2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/139441In this paper we prove the following renewal-type limit theorem. Given an irrational $α$ in (0,1) and R>0, let $q_{n_R}$ be the first denominator of the convergents of $α$ which exceeds R. The main result in the paper is that the ratio $q_{n_R}/R$ has a limiting distribution as R tends to infinity. The existence of the limiting distribution uses mixing of a special flow over the natural extension of the Gauss map.To appear in Ergodic Theory Dynam. SystemsDynamical Systemstext