Level sets of partial maximal digits for Lüroth expansion

2017 ◽  
Vol 13 (10) ◽  
pp. 2777-2790 ◽  
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.

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.

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

Fractals ◽  
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 ◽  
Lüroth Expansion

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 ◽  
2020 ◽  
Vol 28 (04) ◽  
pp. 2050064
Author(s):  
MEIYING LÜ
Keyword(s):  
Growth Rate ◽  
Hausdorff Dimension ◽  
Exponential Rate ◽  
Hausdorff Dimensions ◽  
Lüroth Expansion

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.

scholarly journals The Hausdorff dimension of level sets described by Erdös–Rényi average

2018 ◽  
Vol 458 (1) ◽  
pp. 464-480
Author(s):  
Haibo Chen ◽  
Daoxin Ding ◽  
Xinghuo Long
Keyword(s):  
Hausdorff Dimension ◽  
Level Sets

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.

Applications of Intelligent Agents in Mobile Commerce

2018 ◽  
pp. 566-603
Author(s):  
Suresh Sankaranarayanan ◽  
Subramaniam Ganesan
Keyword(s):  
Communication Network ◽  
Growth Rate ◽  
Intelligent Agents ◽  
Research Work ◽  
Mobile Commerce ◽  
Agent Technology ◽  
Exponential Growth Rate ◽  
Agent Based ◽  
Mobile Phone Usage ◽  
Exponential Rates

Mobile phone usage and its adoption have been growing at exponential rates. It is this exponential growth rate that has led many to predict Mobile Commerce (M-Commerce) as the next major evolution of Electronic Commerce (E-Commerce). While M-Commerce comes rich with unique features, it is currently lacking in usage when compared to traditional e-commerce. There are many challenges that must be addressed in this respect. These challenges are mostly inherent in the mobile devices, communication network, legal and regulatory infrastructure. The use of agents in e-commerce has long been explored in the context, resulting in a number of agent based e-commerce systems. It is not surprising then to note that many researchers believe that the problems that the M-Commerce now faces can be addressed well using agent technology. While there is an abundance of information on the use of agent based systems in other areas, there has been no great surge yet in the use of agent based systems in real world M-Commerce applications. We believe that this slow adoption of this agent technology is due to a lack of standards. There has been a quite an amount of research work carried out in the use of software intelligent agents in the M-Commerce applications like Shopping, Hotel, and Airline industries. These are outlined in the paper with appropriate screenshots and descriptions.

scholarly journals A classification of aperiodic order via spectral metrics and Jarník sets

2018 ◽  
Vol 39 (11) ◽  
pp. 3031-3065 ◽  
Author(s):  
MAIK GRÖGER ◽  
MARC KESSEBÖHMER ◽  
ARNE MOSBACH ◽  
TONY SAMUEL ◽  
MALTE STEFFENS
Keyword(s):  
Hausdorff Dimension ◽  
Level Sets ◽  
Continued Fraction ◽  
Regularity Condition ◽  
Hölder Regularity ◽  
Aperiodic Order ◽  
Holder Regularity

Given an$\unicode[STIX]{x1D6FC}>1$and a$\unicode[STIX]{x1D703}$with unbounded continued fraction entries, we characterize new relations between Sturmian subshifts with slope$\unicode[STIX]{x1D703}$with respect to (i) an$\unicode[STIX]{x1D6FC}$-Hölder regularity condition of a spectral metric, (ii) level sets defined in terms of the Diophantine properties of$\unicode[STIX]{x1D703}$, and (iii) complexity notions which we call$\unicode[STIX]{x1D6FC}$-repetitiveness,$\unicode[STIX]{x1D6FC}$-repulsiveness and$\unicode[STIX]{x1D6FC}$-finiteness—generalizations of the properties known as linear repetitiveness, repulsiveness and power freeness, respectively. We show that the level sets relate naturally to (exact) Jarník sets and prove that their Hausdorff dimension is$2/(\unicode[STIX]{x1D6FC}+1)$.

scholarly journals The Lyapunov spectrum of some parabolic systems

2009 ◽  
Vol 29 (3) ◽  
pp. 919-940 ◽  
Author(s):  
KATRIN GELFERT ◽  
MICHAŁ RAMS
Keyword(s):  
Lyapunov Exponent ◽  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Topological Entropy ◽  
Level Set ◽  
Level Sets ◽  
Parabolic Systems ◽  
Lyapunov Spectrum ◽  
Interval Maps ◽  
Set Of Points

AbstractWe study the Hausdorff dimension for Lyapunov exponents for a class of interval maps which includes several non-hyperbolic situations. We also analyze the level sets of points with given lower and upper Lyapunov exponents and, in particular, with zero lower Lyapunov exponent. We prove that the level set of points with zero exponent has full Hausdorff dimension, but carries no topological entropy.

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