A remark on continued fractions with sequences of partial quotients
The author studies the Hausdorff dimension of the set of real numbers \(x\) such that the regular continued fraction expansion of \(x\) has the properties that (i) all partial quotients lie in some subset \(B \subseteq {\mathbb N}\) and (ii) the \(n\)th partial quotient \(a_n(x)\) satisfies the inequality \(a_n(x) \geq f(n)\), where \(f: {\mathbb N} \rightarrow {\mathbb R}\) is some function which tends to infinity. It was shown by \textit{B.-W. Wang} and \textit{J. Wu} [Bull. Lond. Math. Soc. 40, No. 1, 18--22 (2008; Zbl 1155.11046)] that without condition (ii), the Hausdorff dimension of this set is \(\tau(B)/2\), where \(\tau(B)\) is defined as \(\inf\{s > 0 : \sum_{i=1}^\infty b_i^{-s} < \infty\}\) for \(B = \{b_i\}_{i=1}^\infty\). In this paper, given a function \(f\) tending to infinity, the author constructs a set \(B\) with \(\tau(B) = 1\), but such that the Hausdorff dimension of the corresponding set of real numbers is equal to zero. This shows that the result of \textit{Wang} and \textit{Wu} [loc.cit.] is sharp.
scientific article; zbMATH DE number 5565655