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| 摘要: |
| 本文研究了Lüroth展式字符乘积的部分和Sn(x)=∑i=1n di(x)di+1(x)的度量性质和相关分形集的Hausdorff维数, 其中di(x)表示x∈(0,1)的Lüroth展式的第i个字符.利用对部分和序列的修正和适当分形集的构造, 获得了Sn(x)/nlog2n 依勒贝格测度收敛于1/2并且得到了相关例外集的Hausdorff维数, 扩展了数的展式的维数研究. |
| 关键词: Lüroth展式 字符乘积 部分和 Hausdorff维数 |
| DOI： |
| 分类号:O156 |
| 基金项目:国家自然科学基金~(11701261). |
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| ON THE METRIC PROPERTIES OF THE SUM OF PRODUCTS OF CONSECUTIVE DIGITS IN LüROTH EXPANSIONS AND RELATED DIMENSIONS |
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HU Hui,CHENG Cheng
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| Abstract: |
| In this paper, we consider the metric properties of Sn(x)=∑i=1n di(x)di+1(x) and the Hausdroff dimension of related fractal sets, where di(x) is the i-th digit of the Lüroth expansions of x ∈ [0, 1). By some modification of Sn(x) and the construction of suitable fractal subsets, we prove that Sn(x)/n log2 n converges to 1=2 in Lebesgue measure λ, and we get the Hausdorff dimensions of related exceptional sets. It extends related dimensional results about expansions of numbers. |
| Key words: Lüroth expansions product of digits partial sum Hausdorff dimension |
