A dichotomy between uniform distributions of the Stern-Brocot and the Farey sequence

We employ infinite ergodic theory to show that theeven Stern-Brocot sequence and the Farey sequence are uniformly distributed with respect to certain canonical weightings. As a corollary we derivethe precise asymptotic for the Lebesgue measure of continued fractionsum-level sets as well as connections to asymptotic behaviours of geometricallyand arithmetically restricted Poincaré series. Moreover, we give relations of our main results to elementary observations for the Stern-Brocot tree.