No temporal distributional limit theorem for a.e. irrational translation
[Une translation irrationnelle typique ne satisfait pas de théorème limite temporel]
Dolgopyat, Dmitry1 ; Sarig, Omri2
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)
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=0 n-1 f(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=0 n-1 f(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.

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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) 
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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/

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