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Diophantine Approximations on Fractals

Einsiedler, Manfred ; Fishman, Lior ; Shapira, Uri

In: Geometric and Functional Analysis, 2011, vol. 21, no. 1, p. 14-35

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    Titre
    • Diophantine Approximations on Fractals
    Auteur
    • Einsiedler, Manfred. Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland
    • Fishman, Lior. Department of Mathematics, Brandeis University, 415 South Street MS 050, P.O. Box 9110, 02454, Waltham, MA, USA
    • Shapira, Uri. Departement Mathematik, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland
    Type de document
    • Postprint
    Langue
    • Anglais
    Publié dans
    • Geometric and Functional Analysis, 2011, vol. 21, no. 1, p. 14-35. SP Birkhäuser Verlag Basel
    Autre version électronique
    Identifiant OAI-PMH
    • oai:doc.rero.ch:318563
    Summary
    We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0, 1]2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of $${\{nx\,{\rm mod}\,1\}_{n \in {\mathbb N}}}$$ are uniformly eventually bounded