Dyadic approximation in the middle-third Cantor set

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 w...

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Bibliographic Details
Published in:Selecta mathematica (Basel, Switzerland) Switzerland), 2023-02, Vol.29 (1), Article 11
Main Authors: Allen, Demi, Chow, Sam, Yu, Han
Format: Article
Language:English
Subjects:
Approximation
Mathematical analysis
Mathematics
Mathematics and Statistics
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Summary: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.
ISSN:1022-1824
1420-9020
DOI:10.1007/s00029-022-00814-x