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# Multi angle

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Multi angle Bounded remainder sets for the discrete and continuous irrational rotation

Auteurs : Grepstad, Sigrid (Auteur de la Conférence)
CIRM (Editeur )

Résumé : Let $\alpha$ $\epsilon$ $\mathbb{R}^d$ be a vector whose entries $\alpha_1, . . . , \alpha_d$ and $1$ are linearly independent over the rationals. We say that $S \subset \mathbb{T}^d$ is a bounded remainder set for the sequence of irrational rotations $\lbrace n\alpha\rbrace_{n\geqslant1}$ if the discrepancy
$\sum_{k=1}^{N}1_S (\lbrace k\alpha\rbrace) - N$ $mes(S)$
is bounded in absolute value as $N \to \infty$. In one dimension, Hecke, Ostrowski and Kesten characterized the intervals with this property.
We will discuss the bounded remainder property for sets in higher dimensions. In particular, we will see that parallelotopes spanned by vectors in $\mathbb{Z}\alpha + \mathbb{Z}^d$ have bounded remainder. Moreover, we show that this condition can be established by exploiting a connection between irrational rotation on $\mathbb{T}^d$ and certain cut-and-project sets. If time allows, we will discuss bounded remainder sets for the continuous irrational rotation $\lbrace t \alpha : t$ $\epsilon$ $\mathbb{R}^+\rbrace$ in two dimensions.

Codes MSC :
11J71 - Distribution modulo one
11K06 - General theory of distribution modulo 1
11K38 - Irregularities of distribution, discrepancy

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 01/06/17 Date de captation : 25/05/17 Collection : Research talks Format : MP4 Durée : 00:32:39 Domaine : Number Theory Audience : Chercheurs ; Doctorants , Post - Doctorants Download : http://videos.cirm-math.fr/2017-05-25_Grepstad.mp4 Informations sur la rencontre Nom du congrès : Prime numbers and automatic sequences: determinism and randomness / Nombres premiers et suites automatiques : aléa et déterminismeOrganisteurs Congrès : Dartyge, Cécile ; Drmota, Michael ; Martin, Bruno ; Mauduit, Christian ; Rivat, Joël ; Stoll, ThomasDates : 22/05/17 - 26/05/17 Année de la rencontre : 2017 URL Congrès : http://conferences.cirm-math.fr/1595.htmlCitation DataDOI : 10.24350/CIRM.V.19172203Cite this video as: Grepstad, Sigrid (2017). Bounded remainder sets for the discrete and continuous irrational rotation. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19172203URI : http://dx.doi.org/10.24350/CIRM.V.19172203

### Voir aussi

Bibliographie

1. Grepstad, S., & Larcher, G. (2016). Sets of bounded remainder for the continuous irrational rotation on $[0,1)^2$. Acta Arithmetica, 176(4), 365-395 - http://dx.doi.org/10.4064/aa8453-8-2016

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