Constructive Diophantine approximation in generalized continued fraction Cantor sets
Volume 186 / 2018
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.
