On the intersections of the Besicovitch sets and the Erdos-Renyi sets
On the intersections of the Besicovitch sets and the Erdos-Renyi sets
复制标题
关于贝西科维奇集和鄂尔多斯-仁义集的交集
DOI：
10.1007/s00605-018-1206-7
复制
发表时间：
2019
影响因子：
0.9
通讯作者：
Peng Li
中科院分区：
文献类型：
--
作者：
Zhang Mengjie;Peng Li
我们对实数的两个性质感兴趣：第一个性质是在二进制展开式中具有给定的数字频率的性质，如著名的Besicovitch集；第二个性质是在独立的伯努利试验中具有最长的头部行程的性质，即所谓的ErdöS-Rényi集。2013年，陈和文(数学肛门应用401：29-37，)通过确定以下集合的Hausdorff维度来考虑这两类集合的交集：[0，1]中的集合\Documentclass[12pt]{Minimum}\usepackage{amsath}\usepackage{waysym}\usepackage{amsFonts}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemarin}{-69pt}\Begin{Document}$\Begin{Aligned}\Left\{x)：~\liminf\Limits_{n\right tarrow\inty}\frac{S_{n}(X)}{n}\ge\pha，~\LIM\Limits_{n\right tarrow\infty}\frac{R_{n}(X)}{\log_{2}n}=\beta\right\}，~0\le\pha\le 1，~0\le\beta\le+\，\infty，\end{对准}$$\end{文档}，其中表示第一位数字的总和，是连续一位数字的最大长度，以的二进展开表示。在本论文中，我们通过计算以下集合的Hausdorff维度来补充这一结果：[0，1]中的以下集合：~\LIM\upgreek}\setLong{oddsidemarin}{-69pt}\{Begin文档}$\Begin{Begin{Align}\Left\{x\in[0，1]Limits_{n\right tarrow\inty}\frac{S_{n}(X)}{n}=\α，~\LIM\Limits_{n\right tarrow\infty}\frac{R_{n}(X)}{\log_{2}n}=\beta\right\}，~~0\le\pha\le 1，~0\le\beta\le+\，\infty。\end{已对齐}$$\end{文档}
We are interested in two properties of real numbers: the first one is the property of having given digit frequencies in the binary expansion, such as the well known Besicovitch sets, and the second one is the property of having the longest run of heads in thenindependent Bernoulli trials, that is the so called Erdös–Rényi sets. In 2013, Chen and Wen (J Math Anal Appl 401:29–37, ) considered the intersections of these two kinds of sets by determining the Hausdorff dimension of the sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ x\in [0,1):~\liminf \limits _{n\rightarrow \infty }\frac{S_{n}(x)}{n}\ge \alpha ,~ \lim \limits _{n\rightarrow \infty }\frac{R_{n}(x)}{\log _{2}n}=\beta \right\} ,~~0\le \alpha \le 1,~0\le \beta \le +\,\infty , \end{aligned}$$\end{document}wheredenotes the summation of the firstndigits andis the maximal length of consecutive one digits in the firstnterms of the dyadic expansion of. In the present paper, we complement this result by computing the Hausdorff dimension of the following sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ x\in [0,1):~\lim \limits _{n\rightarrow \infty }\frac{S_{n}(x)}{n}=\alpha ,~\lim \limits _{n\rightarrow \infty }\frac{R_{n}(x)}{\log _{2}n}=\beta \right\} , ~~0\le \alpha \le 1,~0\le \beta \le +\,\infty . \end{aligned}$$\end{document}