Lévy constants of quadratic irrationalities

An irrational number $\alpha$ is said to have Lévy constant $\beta(\alpha)$ if the limit $$\lim_{m \to \infty} \frac 1 m \log q_m(\alpha) =: \beta(\alpha)$$ exists where $q_m(\alpha)$ denotes the denominator of the $m$th convergent of $\alpha$. We give a new proof of the fact that the Lévy constants of quadratic irrationalities are dense in the interval [\log \frac{1+ \sqrt5}2, + \infty)$.