# A Remark on the Growth of the Denominators of Convergents

Research paper by **Jun Wu**

Indexed on: **10 Feb '06**Published on: **10 Feb '06**Published in: **Monatshefte für Mathematik**

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#### Abstract

For \({\rm log}\frac{1+\sqrt{5}}{2}\leq \lambda_\ast \leq \lambda^\ast < \infty\), let E(λ*, λ*) be the set
\( \left\{x\in [0,1):\ \mathop{\lim\inf}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda_{\ast}, \mathop{\lim\sup}_{n \rightarrow \infty}\frac{\log q_n(x)}{n}=\lambda^{\ast}\right\}.\)It has been proved in [1] and [3] that E(λ*, λ*) is an uncountable set. In the present paper, we strengthen this result by showing that
\(\dim E(\lambda_{\ast}, \lambda^{\ast}) \ge \frac{\lambda_{\ast} -\log \frac{1+\sqrt{5}}{2}}{2\lambda^{\ast}}\)where dim denotes the Hausdorff dimension.