A note on continued fractions with sequences of partial quotients over the field of formal power series (Q2902044)

The authors study the metric theory of continued fractions over the field \({\mathbb F}((X^{-1}))\) of formal Laurent series with coefficients in the finite field \({\mathbb F}\) with \(q\) elements. For an element \(x \in I\), the unit ball in \({\mathbb F}((X^{-1}))\), denote by \(A_n(x) \in {\mathbb F}[X]\) the \(n\)th partial quotient of \(x\). Let \({\mathbb B} \subseteq {\mathbb F}[X]\) be an infinite subset, and let \(f : {\mathbb N} \rightarrow {\mathbb N}\) be a function tending to infinity with \(n\). The authors consider the Hausdorff dimension of the set NEWLINE\[NEWLINE E({\mathbb B}, f) = \{x \in I : A_n(x) \in {\mathbb B}, \deg A_n(x) \geq f(n) \,\, \forall n \geq 1\}. NEWLINE\]NEWLINE In view of results of \textit{X. Hu} and \textit{J. Wu} [Acta Arith. 136, No. 3, 201--211 (2009; Zbl 1254.11077)], it is natural to conjecture that NEWLINE\[NEWLINE \dim_H E({\mathbb B}, f) = \alpha_{\mathbb B} = \limsup_{n \rightarrow \infty} {{\log \# \{b\in {\mathbb B} : \deg b = n\}}\over{2n \log q}}. NEWLINE\]NEWLINE However, in the paper under review it is shown that for any \(f\), there is a set \({\mathbb B}\) such that \(\alpha_{\mathbb B} = 1/2\), but \(\dim_H E({\mathbb B}, f) = 0\). On the other hand, if the limit in the definition of \(\alpha_{\mathbb B}\) exists and \(\limsup_{n \rightarrow \infty} f(n)/n = 0\), then the conjectured result is in fact true, and \(\dim_H E({\mathbb B}, f) = \alpha_{\mathbb B}\).NEWLINENEWLINEAs remarked by the authors, conditions on both the asymptotic density of \({\mathbb B}\) and on the growth of \(f\) are needed in order to ensure that the dimension is as expected. It is however not clear that the conditions in the paper are optimal.