On the rate of convergence of continued fraction statistics of random rationals

We show that the statistics of the continued fraction expansion of a randomly chosen rational in the unit interval, with a fixed large denominator $q$, approaches the Gauss-Kuzmin statistics with polynomial rate in $q$. This improves on previous results giving the convergence without rate. As an application of this effective rate of convergence, we show that the statistics of a randomly chosen rational in the unit interval, with a fixed large denominator $q$ and prime numerator, also approaches the Gauss-Kuzmin statistics. Our results are obtained as applications of improved non-escape of mass and equidistribution statements for the geodesic flow on the space $SL_2(\mathbb{R})/SL_2(\mathbb{Z})$.