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

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

2010-09-09

arxiv.org/abs/1009.1823v1

Mathematics

Number Theory

9 pages

Scientific paper

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.

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