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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 )

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    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 rencontre

    Nom du congrès : Prime numbers and automatic sequences: determinism and randomness / Nombres premiers et suites automatiques : aléa et déterminisme
    Organisteurs Congrès : Dartyge, Cécile ; Drmota, Michael ; Martin, Bruno ; Mauduit, Christian ; Rivat, Joël ; Stoll, Thomas
    Dates : 22/05/17 - 26/05/17
    Année de la rencontre : 2017
    URL Congrès : http://conferences.cirm-math.fr/1595.html

    Citation Data

    DOI : 10.24350/CIRM.V.19172203
    Cite this video as: Grepstad, Sigrid (2017). Bounded remainder sets for the discrete and continuous irrational rotation. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19172203
    URI : http://dx.doi.org/10.24350/CIRM.V.19172203

    Voir aussi


    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