The sets of different continued fractions with the same partial quotients (Q2867005)

Every real irrational number \(x \in (0,1)\) admits an expansion as a simple (or regular) continued fraction, NEWLINE\[NEWLINE x = {1 \over {a_1(x) + {1 \over {a_2(x) + {1 \over {a_3(x) \dots}}}}}} NEWLINE\]NEWLINE and as an Engel continued fraction NEWLINE\[NEWLINE x = {1 \over {e_1(x) + {e_1(x) \over {e_2(x) + {e_2 \over {e_3(x) + \dots}}}}}}, NEWLINE\]NEWLINE where \((a_i(x))\) is some sequence of natural numbers and \((e_i(x))\) is a strictly increasing sequence of natural numbers. Conversely, every such sequence gives rise to an irrational number in \((0,1)\).NEWLINENEWLINELet \(B = (b_n)\) be some strictly increasing sequence of natural numbers. In the paper under review, the authors study the sets NEWLINE\[NEWLINE R(B) = \{x \in (0,1) : a_n(x) \in B, a_n(x) \rightarrow \infty\}, NEWLINE\]NEWLINE NEWLINE\[NEWLINE E(B) = \{x \in (0,1) : e_n(x) \in B, a_n(x) \rightarrow \infty\}. NEWLINE\]NEWLINE It is shown that if for some subsequence \((b_{n_i})\) of \(B\), NEWLINE\[NEWLINE \lim_{i \rightarrow \infty} {\log n_i \over \log b_{n_i}} = \alpha, \quad {n_{i+1} \over n_i} \leq M, NEWLINE\]NEWLINE for some \(\alpha, M\), then NEWLINE\[NEWLINE \dim E(B) = 2 \dim R(B) = \alpha = \inf\Bigg\{ \tau > 0: \sum_{n = 1}^\infty {1 \over b_n^\tau} < \infty \Bigg\}. NEWLINE\]NEWLINE Conversely it is shown that a growth condition on \(B\) is indeed required, as for any \(a,c > 1\), the sets \(E(a,c)\) of numbers with \(e_n(x) \geq a^{c^n}\) and \(R(a,c)\) of numbers with \(a_n(x) \geq a^{c^n}\) fail to satisfy the above equation.