On the distribution of Sudler products and Birkhoff sums for the irrational rotation

Borda, Bence

doi:10.5802/aif.3640

On the distribution of Sudler products and Birkhoff sums for the irrational rotation
[À propos de la répartition des produits de Sudler et des sommes de Birkhoff de la rotation irrationnelle]

Borda, Bence ¹

¹ Graz University of Technology, Steyrergasse 30, 8010 Graz (Austria)

Annales de l'Institut Fourier, Online first, 63 p.

Résumé
Abstract

Dans cet article, nous étudions la répartition des valeurs du produit de Sudler $\prod_{n = 1}^{N} | 2 sin (π n α) |$ et du produit diophantien $\prod_{n = 1}^{N} (2 e ∥ n α ∥)$ pour plusieurs types de nombres irrationnels de $α$ , avec $N$ variant dans un long intervalle d’entiers. Pour les irrationnels mal approchables par des nombres rationnels, ces produits ont une forte concentration autour de $N^{1 / 2}$ , et pour certains irrationnels quadratiques, ils vérifient même un théorème central limite. En revanche, pour presque tout $α$ , nous observons un phénomène d’anti-concentration intéressant où les valeurs typiques et extrêmes sont du même ordre de grandeur. Nos méthodes conviennent également à l’étude de la répartition des sommes de Birkhoff $\sum_{n = 1}^{N} f (n α)$ pour les rotations du cercle. En utilisant l’approximation diophantienne et l’analyse de Fourier, nous déterminons la moyenne et le moment d’ordre deux pour une fonction périodique quelconque $f$ à variation bornée, et démontrons (presque) une conjecture de Bromberg et Ulcigrai quant au facteur de normalisation approprié dans un certain théorème limite temporel. Les sommes de Birkhoff vérifient également un théorème central limite pour certains irrationnels quadratiques.

In this paper, we study the value distribution of the Sudler product $\prod_{n = 1}^{N} | 2 sin (π n α) |$ and the Diophantine product $\prod_{n = 1}^{N} (2 e ∥ n α ∥)$ for various irrational $α$ , as $N$ ranges in a long interval of integers. At badly approximable irrationals these products exhibit strong concentration around $N^{1 / 2}$ , and at certain quadratic irrationals they even satisfy a central limit theorem. In contrast, at almost every $α$ we observe an interesting anticoncentration phenomenon when the typical and the extreme values are of the same order of magnitude. Our methods are equally suited for the value distribution of Birkhoff sums $\sum_{n = 1}^{N} f (n α)$ for circle rotations. Using Diophantine approximation and Fourier analysis, we find the first and second moment for an arbitrary periodic $f$ of bounded variation, and (almost) prove a conjecture of Bromberg and Ulcigrai on the appropriate scaling factor in a so-called temporal limit theorem. Birkhoff sums also satisfy a central limit theorem at certain quadratic irrationals.

Reçu le : 2021-03-29
Révisé le : 2023-03-06
Accepté le : 2023-05-15
Première publication : 2024-05-17

DOI : 10.5802/aif.3640

Classification : 11K60, 37A50, 37E10
Keywords: Toral translation, Diophantine approximation, quadratic irrational, Fourier series, central limit theorem, temporal limit theorem.
Mot clés : Translation sur le tore, approximation diophantienne, irrationnel quadratique, séries de Fourier, théorème central limite, théorème limite temporel.

Affiliations des auteurs :

Borda, Bence ¹

¹ Graz University of Technology, Steyrergasse 30, 8010 Graz (Austria)

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Borda, Bence. On the distribution of Sudler products and Birkhoff sums for the irrational rotation. Annales de l'Institut Fourier, Online first, 63 p.

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