The distribution of Jager pairs for continued fraction like mappings of the interval

Andrew Haas and David Molnar

Vol. 217 (2004), No. 1, 101–114

DOI: 10.2140/pjm.2004.217.101

Abstract

Certain ergodic, piecewise Möbius self-mappings of the unit interval, similar to the classical Gauss or Rényi maps, give rise to natural sequences of convergents p_n∕q_n for every associated “irrational” number x. Here we study the metric theory of the approximation sequences 𝜃_n = |q_n||q_nx − p_n|. Following Jager we describe the distribution of pairs (𝜃_n,𝜃_n+1) in a plane domain by deriving their distribution function. As a consequence we get a generalization of the theorem of Bosma, Jager and Wiedijk, referred to as the Lenstra Conjecture, which describes the distribution of the 𝜃_n.

Milestones

Received: 4 October 2002

Published: 1 November 2004