Constructive Diophantine approximation in generalized continued fraction Cantor sets
Leppälä, Kalle; Törmä, Topi (2018-11-05)
Leppälä, K., Törmä, T. (2018) Constructive Diophantine approximation in generalized continued fraction Cantor sets. Acta Arithmetica, 186 (3), 225-241. doi:10.4064/aa180108-15-8
© Instytut Matematyczny PAN, 2018. Published in this repository with the kind permission of the publisher.
https://rightsstatements.org/vocab/InC/1.0/
https://urn.fi/URN:NBN:fi-fe201901091792
Tiivistelmä
Abstract
We study which asymptotic irrationality exponents are possible for numbers in generalized continued fraction Cantor sets\[E_{\mathcal B}^{\mathcal A} =\Biggl\{\frac{a_1}{b_1+\dfrac{a_2}{b_2+\cdots}}\colon a_n \in {\mathcal A},\, b_n \in {\mathcal B} \text{ for all } n \Biggr\},\]where \({\mathcal A}\) and \({\mathcal B}\) are some given finite sets of positive integers. We give sufficient conditions for \(E^{\mathcal A}_{\mathcal B}\) to contain numbers for any possible asymptotic irrationality exponent and show that sets with this property can have arbitrarily small Hausdorff dimension. We also show that it is possible for \(E^{\mathcal A}_{\mathcal B}\) to contain very well approximable numbers even though the asymptotic irrationality exponents of the numbers in \(E^{\mathcal A}_{\mathcal B}\) are bounded.
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