We construct a cross-section to the principal congruence modular flow which is represented as a skew product transformation over the natural extension of the Gauss map. This leads to a new proof of Moeckel's theorem on rational approximants.
Each irrational number $x$ in the unit interval has a continued fraction expansion $[a_1, a_2, ...]$, giving rational numbers $p_n/q_n= [a_1,…, a_k]$, the approximants for $x$. Each approximate is of one of three types: odd/even, even/odd or odd/odd. Moeckel's theorem states that for almost every $x$ the limiting frequency of each of these three types exists. What is unusual in the proof is that this does not follow directly from the ergodic theorem applied to an observable on the Gauss map (the shift on continued fractions): one must first enlarge the space.
Moeckel's approach makes use of the geodesic flow on a three-fold cover of the modular surface, together with a geometric argument for counting the time that geodesics spend in cusps. Ergodicity of the flow is well known, but the counting is somewhat involved. Later Jager and Liardet found a purely ergodic theoretic proof, constructing a skew product over the Gauss map. There the counting is direct, but the proof of ergodicity is more difficult. Our proof unifies the two earlier arguments, inheriting the strong points of each.