Bounded remainder sets for the discrete and continuous irrational rotation | Vidéo | Carmin.tv

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Bounded remainder sets for the discrete and continuous irrational rotation

De Sigrid Grepstad

Apparaît dans les collections : Prime numbers and automatic sequences: determinism and randomness / Nombres premiers et suites automatiques : aléa et déterminisme, Exposés de recherche

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.

Informations sur la vidéo

Date de captation 25/05/2017
Date de publication 01/06/2017
Institut CIRM
Licence CC BY NC ND
Langue Anglais
Audience Chercheurs
Réalisateur(s) Guillaume Hennenfent
Format MP4

Données de citation

DOI 10.24350/CIRM.V.19172203
Citer cette vidéo Grepstad Sigrid (25/05/2017). Bounded remainder sets for the discrete and continuous irrational rotation. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19172203
URL https://dx.doi.org/10.24350/CIRM.V.19172203

Domaine(s)

Théorie des nombres

Bibliographie

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

Codes MSC

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