scholarly journals On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities

2016 ◽  
Vol 11 (1) ◽  
pp. 141-157
Author(s):  
Ladislav Mišík ◽  
Jan Šustek ◽  
Bodo Volkmann
Keyword(s):  
Hausdorff Dimension ◽  
Characteristic Function ◽  
Entropy Function ◽  
Hausdorff Dimensions ◽  
Delta Sigma ◽  
Image Set ◽  
Alpha Beta ◽  
Positive Integers ◽  
Asymptotic Densities

AbstractFor a set A of positive integers a1< a2< · · ·, let d(A), $\overline d (A)$ denote its lower and upper asymptotic densities. The gap density is defined as $\lambda (A) = \lim \;{\rm sup} _{n \to \infty } {{a_{n + 1} } \over {a_n }}$. The paper investigates the class 𝒢(α, β, γ) of all sets A with d(A) = α, $\overline d (A) = \beta $ and λ(A) = γ for given α, β, γ with 0 ≤ α ≤ β ≤ 1 ≤ γ and αγ ≤ β. Using the classical dyadic mapping $\varrho (A) = \sum\nolimits_{n = 1}^\infty {{{\chi _A (n)} \over {2^n }}} $, where χA is the characteristic function of A, the main result of the paper states that the ϱ-image set ϱ𝒢(α, β, γ) has the Hausdorff dimension $$\dim \varrho \cal {G}(\alpha ,\beta ,\gamma ) = \min \left\{ {\delta (\alpha ),\delta (\beta ), { 1 \over \gamma }\mathop {\max }\limits_{\sigma \in [\alpha \gamma ,\beta ]} \delta (\sigma )} \right\},$$ where δ is the entropy function $$\delta (x) = - x\log _2 x - (1 - x)\;\log _2 (1 - x).$$

scholarly journals The Dimension of the Boundary of a Liouville Quantum Gravity Metric Ball

2020 ◽  
Vol 378 (1) ◽  
pp. 625-689 ◽  
Author(s):  
Ewain Gwynne
Keyword(s):  
Hausdorff Dimension ◽  
Quantum Gravity ◽  
Free Field ◽  
Gaussian Free Field ◽  
Hausdorff Dimensions ◽  
Thick Points

Abstract Let $$\gamma \in (0,2)$$ γ ∈ ( 0 , 2 ) , let h be the planar Gaussian free field, and consider the $$\gamma $$ γ -Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a $$\gamma $$ γ -LQG metric ball with respect to the Euclidean (resp. $$\gamma $$ γ -LQG) metric is $$2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}$$ 2 - γ d γ 2 γ + γ 2 + γ 2 2 d γ 2 (resp. $$d_\gamma -1$$ d γ - 1 ), where $$d_\gamma $$ d γ is the Hausdorff dimension of the whole plane with respect to the $$\gamma $$ γ -LQG metric. For $$\gamma = \sqrt{8/3}$$ γ = 8 / 3 , in which case $$d_{\sqrt{8/3}}=4$$ d 8 / 3 = 4 , we get that the essential supremum of Euclidean (resp. $$\sqrt{8/3}$$ 8 / 3 -LQG) dimension of a $$\sqrt{8/3}$$ 8 / 3 -LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and $$\gamma $$ γ -LQG Hausdorff dimensions of the intersection of a $$\gamma $$ γ -LQG ball boundary with the set of metric $$\alpha $$ α -thick points of the field h for each $$\alpha \in \mathbb R$$ α ∈ R . Our results show that the set of $$\gamma /d_\gamma $$ γ / d γ -thick points on the ball boundary has full Euclidean dimension and the set of $$\gamma $$ γ -thick points on the ball boundary has full $$\gamma $$ γ -LQG dimension.

Dimension spectra of lines1

Computability ◽  
2021 ◽  
pp. 1-28
Author(s):  
Neil Lutz ◽  
D.M. Stull
Keyword(s):  
Hausdorff Dimension ◽  
Kolmogorov Complexity ◽  
Euclidean Plane ◽  
Unit Interval ◽  
Packing Dimension ◽  
Hausdorff Dimensions ◽  
Dimension Spectrum ◽  
Algorithmic Dimension

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp ( L ) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim ( a , b ) is equal to the effective packing dimension Dim ( a , b ), then sp ( L ) contains a unit interval. We also show that, if the dimension dim ( a , b ) is at least one, then sp ( L ) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

On Hausdorff dimension monotonicity of a family of dynamical subsets of Rauzy fractals

2015 ◽  
Vol 11 (04) ◽  
pp. 1089-1098 ◽  
Author(s):  
W. Georg Nowak ◽  
Klaus Scheicher ◽  
Víctor F. Sirvent
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Dimensions ◽  
The Third

We consider a family of dynamically defined subsets of Rauzy fractals in the plane. These sets were introduced in the context of the study of symmetries of Rauzy fractals. We prove that their Hausdorff dimensions form an ultimately increasing sequence of numbers converging to 2. These results answer a question stated by the third author in 2012.

scholarly journals Enumeration of self-dual cyclic codes of some specific lengths over finite fields

2018 ◽  
Vol 10 (03) ◽  
pp. 1850031 ◽  
Author(s):  
Supawadee Prugsapitak ◽  
Somphong Jitman
Keyword(s):  
Characteristic Function ◽  
Finite Field ◽  
Finite Fields ◽  
Linear Codes ◽  
Cyclic Code ◽  
Cyclic Codes ◽  
Efficient Algorithms ◽  
Important Class ◽  
Field Characteristic ◽  
Positive Integers

Self-dual cyclic codes form an important class of linear codes. It has been shown that there exists a self-dual cyclic code of length [Formula: see text] over a finite field if and only if [Formula: see text] and the field characteristic are even. The enumeration of such codes has been given under both the Euclidean and Hermitian products. However, in each case, the formula for self-dual cyclic codes of length [Formula: see text] over a finite field contains a characteristic function which is not easily computed. In this paper, we focus on more efficient ways to enumerate self-dual cyclic codes of lengths [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are positive integers. Some number theoretical tools are established. Based on these results, alternative formulas and efficient algorithms to determine the number of self-dual cyclic codes of such lengths are provided.

scholarly journals Hausdorff Dimension of Escaping Sets of Nevanlinna Functions

2019 ◽  
Author(s):  
Weiwei Cui
Keyword(s):  
Hausdorff Dimension ◽  
Differential Equations ◽  
Complex Plane ◽  
Meromorphic Functions ◽  
Second Order ◽  
Hausdorff Dimensions ◽  
Second Order Differential Equations ◽  
Schwarzian Derivatives ◽  
Nevanlinna Functions

Abstract We determine the exact values of Hausdorff dimensions of escaping sets of meromorphic functions with polynomial Schwarzian derivatives. This will follow from the relation between these functions and the second-order differential equations in the complex plane.

Hausdorff dimension of the set of almost convergent sequences

2022 ◽  
pp. 1-7
Author(s):  
Alexandr Usachev
Keyword(s):  
Hausdorff Dimension ◽  
Summation Method ◽  
Binary Representation ◽  
Hausdorff Dimensions ◽  
Summation Methods ◽  
Wide Range ◽  
Matrix Summation ◽  
Convergent Sequences

Abstract The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.

On Weierstrass-like functions and random recurrent sets

1989 ◽  
Vol 106 (2) ◽  
pp. 325-342 ◽  
Author(s):  
Tim Bedford
Keyword(s):  
Hausdorff Dimension ◽  
Random Matrix ◽  
Hausdorff Dimensions ◽  
Random Matrix Products ◽  
Matrix Products ◽  
Recurrent Set

AbstractA construction of Weierstrass-like functions using recurrent sets is described, and the Hausdorff dimensions of the graphs computed. An important part of the proof is the notion of a globally random recurrent set. The Hausdorff dimension of a class of such sets is calculated using techniques of random matrix products.

scholarly journals On the value distribution of the differential polynomial Afn f(k) + Bfn+1 -1

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 579-589
Author(s):  
Pulak Sahoo ◽  
Anjan Sarkar
Keyword(s):  
Meromorphic Function ◽  
Characteristic Function ◽  
Recent Result ◽  
Counting Function ◽  
Differential Polynomial ◽  
Value Distribution ◽  
Nevanlinna Characteristic ◽  
Transcendental Meromorphic Function ◽  
Positive Integers

In the paper, we study the value distribution of the differential polynomial Afn f(k) + Bf n+1 -1, where f is a transcendental meromorphic function and n(? 2),k(?2) are positive integers. We prove an inequality for the Nevanlinna characteristic function T(r,f) in terms of reduced counting function only. The result of the paper not only improves the result due to Q.D. Zhang [J. Chengdu Ins. Meteor., 20(1992), 12-20], also partially improves a recent result of H. Karmakar and P. Sahoo [Results Math., (2018),73:98].

ON THE INCREASING PARTIAL QUOTIENTS OF CONTINUED FRACTIONS OF POINTS IN THE PLANE

2021 ◽  
pp. 1-8
Author(s):  
MEIYING LÜ ◽  
ZHENLIANG ZHANG
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Large N ◽  
Partial Quotients ◽  
Positive Integers

Abstract For any x in $[0,1)$ , let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be its continued fraction. Let $\psi :\mathbb {N}\to \mathbb {R}^+$ be such that $\psi (n) \to \infty $ as $n\to \infty $ . For any positive integers s and t, we study the set $$ \begin{align*}E(\psi)=\{(x,y)\in [0,1)^2: \max\{a_{sn}(x), a_{tn}(y)\}\ge \psi(n) \ {\text{for all sufficiently large}}\ n\in \mathbb{N}\} \end{align*} $$ and determine its Hausdorff dimension.

scholarly journals Engel Series Expansions of Laurent Series and Hausdorff Dimensions

2003 ◽  
Vol 75 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Jun Wu
Keyword(s):  
Hausdorff Dimension ◽  
Positive Integer ◽  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Laurent Series ◽  
Series Expansions ◽  
Formal Laurent Series ◽  
Hausdorff Dimensions ◽  
Formal Power

AbstractFor any positive integer q≧2, let Fq be a finite field with q elements, Fq ((z-1)) be the field of all formal Laurent series in an inderminate z, I denote the valuation ideal z-1Fq [[z-1]] in the ring of formal power series Fq ((z-1)) normalized by P(l) = 1. For any x ∈ I, let the series be the Engel expansin of Laurent series of x. Grabner and Knopfmacher have shown that the P-measure of the set A(α) = {x ∞ I: limn→∞ deg an(x)/n = ά} is l when α = q/(q -l), where deg an(x) is the degree of polynomial an(x). In this paper, we prove that for any α ≧ l, A(α) has Hausdorff dimension l. Among other thing we also show that for any integer m, the following set B(m) = {x ∈ l: deg an+1(x) - deg an(x) = m for any n ≧ l} has Hausdorff dimension 1.

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