PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Constructive Diophantine approximation in generalized continued fraction Cantor sets

Volume 186 / 2018

Kalle Leppälä, Topi Törmä Acta Arithmetica 186 (2018), 225-241 MSC: Primary 11J82, 11J70; Secondary 11K50. DOI: 10.4064/aa180108-15-8 Published online: 5 November 2018


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.


  • Kalle LeppäläDepartment of Mathematics
    Aarhus University
    Ny Munkegade 118
    DK-8000 Aarhus C, Denmark
    Aarhus University
    Bartholins Allé 6
    DK-8000 Aarhus C, Denmark
  • Topi TörmäDepartment of Mathematical Sciences
    University of Oulu
    P.O. Box 8000
    FI-90014 University of Oulu, Finland

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image