No temporal distributional limit theorem for a.e. irrational translation
[Une translation irrationnelle typique ne satisfait pas de théorème limite temporel]
Annales Henri Lebesgue, Tome 1 (2018), pp. 127-148.

Bromberg et Ulcigrai ont construit des fonctions lisses par morceaux sur le cercle pour lesquelles l’ensemble des α tels que la somme k=0n-1f(x+kαmod1) satisfait un théorème limite temporel le long de l’orbite de presque tout x est un ensemble de dimension de Hausdorff 1. Nous montrons que cet ensemble est de mesure nulle.

Bromberg and Ulcigrai constructed piecewise smooth functions on the circle such that the set of α for which the sum k=0n-1f(x+kαmod1) satisfies a temporal distributional limit theorem along the orbit of a.e. x has Hausdorff dimension one. We show that the Lebesgue measure of this set is equal to zero.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/ahl.4
Classification : 37D25, 37D35
Dolgopyat, Dmitry 1 ; Sarig, Omri 2

1 Department of Mathematics University of Maryland at College Park College Park, MD 20742 (USA)
2 Faculty of Mathematics and Computer Science The Weizmann Institute of Science 234 Herzl Street 7610001 Rehovot (Israel)
@article{AHL_2018__1__127_0,
     author = {Dolgopyat, Dmitry and Sarig, Omri},
     title = {No temporal distributional limit theorem for a.e. irrational translation},
     journal = {Annales Henri Lebesgue},
     pages = {127--148},
     publisher = {\'ENS Rennes},
     volume = {1},
     year = {2018},
     doi = {10.5802/ahl.4},
     mrnumber = {3963288},
     zbl = {1420.37005},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ahl.4/}
}
TY  - JOUR
AU  - Dolgopyat, Dmitry
AU  - Sarig, Omri
TI  - No temporal distributional limit theorem for a.e. irrational translation
JO  - Annales Henri Lebesgue
PY  - 2018
SP  - 127
EP  - 148
VL  - 1
PB  - ÉNS Rennes
UR  - http://www.numdam.org/articles/10.5802/ahl.4/
DO  - 10.5802/ahl.4
LA  - en
ID  - AHL_2018__1__127_0
ER  - 
%0 Journal Article
%A Dolgopyat, Dmitry
%A Sarig, Omri
%T No temporal distributional limit theorem for a.e. irrational translation
%J Annales Henri Lebesgue
%D 2018
%P 127-148
%V 1
%I ÉNS Rennes
%U http://www.numdam.org/articles/10.5802/ahl.4/
%R 10.5802/ahl.4
%G en
%F AHL_2018__1__127_0
Dolgopyat, Dmitry; Sarig, Omri. No temporal distributional limit theorem for a.e. irrational translation. Annales Henri Lebesgue, Tome 1 (2018), pp. 127-148. doi : 10.5802/ahl.4. http://www.numdam.org/articles/10.5802/ahl.4/

[ADDS15] Avila, Artur; Dolgopyat, Dmitry; Duryev, Eduard; Sarig, Omri The visits to zero of a random walk driven by an irrational rotation, Isr. J. Math., Volume 207 (2015) no. 2, pp. 653-717 | DOI | MR | Zbl

[ADU93] Aaronson, Jon; Denker, Manfred; Urbański, Mariusz Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Am. Math. Soc., Volume 337 (1993) no. 2, pp. 495-548 | DOI | MR | Zbl

[AK82] Aaronson, Jon; Keane, Michael The visits to zero of some deterministic random walks, Proc. Lond. Math. Soc., Volume 44 (1982) no. 3, pp. 535-553 | DOI | MR | Zbl

[Bec94] Beck, József Probabilistic Diophantine approximation. I. Kronecker sequences, Ann. Math., Volume 140 (1994) no. 1, pp. 109-160 | DOI | MR

[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 | MR | 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 | MR | Zbl

[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 | MR | Zbl

[CK76] Conze, Jean-Pierre; Keane, Michael Ergodicité d’un flot cylindrique, Séminaire de Probabilités, I (Univ. Rennes, Rennes, 1976), Université Rennes, 1976 (Exp. no. 5, 7 pages) | MR

[DS17a] Dolgopyat, Dmitry; Sarig, Omri Quenched and annealed temporal limit theorems for circle rotations (2017) (28 pages, preprint)

[DS17b] Dolgopyat, Dmitry; Sarig, Omri Temporal distributional limit theorems for dynamical systems, J. Stat. Phys., Volume 166 (2017) no. 3-4, pp. 680-713 | DOI | MR | Zbl

[DS18] Dolgopyat, Dmitry; Sarig, Omri Asymptotic windings of horocycles, Isr. J. Math., Volume 228 (2018) no. 1, pp. 119-176 | DOI | MR | Zbl

[DV86] Diamond, Harold G.; Vaaler, Jeffrey D. Estimates for partial sums of continued fraction partial quotients, Pac. J. Math., Volume 122 (1986) no. 1, pp. 73-82 | DOI | MR | Zbl

[HW08] Hardy, Godfrey Harold; Wright, Edward Maitland An introduction to the theory of numbers, Oxford University Press, 2008, xxii+621 pages (Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles) | MR

[Jar29] Jarník, Vojtech Zur metrischen Theorie der diophantischen Approximationen, Prace Mat.-Fiz., Volume 36 (1929) no. 1, pp. 91-106 | Zbl

[Khi24] Khintchine, Alexander Ya. Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen, Math. Ann., Volume 92 (1924) no. 1-2, pp. 115-125 | DOI | MR | Zbl

[Khi63] Khintchine, Alexander Ya. Continued fractions, P. Noordhoff, Ltd., 1963, iii+101 pages (translated by Peter Wynn) | MR | Zbl

[PS17] Paquette, Elliot; Son, Younghwan Birkhoff sum fluctuations in substitution dynamical systems, Ergodic Theory Dyn. Syst. (2017), 35 pages | DOI | Zbl

[Sch78] Schmidt, Klaus A cylinder flow arising from irregularity of distribution, Compos. Math., Volume 36 (1978) no. 3, pp. 225-232 | Numdam | MR | Zbl

[Sul82] Sullivan, Dennis Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math., Volume 149 (1982) no. 3-4, pp. 215-237 | DOI | MR | Zbl

  • Frühwirth, Lorenz; Hauke, Manuel On Birkhoff sums that satisfy no temporal distributional limit theorem for almost every irrational, Annales Henri Lebesgue, Volume 7 (2024), p. 251 | DOI:10.5802/ahl.200
  • Borda, Bence On the distribution of Sudler products and Birkhoff sums for the irrational rotation, Annales de l'Institut Fourier, Volume 74 (2024) no. 5, p. 2013 | DOI:10.5802/aif.3640
  • Frühwirth, Lorenz; Hauke, Manuel On the metric upper density of Birkhoff sums for irrational rotations, Nonlinearity, Volume 36 (2023) no. 12, p. 7065 | DOI:10.1088/1361-6544/ad086f
  • Bountis, Anastasios; Veerman, J.J.P.; Vivaldi, Franco Cauchy distributions for the integrable standard map, Physics Letters A, Volume 384 (2020) no. 26, p. 126659 | DOI:10.1016/j.physleta.2020.126659
  • Marmi, Stefano; Ulcigrai, Corinna; Yoccoz, Jean-Christophe On Roth type conditions, duality and central Birkhoff sums for i.e.m, arXiv (2019) | DOI:10.48550/arxiv.1901.09191 | arXiv:1901.09191
  • Bromberg, Michael; Ulcigrai, Corinna A temporal central limit theorem for real-valued cocycles over rotations, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Volume 54 (2018) no. 4 | DOI:10.1214/17-aihp872

Cité par 6 documents. Sources : Crossref, NASA ADS