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.

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.

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].

ON THE LARGEST DEGREE OF THE PARTIAL QUOTIENTS IN CONTINUED FRACTION EXPANSIONS OVER THE FIELD OF FORMAL LAURENT SERIES

2013 ◽  
Vol 09 (05) ◽  
pp. 1237-1247 ◽  
Author(s):  
LUMING SHEN ◽  
JIAN XU ◽  
HUIPING JING
Keyword(s):  
Continued Fraction ◽  
Laurent Series ◽  
Type Theorem ◽  
Formal Laurent Series ◽  
Hausdorff Dimensions ◽  
Exceptional Sets ◽  
Iterated Logarithm ◽  
Partial Quotients ◽  
Almost All ◽  
Continued Fraction Expansions

For x ∈ I, let [A1(x), A2(x), …] be the continued fraction expansions over the field of Laurent series, write Ln(x) ≔ max { deg A1(x), deg A2(x), …, deg An(x)}, which is called the largest degree of partial quotients. In this paper, we give an iterated logarithm type theorem for Ln(x), and by which, we get that for P-almost all x ∈ I, [Formula: see text]. Also the Hausdorff dimensions of the related exceptional sets are determined.

scholarly journals Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

2017 ◽  
Vol 15 (1) ◽  
pp. 1517-1529
Author(s):  
Zhao Feng
Keyword(s):  
Short Intervals ◽  
Exceptional Sets ◽  
Goldbach Problem ◽  
Almost All

Abstract In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., $\begin{array}{} N=p_1^3+ \ldots +p_j^3 \end{array} $ with $\begin{array}{} |p_i-(N/j)^{1/3}|\leq N^{1/3- \delta +\varepsilon} (1\leq i\leq j), \end{array} $ for some $\begin{array}{} 0 \lt \delta\leq\frac{1}{90}. \end{array} $ Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.

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].

scholarly journals Higher Order Oscillation and Uniform Distribution

2019 ◽  
Vol 14 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Shigeki Akiyama ◽  
Yunping Jiang
Keyword(s):  
Number Theory ◽  
Real Number ◽  
Uniform Distribution ◽  
Differentiable Function ◽  
Mild Condition ◽  
Higher Order ◽  
Real Numbers ◽  
Mobius Function ◽  
Real Polynomials ◽  
Almost All

AbstractIt is known that the Möbius function in number theory is higher order oscillating. In this paper we show that there is another kind of higher order oscillating sequences in the form (e2πiαβn g(β))n∈𝕅, for a non-decreasing twice differentiable function g with a mild condition. This follows the result we prove in this paper that for a fixed non-zero real number α and almost all real numbers β> 1 (alternatively, for a fixed real number β> 1 and almost all real numbers α) and for all real polynomials Q(x), sequences (αβng(β)+ Q(n)) n∈𝕅 are uniformly distributed modulo 1.

scholarly journals Problems in Strong Uniform Distribution

2014 ◽  
Vol 59 (1) ◽  
pp. 51-64
Author(s):  
Kwo Chan ◽  
Radhakrishnan Nair
Keyword(s):  
Real Number ◽  
Uniform Distribution ◽  
Lebesgue Measure ◽  
Fractional Part ◽  
Positive Lebesgue Measure ◽  
The Real ◽  
Almost All

Abstract In 1923 A. Khinchin asked if given any B ⊆ [0, 1) of positive Lebesgue measure, we have #{n : 1 ≤ n ≤ N : {nx} ∈ B} → |B| for almost all x with respect to Lebesgue measure. Here {y} denotes the fractional part of the real number y and |A| denotes the Lebesgue measure of the set A in [0, 1). In 1970 J. Marstrand showed the answer is no. In this paper the authors survey contributions to this subject since then.

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 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].

Waterborne Outbreaks in Sweden – Causes and Etiology

1986 ◽  
Vol 18 (10) ◽  
pp. 185-190 ◽  
Author(s):  
Y. Andersson ◽  
T. A. Stenström
Keyword(s):  
Real Number ◽  
Unknown Etiology ◽  
Raw Water ◽  
Actual Number ◽  
Single Family ◽  
The Real ◽  
Almost All ◽  
Community Outbreaks ◽  
Microbial Agents

Thirty-two waterborne outbreaks in Sweden are known during 1975 - 1984, affecting nearly 12 000 people. These range from single family outbreaks to community outbreaks affecting up to 3 000 people. Microbial agents have been isolated in about 40 % of the outbreaks, the rest are of unknown etiology. Epidemiological investigations have shown that only a fraction of the actual number of cases are initially reported. The real number as judged from epidemiological follow-up investigations was in many instances tenfold higher. In almost all the cases, the cause of the outbreaks are technical deficiencies like back-siphonage of wastewater along drainage pipes, broken sewerage or sudden pollution of raw water intakes coinciding with malfunction of chlorination.

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