Sequences and dynamical systems associated with canonical approximation by rationals

TL;DR

This paper investigates the statistical properties of subsequences derived from rational approximations of irrational numbers, utilizing advanced methods from number theory and dynamical systems to deepen understanding of approximation behaviors.

Contribution

It introduces new insights into the metrical properties of approximation subsequences, extending previous work on continued fractions and rational approximation sequences.

Findings

Established new metrical results for subsequences of rational approximants.

Extended the Doblin-Lenstra conjecture proof to broader contexts.

Analyzed the dynamical systems linked to continued fraction approximations.

Abstract

We study metrical properties of various subsequences associated to the sequence of rational approximants coming from the continued fraction of an irrational number. Our methods build upon Bosma, Jager and Wiedijk's proof of the Doblin-Lenstra conjecture as well as Jager's subsequent treatment of the sequence of approximation pairs.