scholarly journals Hausdorff dimension of an exceptional set in the theory of continued fractions

Nonlinearity ◽  
2020 ◽  
Vol 33 (6) ◽  
pp. 2615-2639 ◽  
Author(s):  
Ayreena Bakhtawar ◽  
Philip Bos ◽  
Mumtaz Hussain
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Exceptional Set

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.

scholarly journals Exceptional sets for self-similar fractals in Carnot groups

2010 ◽  
Vol 149 (1) ◽  
pp. 147-172 ◽  
Author(s):  
ZOLTÁN M. BALOGH ◽  
RETO BERGER ◽  
ROBERTO MONTI ◽  
JEREMY T. TYSON
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Iterated Function Systems ◽  
Carnot Groups ◽  
Exceptional Set ◽  
Carathéodory Metric ◽  
Exceptional Sets ◽  
Iterated Function ◽  
Self Similar ◽  
Similarity Dimension

AbstractWe consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

scholarly journals THE EXCEPTIONAL SETS ON THE RUN-LENGTH FUNCTION OF β-EXPANSIONS

Fractals ◽  
2017 ◽  
Vol 25 (06) ◽  
pp. 1750060 ◽  
Author(s):  
LIXUAN ZHENG ◽  
MIN WU ◽  
BING LI
Keyword(s):  
Hausdorff Dimension ◽  
Maximal Length ◽  
Length Function ◽  
Exceptional Set ◽  
Run Length ◽  
Exceptional Sets ◽  
Monotonically Increasing Function ◽  
Increasing Function

Let [Formula: see text] and the run-length function [Formula: see text] be the maximal length of consecutive zeros amongst the first [Formula: see text] digits in the [Formula: see text]-expansion of [Formula: see text]. The exceptional set [Formula: see text] is investigated, where [Formula: see text] is a monotonically increasing function with [Formula: see text]. We prove that the set [Formula: see text] is either empty or of full Hausdorff dimension and residual in [Formula: see text] according to the increasing rate of [Formula: see text].

A DIMENSIONAL RESULT ON THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

2021 ◽  
pp. 1-29
Author(s):  
LINGLING HUANG ◽  
CHAO MA
Keyword(s):  
Growth Rate ◽  
Hausdorff Dimension ◽  
Natural Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Nonnegative Number ◽  
Partial Quotients

Abstract This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

THE DISTAL AND ASYMPTOTIC SETS FOR CONTINUED FRACTIONS

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950139
Author(s):  
WEIBIN LIU ◽  
SHUAILING WANG
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Underlying Space ◽  
And Topology

For continued fraction dynamical system [Formula: see text], we give a classification of the underlying space [Formula: see text] according to the orbit of a given point [Formula: see text]. The sizes of all classes are determined from the viewpoints of measure, Hausdorff dimension and topology. For instance, the Hausdorff dimension of the distal set of [Formula: see text] is one and the Hausdorff dimension of the asymptotic set is either zero or [Formula: see text] according to [Formula: see text] is rational or not.

scholarly journals Good’s theorem for Hurwitz continued fractions

2020 ◽  
Vol 16 (07) ◽  
pp. 1433-1447
Author(s):  
Gerardo Gonzalez Robert
Keyword(s):  
Hausdorff Dimension ◽  
Complex Plane ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Analogous Result ◽  
Complex Numbers ◽  
Real Numbers ◽  
Dimension Formula

Good’s Theorem for regular continued fraction states that the set of real numbers [Formula: see text] such that [Formula: see text] has Hausdorff dimension [Formula: see text]. We show an analogous result for the complex plane and Hurwitz Continued Fractions: the set of complex numbers whose Hurwitz Continued fraction [Formula: see text] satisfies [Formula: see text] has Hausdorff dimension [Formula: see text], half of the ambient space’s dimension.

scholarly journals Dimension Estimates for Certain Sets of Infinite Complex Continued Fractions

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
J. Neunhäuserer
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Gaussian Integers ◽  
Complex Continued Fractions

We prove upper and lower estimates on the Hausdorff dimension of sets of infinite complex continued fractions with finitely many prescribed Gaussian integers. Particulary we will conclude that the dimension of theses sets is not zero or two and there are such sets with dimension greater than one and smaller than one.

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