sequences. Theory, Applications, Generalizations, Cambridge University Press, 2003 | DOI | Zbl

[Bec10] Beck, József Randomness of the square root of 2 and the giant leap. I, Period. Math. Hung., Volume 60 (2010) no. 2, pp. 137-242 | DOI | Zbl

[Bec11] Beck, József Randomness of the square root of 2 and the giant leap. II, Period. Math. Hung., Volume 62 (2011) no. 2, pp. 127-246 | DOI | Zbl

[Bec14] Beck, József Probabilistic Diophantine approximation. Randomness in lattice point counting, Springer Monographs in Mathematics, Springer, 2014 | DOI | Zbl

[Bil95] Billingsley, Patrick Probability and measure, John Wiley & Sons, 1995 | Zbl

[Bor23] Borda, Bence On the distribution of Sudler products and Birkhoff sums for the irrational rotation (2023) to appear in Annales de l’Institut Fourier (Grenoble) | arXiv

[BU18] Bromberg, Michael; Ulcigrai, Corinna A temporal central limit theorem for real-valued cocycles over rotations, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 54 (2018) no. 4, pp. 2304-2334 | DOI | Zbl

[Cas50] Cassels, John W. S. Some metrical theorems in Diophantine approximation I, Math. Proc. Camb. Philos. Soc., Volume 46 (1950), pp. 209-218 | Zbl

[DF15] Dolgopyat, Dmitry; Fayad, Basam Limit theorems for toral translations, Volume 89 (2015), pp. 227-277 | DOI | Zbl

[DS41] Duffin, Richard J.; Schaeffer, Albert C. Khintchine’s problem in metric Diophantine approximation, Duke Math. J., Volume 8 (1941), pp. 243-255 | DOI | Zbl

[DS17] Dolgopyat, Dmitry; Sarig, Omri M. Temporal distributional limit theorems for dynamical systems, J. Stat. Phys., Volume 166 (2017), pp. 680-713 | DOI | Zbl

[DS18] Dolgopyat, Dmitry; Sarig, Omri M. No temporal distributional limit theorem for a.e. irrational translation, Ann. Henri Lebesgue, Volume 1 (2018), pp. 127-148 | DOI | Zbl

[DS20] Dolgopyat, Dmitry; Sarig, Omri M. Quenched and annealed temporal limit theorems for circle rotations, Some aspects of the theory of dynamical systems: a tribute to Jean-Christophe Yoccoz (Astérisque), Volume 415, Société Mathématique de France, 2020, pp. 59-85 | DOI | Zbl

[DV86] Diamond, Harold G.; Vaaler, Jeffrey D. Estimates for partial sums of continued fraction partial quotients, Pac. J. Math., Volume 122 (1986), pp. 73-82 | DOI | Zbl

[FH23] Frühwirth, Lorenz; Hauke, Manuel On the metric upper density of Birkhoff sums for irrational rotations, Nonlinearity, Volume 36 (2023), pp. 7065-7104 | DOI

[Her79] Herman, Michael R. Sur la Conjugaison Différentiable des Difféomorphismes du Cercle à des Rotations, Publ. Math., Inst. Hautes Étud. Sci., Volume 49 (1979), pp. 5-233 | DOI | Zbl

[Kes60] Kesten, Harry Uniform distribution mod 1, Ann. Math., Volume 71 (1960), pp. 445-471 | DOI | Zbl

[KN74] Kuipers, Lauwerens; Niederreiter, Harald Uniform Distribution of Sequences, Pure and Applied Mathematics, John Wiley & Sons, 1974 | Zbl

[RS92] Rockett, Andrew M.; Szüsz, Peter Continued fractions, World Scientific, 1992 | Zbl

[Sch78] Schmidt, Klaus A cylinder flow arising from irregularity of distribution, Compos. Math., Volume 36 (1978), pp. 225-232 | Zbl