Diophantine Approximation by Negative Continued Fraction

We are interested in the statistical behavior of a certain continued fraction whose associated dynamical system has infinite invariant measure. show growth rate denominator $Q_n$ $n$-th convergent negative expansion $x$ and approximation: $$ \frac{\log{n}}{n}\log{\left|x-\frac{P_n}{Q_n}\right|}\rightarrow -\frac{\pi^2}{3} \quad \text{in measure} for $x$. In course proof, we reprove related results that arithmetic mean digits converges to 3 measure, although limit inferior is 2, superior almost everywhere.