A note on the largest digits in Lüroth expansion

Abstract

It is well known that every x ∈ (0, 1] can be expanded into an infinite Lüroth series with the form of [Formula: see text] where d n (x) ≥ 2 and is called the nth digits of x for each n ≥ 1. In [Representations of Real Numbers by Infinite Series, Lecture Notes in Mathematics, Vol. 502 (Springer, New York, 1976)], Galambos showed that for Lebesgue almost all x ∈ (0, 1], [Formula: see text], where L n (x) = max {d 1 (x), …, d n (x)} denotes the largest digit among the first n ones of x. In this paper, we consider the Hausdorff dimension of the set [Formula: see text] for any α ≥ 0.