Uniform Diophantine approximation related to beta-transformations

• Wanlou Wu ^[1] ; Yingqing Zhang ^[1]
  1. [1] Jiangsu Normal University

     Jiangsu Normal University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 6, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Let Tβ (β > 2) be the β-transformation on (0, 1]. Fix some x0 ∈ [0, 1] whose β-expansion does not include the characters 0 and β − 1, given a nonnegative real number ˆv , we compute the Hausdorff dimension of the set of all real numbers x ∈ (0, 1] with the property that, for every sufficiently large integer N, there is an integer n ∈ [1, N] such that the distance between T n β x and x0 is at most equal to β −Nˆv . This work generalizes the result of Bugeaud and Liao [9] where only x0 = 0 was taken into consideration.

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