Dyadic approximation in the middle-third Cantor set

• Demi Allen ^[1] ; Sam Chow ^[2] ; Han Yu ^[3]
  1. [1] University of Exeter

     University of Exeter

     Exeter District, Reino Unido

     
  2. [2] University of Warwick

     University of Warwick

     Reino Unido

     
  3. [3] Centre for Mathematical Sciences, UK

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 29, Nº. 1, 2023
• Idioma: inglés
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  • In this paper, we study the metric theory of dyadic approximation in the middle-third Cantor set. This theory complements earlier work of Levesley et al. (Math Ann 338(1):97–118, 2007), who investigated the problem of approximation in the Cantor set by triadic rationals. We find that the behaviour when we consider dyadic approximation in the Cantor set is substantially different to considering triadic approximation in the Cantor set. In some sense, this difference in behaviour is a manifestation of Furstenberg’s times 2 times 3 phenomenon from dynamical systems, which asserts that the base 2 and base 3 expansions of a number are not both structured.