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.

Authors