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 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.

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 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.

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.

A dimension formula for endomorphisms—the Belykh family

1998 ◽  
Vol 18 (5) ◽  
pp. 1283-1309 ◽  
Author(s):  
J. SCHMELING
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Parameter Family ◽  
Full Measure ◽  
Srb Measure ◽  
Dimension Formula

In [10] we considered a class of hyperbolic endomorphisms and asked the question whether there exists a physically motivated invariant measure (SRB measure) and if so we gave a criterion when the map is invertible on a set of full measure. In this work we want to consider a particular example of this class — in fact a three-parameter family of those — and prove that a.s. the criterion is fulfilled. From this it follows that the Young formulae for the Hausdorff dimension of the SRB measure holds.

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.

A dimensional result in continued fractions

2014 ◽  
Vol 10 (04) ◽  
pp. 849-857 ◽  
Author(s):  
Yu Sun ◽  
Jun Wu
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Dimensional Theory ◽  
Cambridge Philos ◽  
Dimension Formula ◽  
Growth Properties

Given x ∈ (0, 1), let [a1(x), a2(x), a3(x),…] be the continued fraction expansion of x and [Formula: see text] be the sequence of rational convergents. Good [The fractional dimensional theory of continued fractions, Math. Proc. Cambridge Philos. Soc.37 (1941) 199–228] discussed the growth properties of {an(x), n ≥ 1} and proved that for any β > 0, the set [Formula: see text] is of Hausdorff dimension [Formula: see text]. In this paper, we consider, for any β > 0, the set [Formula: see text] and show that the Hausdorff dimension of F(β) is [Formula: see text].

MULTIFRACTAL ANALYSIS OF ONE-DIMENSIONAL BIASED WALKS

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850030 ◽  
Author(s):  
YUFEI CHEN ◽  
MEIFENG DAI ◽  
XIAOQIAN WANG ◽  
YU SUN ◽  
WEIYI SU
Keyword(s):  
Random Walk ◽  
Hausdorff Dimension ◽  
Multifractal Analysis ◽  
Infinite Sequence ◽  
Real Numbers ◽  
One Dimensional ◽  
Dimension Formula ◽  
Sum Of Digits ◽  
Packing Dimensions ◽  
Dyadic System

For an infinite sequence [Formula: see text] of [Formula: see text] and [Formula: see text] with probability [Formula: see text] and [Formula: see text], we mainly study the multifractal analysis of one-dimensional biased walks. Let [Formula: see text] and [Formula: see text]. The Hausdorff and packing dimensions of the sets [Formula: see text] are [Formula: see text], which is the development of the theorem of Besicovitch [On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1934) 321–330] on random walk, saying that: For any [Formula: see text], the set [Formula: see text] has Hausdorff dimension [Formula: see text].

scholarly journals A note on the Hausdorff dimension of the singular set of solutions to elasticity type systems

2019 ◽  
Vol 21 (06) ◽  
pp. 1950026 ◽  
Author(s):  
Sergio Conti ◽  
Matteo Focardi ◽  
Flaviana Iurlano
Keyword(s):  
Hausdorff Dimension ◽  
Constitutive Relations ◽  
Type Systems ◽  
Singular Set ◽  
Dimension Formula ◽  
Linear Framework ◽  
Autonomous Case ◽  
Strong Formulation ◽  
Nonlinear Constitutive Relations ◽  
Full Regularity

We prove partial regularity for minimizers to elasticity type energies with [Formula: see text]-growth, [Formula: see text], in a geometrically linear framework in dimension [Formula: see text]. Therefore, the energies we consider depend on the symmetrized gradient of the displacement field. It is an open problem in such a setting either to establish full regularity or to provide counterexamples. In particular, we give an estimate on the Hausdorff dimension of the potential singular set by proving that is strictly less than [Formula: see text], and actually [Formula: see text] in the autonomous case (full regularity is well-known in dimension [Formula: see text]). The latter result is instrumental to establish existence for the strong formulation of Griffith type models in brittle fracture with nonlinear constitutive relations, accounting for damage and plasticity in space dimensions [Formula: see text] and [Formula: see text].

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