On approximation by random Lüroth expansions

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.