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Start Over Subject Hausdorff dimension

2. Hausdorff dimension of the maximal run-length in dyadic expansion

Creator:
Zou, Ruibiao
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
run-length function, Hausdorff dimension, and dyadic expansion
Language:
English
Description:
For any $x\in [0,1)$, let $x=[\epsilon _1,\epsilon _2,\cdots ,]$ be its dyadic expansion. Call $r_n(x):=\max \{j\geq 1\colon \epsilon _{i+1}=\cdots =\epsilon _{i+j}=1$, $0\leq i\leq n-j\}$ the $n$-th maximal run-length function of $x$. P. Erdös and A. Rényi showed that $\lim _{n\to \infty }{r_n(x)}/{\log _2 n}=1$ almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose run-length function assumes on other possible asymptotic behaviors than $\log _2 n$, is quantified by their Hausdorff dimension.
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4. Some dimensional results for a class of special homogeneous Moran sets

Creator:
Hu, Xiaomei
Format:
print, bez média, and svazek
Type:
model:article and TEXT
Subject:
fraktály, geometrie, mathematics, fractals, geometry, Hausdorffova dimenze, homogeneous Moran set, {mk}-Moran set, {mk}-quasi homogeneous Cantor set, Hausdorff dimension, 13, and 51
Language:
English
Description:
We construct a class of special homogeneous Moran sets, called {mk}-quasi homogeneous Cantor sets, and discuss their Hausdorff dimensions. By adjusting the value of {mk}k\geqslant 1, we constructively prove the intermediate value theorem for the homogeneous Moran set. Moreover, we obtain a sufficient condition for the Hausdorff dimension of ho- mogeneous Moran sets to assume the minimum value, which expands earlier works., Xiaomei Hu., and Obsahuje seznam literatury
Rights:
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5. Statistical convergence of infinite series

Creator:
Dindoš, M., Šalát, Tibor, and Toma, Vladimír
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
statistical convergence, set of the first category, Hausdorff dimension, and homogeneous set
Language:
English
Description:
In this paper we use the notion of statistical convergence of infinite series naturally introduced as the statistical convergence of the sequence of the partial sums of the series. We will discuss some questions related to the convergence of subseries of a given series.
Rights:
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6. Statistical convergence of subsequences of a given sequence

Creator:
Mačaj, M. and Šalát, T.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
asymptotic density, statistical convergence, Lebesgue measure, Hausdorff dimension, and Baire category
Language:
English
Description:
This paper is closely related to the paper of Harry I. Miller: Measure theoretical subsequence characterization of statistical convergence, Trans. Amer. Math. Soc. 347 (1995), 1811–1819 and contains a general investigation of statistical convergence of subsequences of an arbitrary sequence from the point of view of Lebesgue measure, Hausdorff dimensions and Baire’s categories.
Rights:
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7. The efficiency of approximating real numbers by Lüroth expansion

Creator:
Cao, Chunyun, Wu, Jun, and Zhang, Zhenliang
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Lüroth expansion, optimal approximation, and Hausdorff dimension
Language:
English
Description:
For any $x\in (0,1]$, let $$ x=\frac {1}{d_1}+\frac {1}{d_1(d_1-1)d_2}+\dots +\frac {1}{d_1(d_1-1) \dots d_{n-1}(d_{n-1}-1)d_{n}}+\dots $$ be its Lüroth expansion. Denote by ${P_n(x)}/{Q_n(x)}$ the partial sum of the first $n$ terms in the above series and call it the $n$th convergent of $x$ in the Lüroth expansion. This paper is concerned with the efficiency of approximating real numbers by their convergents $\{{P_n(x)}/{Q_n(x)}\}_{n\ge 1}$ in the Lüroth expansion. It is shown that almost no points can have convergents as the optimal approximation for infinitely many times in the Lüroth expansion. Consequently, Hausdorff dimension is introduced to quantify the set of real numbers which can be well approximated by their convergents in the Lüroth expansion, namely the following Jarník-like set: $\{x\in (0,1]\colon |x-{P_n(x)}/{Q_n(x)}|<{1}/{Q_n(x)^{\nu +1}} \text{infinitely often}\}$ for any $\nu \ge 1$.
Rights:
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8. The fractional dimensional theory in Lüroth expansion

Creator:
Shen, Luming and Fang, Kui
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Lüroth series, Cantor set, and Hausdorff dimension
Language:
English
Description:
It is well known that every $x\in (0,1]$ can be expanded to an infinite Lüroth series in the form of $$x=\frac {1}{d_1(x)}+\cdots +\frac {1}{d_1(x)(d_1(x)-1)\cdots d_{n-1}(x)(d_{n-1}(x)-1)d_n(x)}+\cdots , $$ where $d_n(x)\geq 2$ for all $n\geq 1$. In this paper, sets of points with some restrictions on the digits in Lüroth series expansions are considered. Mainly, the Hausdorff dimensions of the Cantor sets $$ F_{\phi }=\{x\in (0,1]\colon d_n(x)\geq \phi (n), \ \forall n\geq 1\} $$are completely determined, where $\phi $ is an integer-valued function defined on $\mathbb N$, and $\phi (n)\to \infty $ as $n\to \infty $.
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