Constructive Diophantine approximation in generalized continued fraction Cantor sets 

Author:  Leppälä, Kalle^{1,2}; Törmä, Topi^{3} 
Organizations: 
^{1}Department of Mathematics, Aarhus University, Ny Munkegade 118, DK8000 Aarhus C, Denmark ^{2}Present address: iPsych Aarhus University, Bartholins Allé 6, DK8000 Aarhus C, Denmark ^{3}Department of Mathematical Sciences, University of Oulu, P.O. Box 8000, FI90014 University of Oulu, Finland 
Format:  article 
Version:  accepted version 
Access:  open 
Online Access:  PDF Full Text (PDF, 0.3 MB) 
Persistent link:  http://urn.fi/urn:nbn:fife201901091792 
Language:  English 
Published: 
Polish Academy of Sciences, Institute of Mathematics,
2018

Publish Date:  20190109 
Description: 
AbstractWe 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. see all

Series: 
Acta arithmetica 
ISSN:  00651036 
ISSNE:  17306264 
ISSNL:  00651036 
Volume:  186 
Pages:  225  241 
DOI:  10.4064/aa180108158 
OADOI:  https://oadoi.org/10.4064/aa180108158 
Type of Publication: 
A1 Journal article – refereed 
Field of Science: 
111 Mathematics 
Subjects:  
Copyright information: 
© Instytut Matematyczny PAN, 2018. Published in this repository with the kind permission of the publisher. 