Constructive Diophantine approximation in generalized continued fraction Cantor sets

Abstract

<div lang="en" class="abs"><p class="abs_header">Abstract</p><p>We study which asymptotic irrationality exponents are possible for numbers in <em>generalized continued fraction Cantor sets</em>\[E_{\mathcal B}^{\mathcal A} =\Biggl\{\frac{a_1}{b_1+\dfrac{a_2}{b_2+\cdots}}\colon a_n \in {\mathcal A},\, b_n \in {\mathcal B} \text{ for all } n \Biggr\},\]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.</p></div>