On approximation by random Lüroth expansions

International Journal of Number Theory, vol. 18, pp. 1013–1046

Abstract

In this paper, we employ a random dynamical systems approach to study generalized Lüroth series expansions of numbers in the unit interval. We prove that for each [Formula: see text] with [Formula: see text] Lebesgue almost all numbers in [Formula: see text] have uncountably many universal generalized Lüroth series expansions with digits less than or equal to [Formula: see text], so expansions in which each possible block of digits occurs. In particular this means that Lebesgue almost all [Formula: see text] have uncountably many universal generalized Lüroth series expansions using finitely many digits only. For [Formula: see text] we show that typically the speed of convergence to an irrational number [Formula: see text] of the corresponding sequence of Lüroth approximants is equal to that of the standard Lüroth approximants. For other rational values of [Formula: see text] we use stationary measures to study the typical speed of convergence of the approximants and the digit frequencies.

Authors 2

1. Charlene Kalle Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Leiden University, Niels Bohrweg 1, 2333CA Leiden, The Netherlands

2. Marta Maggioni Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, Leiden University, Niels Bohrweg 1, 2333CA Leiden, The Netherlands