We consider the statistics of the continued fraction expansion of a randomly chosen rational in the unit interval with a fixed large denominator q. We show that the statistics approach the Gauss-Kuzmin statistics as q goes to infinity, with probability approaching 1 at a polynomial rate in q.
This improves on previous results giving the convergence without rate.
Our results are obtained as an application of an equidistribution statement for divergent orbits of the geodesic flow on the space SL(2,R)/SL(2,Z) and uses bounds on entropy of invariant measures that spend a large amount of time in the cusp.
Joint work with Ofir David, Taehyeong Kim, and Uri Shapira.