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

### Constructive Diophantine approximation in generalized continued fraction Cantor sets

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Author: Leppälä, Kalle1,2; Törmä, Topi3
Organizations: 1Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark
2Present address: iPsych Aarhus University, Bartholins Allé 6, DK-8000 Aarhus C, Denmark
3Department of Mathematical Sciences, University of Oulu, P.O. Box 8000, FI-90014 University of Oulu, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.3 MB)
Language: English
Published: Polish Academy of Sciences, Institute of Mathematics, 2018
Publish Date: 2019-01-09
Description:

# 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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Series: Acta arithmetica
ISSN: 0065-1036
ISSN-E: 1730-6264
ISSN-L: 0065-1036
Volume: 186
Pages: 225 - 241
DOI: 10.4064/aa180108-15-8