On the Growth of the Denominators of Convergents

Abstract

Let log $\log \left( {\left( {1 + \sqrt 5 } \right)/2} \right) \underline{\underline < } X\underline{\underline < } Y$. We prove that there exist non-denumerably many pairwise not equivalent irrational numbers α such that $\mathop {\underline {\lim } }\limits_{n \to \infty } (1/n) log q_n (\alpha ) = X$ and $\overline {\mathop {\lim }\limits_{n \to \infty } } (1/n) log q_n (\alpha ) = Y$ where qn(α) denotes the denominator of the nth convergent of α.