Exceptional sets related to the largest digits in Lüroth expansions

Let [Formula: see text] denote the largest digit of the first [Formula: see text] terms in the Lüroth expansion of [Formula: see text]. Shen, Yu and Zhou, A note on the largest digits in Luroth expansion, Int. J. Number Theory 10 (2014) 1015–1023 considered the level sets [Formula: see text] and proved that each [Formula: see text] has full Hausdorff dimension. In this paper, we investigate the Hausdorff dimension of the following refined exceptional set: [Formula: see text] and show that [Formula: see text] has full Hausdorff dimension for each pair [Formula: see text] with [Formula: see text]. Combining the two results, [Formula: see text] can be decomposed into the disjoint union of uncountably many sets with full Hausdorff dimension.

Relative convergence speed of Lüroth expansions and Hausdorff dimension

For any [Formula: see text] in [Formula: see text], let [Formula: see text] be the Lüroth expansion of [Formula: see text]. In this paper, we study the relative convergence speed of its convergents [Formula: see text] to the rate of growth of digits in the Lüroth expansion of an irrational number. For any [Formula: see text] in [Formula: see text], the sets [Formula: see text] and [Formula: see text] are proved to be of same Hausdorff dimension [Formula: see text]. Furthermore, for any [Formula: see text] in [Formula: see text] with [Formula: see text], the Hausdorff dimension of the set [Formula: see text] [Formula: see text] is proved to be either [Formula: see text] or [Formula: see text] according as [Formula: see text] or not.

Dimension theory of the product of partial quotients in Lüroth expansions

For [Formula: see text] let [Formula: see text] be its Lüroth expansion and [Formula: see text] be the sequence of convergents of [Formula: see text] Define the sets [Formula: see text] [Formula: see text] and [Formula: see text] where [Formula: see text] is a positive function. In this paper, we calculate the Lebesgue measure of the set [Formula: see text] and the Hausdorff dimension of the sets [Formula: see text] and [Formula: see text].

A NOTE ON THE RELATIVE GROWTH RATE OF THE MAXIMAL DIGITS IN LÜROTH EXPANSIONS

This paper is concerned with the growth rate of the maximal digits relative to the rate of approximation of the number by its convergents, as well as relative to the rate of the sum of digits for the Lüroth expansion of an irrational number. The Hausdorff dimension of the sets of points with a given relative growth rate is proved to be full.

THE GROWTH RATE OF THE DIGITS IN THE LÜROTH EXPANSIONS

For any [Formula: see text], let [Formula: see text] be its Lüroth expansion with digits [Formula: see text]. This paper is concerned with the growth rate of the digits in the Lüroth expansions. Let [Formula: see text] be a function satisfying [Formula: see text] as [Formula: see text] and [Formula: see text]. In this paper, we consider the set [Formula: see text] and we quantify the size of [Formula: see text] in the sense of Hausdorff dimension. As applications, we get the Hausdorff dimensions of the sets of points for which [Formula: see text] grows with polynomial and exponential rate.

FINITE PATTERN PROBLEMS RELATED TO LÜROTH EXPANSION

Let [Formula: see text] be a set of countable many functions, where every function [Formula: see text] satisfies [Formula: see text] and [Formula: see text] as [Formula: see text]. This paper studies the size of the set consisting of those numbers whose Lüroth digit sequence is strictly increasing and contains any finite pattern of [Formula: see text]. We prove that the Hausdorff dimension of such set is [Formula: see text] and give several applications.

Level sets of partial maximal digits for Lüroth expansion

Let [Formula: see text] be the Lüroth expansion of [Formula: see text]. This paper is concerned with the growth rate of the partial maximum [Formula: see text]. We completely determined the Hausdorff dimension of the set [Formula: see text] when [Formula: see text] tends to infinity with polynomial or exponential rates.