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H 1 Algebraic sums and products of univoque bases

Auteurs : Dajani, Karma (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : Given $x\in(0, 1]$, let ${\mathcal U}(x)$ be the set of bases $\beta\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^{\infty}{{d_i}/{\beta^i}}$. In 2014, Lü, Tan and Wu proved that ${\mathcal U}(x)$ is a Lebesgue null set of full Hausdorff dimension. In this talk, we will show that the algebraic sum ${\mathcal U}(x)+\lambda {\mathcal U}(x)$, and the product ${\mathcal U}(x)\cdot {\mathcal U}(x)^{\lambda}$ contain an interval for all $x\in (0, 1]$ and $\lambda\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters associated with the family of symmetric binary expansions studied recently by the first speaker and C. Kalle.
    This is joint work with V. Komornik, D. Kong and W. Li.

    Keywords : algebraic differences; non-integer base expansions, univoque bases; thickness; Cantor sets; non-matching parameters

    Codes MSC :
    11A63 - Radix representation; digital problems
    28A80 - Fractals
    37B10 - Symbolic dynamics

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 07/12/2017
      Date de captation : 05/12/2017
      Collection : Research talks
      Format : MP4
      Durée : 00:58:21
      Domaine : Number Theory ; Dynamical Systems & ODE
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2017-12-05_Dajani.mp4

    Informations sur la rencontre

    Nom du congrès : Jean-Morlet chair: Tiling and recurrence / Chaire Jean-Morlet : Pavages et récurrence
    Organisteurs Congrès : Akiyama, Shigeki ; Arnoux, Pierre
    Dates : 04/12/2017 - 08/12/2017
    Année de la rencontre : 2017
    URL Congrès : https://akiyama-arnoux.weebly.com/conference.html

    Citation Data

    DOI : 10.24350/CIRM.V.19249903
    Cite this video as: Dajani, Karma (2017). Algebraic sums and products of univoque bases. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19249903
    URI : http://dx.doi.org/10.24350/CIRM.V.19249903


    Voir aussi

    Bibliographie

    1. Dajani, K., Komornik, V., Kong, D., & Li, W. (2017). Algebraic sums and products of univoque bases. - https://arxiv.org/abs/1710.03291

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