ON POINTS CONTAIN ARITHMETIC PROGRESSIONS IN THEIR LUROTH EXPANSION

For any x ∈ （0, 1] （except at most countably many points）, there exists a unique sequence {dn（x）}n≥1 of integers, called the digit sequence of x, such that x =∞ ∑j=1 1/d1（x）（d1（x）-1）……dj-1（x）（dj-1（x）-1）dj（x）. The dexter infinite series expansion is called the Liiroth expansion of x. This paper is co...

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Bibliographic Details
Published inActa mathematica scientia Vol. 36; no. 1; pp. 257 - 264
Main Author 张振亮 曹春云
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 2016
Subjects
11K55
28A80
arithmetic progression
COM
Hausdorff dimension
Hausdorff维数
Lüroth expansion
严格递增
可数
数字序列
整数
算术级数
级数展开
28A80
Hausdorff dimension
11K55
Lüroth expansion
arithmetic progression
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Summary:For any x ∈ （0, 1] （except at most countably many points）, there exists a unique sequence {dn（x）}n≥1 of integers, called the digit sequence of x, such that x =∞ ∑j=1 1/d1（x）（d1（x）-1）……dj-1（x）（dj-1（x）-1）dj（x）. The dexter infinite series expansion is called the Liiroth expansion of x. This paper is con- cerned with the size of the set of points x whose digit sequence in its Liiroth expansion is strictly increasing and contains arbitrarily long arithmetic progressions with arbitrary com- mon difference. More precisely, we determine the Hausdorff dimension of the above set.
Bibliography:Luroth expansion; arithmetic progression; Hausdorff dimension
42-1227/O
For any x ∈ （0, 1] （except at most countably many points）, there exists a unique sequence {dn（x）}n≥1 of integers, called the digit sequence of x, such that x =∞ ∑j=1 1/d1（x）（d1（x）-1）……dj-1（x）（dj-1（x）-1）dj（x）. The dexter infinite series expansion is called the Liiroth expansion of x. This paper is con- cerned with the size of the set of points x whose digit sequence in its Liiroth expansion is strictly increasing and contains arbitrarily long arithmetic progressions with arbitrary com- mon difference. More precisely, we determine the Hausdorff dimension of the above set.
ISSN:0252-9602
1572-9087
DOI:10.1016/S0252-9602(15)30093-X