scholarly journals On the maximal run-length function in the Lüroth expansion

2018 ◽  
Vol 68 (1) ◽  
pp. 277-291
Author(s):  
Yu Sun ◽  
Jian Xu
Keyword(s):  
Length Function ◽  
Run Length ◽  
Lüroth Expansion

EXCEPTIONAL SETS RELATED TO THE RUN-LENGTH FUNCTION OF BETA-EXPANSIONS

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850049 ◽  
Author(s):  
LULU FANG ◽  
KUNKUN SONG ◽  
MIN WU
Keyword(s):  
Limit Theorem ◽  
Hausdorff Dimension ◽  
Maximal Length ◽  
Length Function ◽  
Run Length ◽  
Exceptional Sets ◽  
Consecutive Zero ◽  
Dyadic Expansion ◽  
Almost All ◽  
Increasing Function

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].

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].

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

2018 ◽  
Vol 14 (07) ◽  
pp. 1919-1934 ◽  
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].

scholarly journals The run-length function of the β -expansion of the unit

2017 ◽  
Vol 177 ◽  
pp. 248-262 ◽  
Author(s):  
Chunyun Cao ◽  
Yuanhong Chen
Keyword(s):  
Length Function ◽  
Run Length

scholarly journals A note on the run length function for intermittent maps

2019 ◽  
Vol 472 (1) ◽  
pp. 937-946
Author(s):  
Hongfei Cui ◽  
Lulu Fang ◽  
Yiwei Zhang
Keyword(s):  
Length Function ◽  
Run Length

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

2019 ◽  
Vol 108 (1) ◽  
pp. 33-45 ◽  
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 $0\leq \unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D6FD}\leq 1$, $0\leq \unicode[STIX]{x1D6FE}\leq +\infty$.

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