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

TL;DR

This paper uses infinite ergodic theory to demonstrate the uniform distribution of the Stern-Brocot and Farey sequences mod 1, deriving asymptotics for continued fraction sets and linking to Poincaré series.

Contribution

It introduces a novel ergodic-theoretic approach to analyze the distribution properties of these sequences and their connections to number theory and dynamical systems.

Findings

Uniform distribution of sequences mod 1 established

Asymptotic formulas for continued fraction sum-level sets derived

Connections made to Poincaré series and Stern-Brocot tree

Abstract

We employ infinite ergodic theory to show that the even Stern-Brocot sequence and the Farey sequence are uniformly distributed mod 1 with respect to certain canonical weightings. As a corollary we derive the precise asymptotic for the Lebesgue measure of continued fraction sum-level sets as well as connections to asymptotic behaviours of geometrically and arithmetically restricted Poincar\'e series. Moreover, we give relations of our main results to elementary observations for the Stern-Brocot tree.