scholarly journals On the longest block in Lüroth expansion

2018 ◽  
Vol 457 (1) ◽  
pp. 522-532 ◽  
Author(s):  
Jinjun Li ◽  
Min Wu ◽  
Xiangfeng Yang
Keyword(s):  
Lüroth Expansion

Exceptional sets related to the largest digits in Lüroth expansions

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 ◽  
Lüroth Expansion

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

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 ◽  
Lüroth Expansion

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

2014 ◽  
Vol 10 (04) ◽  
pp. 1015-1023 ◽  
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.

scholarly journals On the maximal run-length function in the Lüroth expansion

2018 ◽  
Vol 68 (1) ◽  
pp. 277-291
Author(s):  
Yu Sun ◽  
Jian Xu
Keyword(s):  
Length Function ◽  
Run Length ◽  
Lüroth Expansion

scholarly journals On Gauss problem for the Lüroth expansion

2013 ◽  
Vol 24 (2) ◽  
pp. 382-390 ◽  
Author(s):  
Marius Iosifescu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Lüroth Expansion

FINITE PATTERN PROBLEMS RELATED TO LÜROTH EXPANSION

Fractals ◽  
2020 ◽  
Vol 28 (03) ◽  
pp. 2050048
Author(s):  
ZHIGANG TIAN ◽  
MIN WU ◽  
MAN-LI LOU
Keyword(s):  
Hausdorff Dimension ◽  
Digit Sequence ◽  
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.

scholarly journals Frequency of digits in the Lüroth expansion

2009 ◽  
Vol 129 (6) ◽  
pp. 1479-1490 ◽  
Author(s):  
Luis Barreira ◽  
Godofredo Iommi
Keyword(s):  
Lüroth Expansion
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