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].

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

Fractals ◽  
2018 ◽  
Vol 26 (05) ◽  
pp. 1850074 ◽  
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.

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].

On the intersections of exceptional sets in Borel’s normal number theorem and Erdös–Rényi limit theorem

2020 ◽  
pp. 1-14
Author(s):  
Mengjie Zhang
Keyword(s):  
Limit Theorem ◽  
Hausdorff Dimension ◽  
The Other ◽  
Partial Summation ◽  
Binary Expansion ◽  
Exceptional Sets ◽  
Normal Number ◽  
Almost All ◽  
Set Of Points ◽  
Increasing Function

For any [Formula: see text], let [Formula: see text] be the partial summation of the first [Formula: see text] digits in the binary expansion of [Formula: see text] and [Formula: see text] be its run-length function. The classical Borel’s normal number theorem tells us that for almost all [Formula: see text], the limit of [Formula: see text] as [Formula: see text] goes to infinity is one half. On the other hand, the Erdös–Rényi limit theorem shows that [Formula: see text] increases to infinity with the logarithmic speed [Formula: see text] as [Formula: see text] for almost every [Formula: see text] in [Formula: see text]. In this paper, we are interested in the intersections of exceptional sets arising in the above two famous theorems. More precisely, for any [Formula: see text] and [Formula: see text], we completely determine the Hausdorff dimension of the following set: [Formula: see text] where [Formula: see text] and [Formula: see text] After some minor modifications, our result still holds if we replace the denominator [Formula: see text] in [Formula: see text] with any increasing function [Formula: see text] satisfying [Formula: see text] tending to [Formula: see text] and [Formula: see text]. As a result, we also obtain that the set of points for which neither the sequence [Formula: see text] nor [Formula: see text] converges has full Hausdorff dimension.

On the maximal length of consecutive zero digits of β-expansions

2016 ◽  
Vol 12 (03) ◽  
pp. 625-633 ◽  
Author(s):  
Xin Tong ◽  
Yueli Yu ◽  
Yanfen Zhao
Keyword(s):  
Real Number ◽  
Maximal Length ◽  
Hausdorff Dimensions ◽  
Exceptional Sets ◽  
Consecutive Zero ◽  
Almost All

Let [Formula: see text] be a real number. For any [Formula: see text], let [Formula: see text] be the maximal length of consecutive zero digits between the first [Formula: see text] digits of [Formula: see text]’s [Formula: see text]-expansion. We prove that for Lebesgue almost all [Formula: see text], [Formula: see text]. Also the Hausdorff dimensions of the related exceptional sets are determined.

Exceptional sets for self-affine fractals

2008 ◽  
Vol 145 (3) ◽  
pp. 669-684 ◽  
Author(s):  
KENNETH FALCONER ◽  
JUN MIAO
Keyword(s):  
Hausdorff Dimension ◽  
Value Function ◽  
Singular Value ◽  
Exceptional Sets ◽  
Fourier Dimension ◽  
Almost All

AbstractUnder certain conditions the ‘singular value function’ formula gives the Hausdorff dimension of self-affine fractals for almost all parameters in a family. We show that the size of the set of exceptional parameters is small both in the sense of Hausdorff dimension and Fourier dimension.

ON SOME EXCEPTIONAL SETS IN ENGEL EXPANSIONS AND HAUSDORFF DIMENSIONS

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050140
Author(s):  
JIA LIU
Keyword(s):  
Hausdorff Dimension ◽  
Lower Limit ◽  
Infinite Series ◽  
Three Dimensions ◽  
Hausdorff Dimensions ◽  
Exceptional Sets ◽  
Upper Limit ◽  
Engel Expansions ◽  
Increasing Function

For any [Formula: see text], let the infinite series [Formula: see text] be the Engel expansion of [Formula: see text]. Suppose [Formula: see text] is a strictly increasing function with [Formula: see text] and let [Formula: see text], [Formula: see text] and [Formula: see text] be defined as the sets of numbers [Formula: see text] for which the limit, upper limit and lower limit of [Formula: see text] is equal to [Formula: see text]. In this paper, we qualify the size of the set [Formula: see text], [Formula: see text] and [Formula: see text] in the sense of Hausdorff dimension and show that these three dimensions can be different.

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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