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.

Exceptional sets related to the largest digits in Lüroth expansions

10.1142/s1793042122500737 ◽

2021 ◽

pp. 1-15

Author(s):

Shuyi Lin ◽

Jinjun Li ◽

Manli Lou

Keyword(s):

Number Theory ◽

Hausdorff Dimension ◽

Level Sets ◽

Disjoint Union ◽

Exceptional Set ◽

Exceptional Sets ◽

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

10.1142/s1793042122500518 ◽

2021 ◽

pp. 1-21

Author(s):

Xiaoyan Tan ◽

Jia Liu ◽

Zhenliang Zhang

Keyword(s):

Hausdorff Dimension ◽

Irrational Number ◽

Convergence Speed ◽

Rate Of Growth ◽

Dimension Formula ◽

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.

A note on the largest digits in Lüroth expansion

10.1142/s1793042114500122 ◽

2014 ◽

Vol 10 (04) ◽

pp. 1015-1023 ◽

Cited By ~ 4

Author(s):

Luming Shen ◽

Yiying Yu ◽

Yuxin Zhou

Keyword(s):

New York ◽

Hausdorff Dimension ◽

Infinite Series ◽

Real Numbers ◽

Lecture Notes ◽

Lüroth Expansion ◽

Almost All

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 dn(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 Ln(x) = max {d1(x), …, dn(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.

Dimension theory of the product of partial quotients in Lüroth expansions

10.1142/s1793042121500287 ◽

2020 ◽

pp. 1-16

Author(s):

Bo Tan ◽

Qinglong Zhou

Keyword(s):

Hausdorff Dimension ◽

Lebesgue Measure ◽

Positive Function ◽

Dimension Theory ◽

Lüroth Expansion ◽

Partial Quotients

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

Fractals ◽

10.1142/s0218348x20501169 ◽

2020 ◽

Vol 28 (06) ◽

pp. 2050116

Author(s):

XIAOYAN TAN ◽

KANGJIE HE

Keyword(s):

Growth Rate ◽

Hausdorff Dimension ◽

Relative Growth Rate ◽

Irrational Number ◽

Relative Growth ◽

Rate Of Approximation ◽

Sum Of Digits ◽

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

Fractals ◽

10.1142/s0218348x20500644 ◽

2020 ◽

Vol 28 (04) ◽

pp. 2050064

Author(s):

MEIYING LÜ

Keyword(s):

Growth Rate ◽

Hausdorff Dimension ◽

Exponential Rate ◽

Hausdorff Dimensions ◽

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.

Level sets of partial maximal digits for Lüroth expansion

10.1142/s1793042117501536 ◽

2017 ◽

Vol 13 (10) ◽

pp. 2777-2790 ◽

Cited By ~ 2

Author(s):

Kunkun Song ◽

Lulu Fang ◽

Jihua Ma

Keyword(s):

Growth Rate ◽

Hausdorff Dimension ◽

Level Sets ◽

Lüroth Expansion ◽

Exponential Rates

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.