Hausdorff dimension of the maximal run-length in dyadic expansion

Let [Formula: see text] and [Formula: see text] be real numbers. The run-length function of [Formula: see text]-expansions denoted by [Formula: see text] is defined as the maximal length of consecutive zeros in the first [Formula: see text] digits of the [Formula: see text]-expansion of [Formula: see text]. It is known that for Lebesgue almost all [Formula: see text], [Formula: see text] increases to infinity with the logarithmic speed [Formula: see text] as [Formula: see text] goes to infinity. In this paper, we calculate the Hausdorff dimension of the subtle set for which [Formula: see text] grows to infinity with other speeds. More precisely, we prove that for any [Formula: see text], the set [Formula: see text] has full Hausdorff dimension, where [Formula: see text] is a strictly increasing function satisfying that [Formula: see text] is non-increasing, [Formula: see text] and [Formula: see text] as [Formula: see text]. This result significantly extends the existing results in this topic, such as the results in [J.-H. Ma, S.-Y. Wen and Z.-Y. Wen, Egoroff’s theorem and maximal run length, Monatsh. Math. 151(4) (2007) 287–292; R.-B. Zou, Hausdorff dimension of the maximal run-length in dyadic expansion, Czechoslovak Math. J. 61(4) (2011) 881–888; J.-J. Li and M. Wu, On exceptional sets in Erdős–Rényi limit theorem, J. Math. Anal. Appl. 436(1) (2016) 355–365; J.-J. Li and M. Wu, On exceptional sets in Erdős–Rényi limit theorem revisited, Monatsh. Math. 182(4) (2017) 865–875; Y. Sun and J. Xu, A remark on exceptional sets in Erdős–Rényi limit theorem, Monatsh. Math. 184(2) (2017) 291–296; X. Tong, Y.-L. Yu and Y.-F. Zhao, On the maximal length of consecutive zero digits of [Formula: see text]-expansions, Int. J. Number Theory 12(3) (2016) 625–633; J. Liu, and M.-Y. Lü, Hausdorff dimension of some sets arising by the run-length function of [Formula: see text]-expansions, J. Math. Anal. Appl. 455(1) (2017) 832–841; L.-X. Zheng, M. Wu and B. Li, The exceptional sets on the run-length function of [Formula: see text]-expansions, Fractals 25(6) (2017) 1750060; X. Gao, H. Hu and Z.-H. Li, A result on the maximal length of consecutive 0 digits in [Formula: see text]-expansions, Turkish J. Math. 42(2) (2018) 656–665, doi: 10.3906/mat-1704-119].

Download Full-text

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

Fractals ◽

10.1142/s0218348x17500608 ◽

2017 ◽

Vol 25 (06) ◽

pp. 1750060 ◽

Cited By ~ 5

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].

Download Full-text

HAUSDORFF DIMENSION OF THE EXCEPTIONAL SETS CONCERNING THE RUN-LENGTH FUNCTION OF BETA-EXPANSION

Fractals ◽

10.1142/s0218348x18500743 ◽

2018 ◽

Vol 26 (05) ◽

pp. 1850074 ◽

Cited By ~ 1

Author(s):

MENGJIE ZHANG

Keyword(s):

Number Theory ◽

Hausdorff Dimension ◽

Real Number ◽

Maximal Length ◽

Run Length ◽

Exceptional Sets ◽

Number Formula ◽

Consecutive Zero ◽

Almost All ◽

Increasing Function

For any real number [Formula: see text], and any [Formula: see text], let [Formula: see text] be the maximal length of consecutive zeros in the first [Formula: see text] digits of the [Formula: see text]-expansion of [Formula: see text]. Recently, Tong, Yu and Zhao [On the length of consecutive zero digits of [Formula: see text]-expansions, Int. J. Number Theory 12 (2016) 625–633] proved that for any [Formula: see text], for Lebesgue almost all [Formula: see text], [Formula: see text] In this paper, we quantify the size of the set of [Formula: see text] for which [Formula: see text] grows to infinity in a general speed. More precisely, for any increasing function [Formula: see text] with [Formula: see text] tending to [Formula: see text] and [Formula: see text], we show that for any [Formula: see text], the set [Formula: see text] has full Hausdorff dimension.

Download Full-text

On the exceptional sets in Erdös–Rényi limit theorem of β-expansion

International Journal of Number Theory ◽

10.1142/s1793042118501142 ◽

2018 ◽

Vol 14 (07) ◽

pp. 1919-1934 ◽

Cited By ~ 1

Author(s):

Jia Liu ◽

Meiying Lü ◽

Zhenliang Zhang

Keyword(s):

Limit Theorem ◽

Hausdorff Dimension ◽

Real Number ◽

Length Function ◽

Run Length ◽

Exceptional Sets ◽

Longest Run

Let [Formula: see text] be a real number. For any [Formula: see text], the run-length function [Formula: see text] is defined as the length of the longest run of 0’s amongst the first [Formula: see text] digits in the [Formula: see text]-expansion of [Formula: see text]. Let [Formula: see text] be a non-decreasing sequence of integers and [Formula: see text], we define [Formula: see text] In this paper, we show that the set [Formula: see text] has full Hausdorff dimension under the condition that [Formula: see text].

Download Full-text

A NOTE ON THE INTERSECTIONS OF THE BESICOVITCH SETS AND ERDŐS–RÉNYI SETS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788718000435 ◽

2019 ◽

Vol 108 (1) ◽

pp. 33-45 ◽

Cited By ~ 1

Author(s):

JINJUN LI ◽

MIN WU

Keyword(s):

Positive Integer ◽

Length Function ◽

Hausdorff Dimensions ◽

Run Length ◽

Dyadic Expansion ◽

Image Position ◽

Besicovitch Sets

For $x\in (0,1]$ and a positive integer $n,$ let $S_{\!n}(x)$ denote the summation of the first $n$ digits in the dyadic expansion of $x$ and let $r_{n}(x)$ denote the run-length function. In this paper, we obtain the Hausdorff dimensions of the following sets: $$\begin{eqnarray}\bigg\{x\in (0,1]:\liminf _{n\rightarrow \infty }\frac{S_{\!n}(x)}{n}=\unicode[STIX]{x1D6FC},\limsup _{n\rightarrow \infty }\frac{S_{\!n}(x)}{n}=\unicode[STIX]{x1D6FD},\lim _{n\rightarrow \infty }\frac{r_{n}(x)}{\log _{2}n}=\unicode[STIX]{x1D6FE}\bigg\},\end{eqnarray}$$ where $0\leq \unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D6FD}\leq 1$, $0\leq \unicode[STIX]{x1D6FE}\leq +\infty$.

Download Full-text

Hausdorff dimension of some sets arising by the run-length function of β-expansions

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2017.06.001 ◽

2017 ◽

Vol 455 (1) ◽

pp. 832-841 ◽

Cited By ~ 3

Author(s):

Jia Liu ◽

Meiying Lü

Keyword(s):

Hausdorff Dimension ◽

Length Function ◽

Run Length

Download Full-text

A Novel Meteosat Second Generation Image Compression Method Based on Radon Transform, Linear Predictive Coding with Filtering and Sorted Run Length Coding

International Review on Computers and Software (IRECOS) ◽

10.15866/irecos.v10i4.5704 ◽

2015 ◽

Vol 10 (4) ◽

pp. 438 ◽

Cited By ~ 1

Author(s):

Mehdi Cherifi ◽

Mourad Lahdir ◽

Soltane Ameur

Keyword(s):

Image Compression ◽

Radon Transform ◽

Second Generation ◽

Predictive Coding ◽

Compression Method ◽

Linear Predictive Coding ◽

Run Length ◽

Run Length Coding

Download Full-text

Characterization of Pancreas at Diabetic Patients in CT images using Texture Analysis

International Journal of Advanced Research in Computer Science and Software Engineering ◽

10.23956/ijarcsse.v7i7.88 ◽

2017 ◽

Vol 7 (7) ◽

pp. 8

Author(s):

Mona E. Elbashier ◽

Suhaib Alameen ◽

Caroline Edward Ayad ◽

Mohamed E. M. Gar-Elnabi

Keyword(s):

Size Distribution ◽

Texture Analysis ◽

Classification Accuracy ◽

Ct Images ◽

Diabetic Patients ◽

Gray Level ◽

Grey Level ◽

Run Length ◽

Interactive Data

This study concern to characterize the pancreas areato head, body and tail using Gray Level Run Length Matrix (GLRLM) and extract classification features from CT images. The GLRLM techniques included eleven’s features. To find the gray level distribution in CT images it complements the GLRLM features extracted from CT images with runs of gray level in pixels and estimate the size distribution of thesubpatterns. analyzing the image with Interactive Data Language IDL software to measure the grey level distribution of images. The results show that the Gray Level Run Length Matrix and features give classification accuracy of pancreashead 89.2%, body 93.6 and the tail classification accuracy 93.5%. The overall classification accuracy of pancreas area 92.0%.These relationships are stored in a Texture Dictionary that can be later used to automatically annotate new CT images with the appropriate pancreas area names.

Download Full-text