On the growth of the denominators of convergents (Q1307418)

The following result is proved. Let \(\log \frac{1+\sqrt 5}2\leq X\leq Y<\infty\). Then there exist non-enumerable many pairwise not equivalent numbers \(\alpha\) such that \[ \lim_{\overline{n\to \infty}} \frac 1n \log q_n =X\quad \text{and}\quad \overline{\lim_{n\to\infty}} \frac 1n \log q_n= Y, \] where \(q_n\) denotes the denominator of the \(n^{th}\) convergent of \(\alpha\).