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On the intersections of the Besicovitch sets and exceptional sets in the Erdős–Rényi limit theorem

  • J. Li
  • M. WuEmail author
Article
  • 30 Downloads

Abstract

For \({x \in [0,1)}\), let Sn(x) denote the summation of the first n digits in the dyadic expansion of x and rn(x) denote the run-length function. Let H denote the set of monotonically increasing functions \({\varphi \colon \mathbb{N} \to (0, +\infty)}\) with \({\lim_{n\to\infty} \varphi(n)=+\infty}\). For any \({\varphi \in H}\) and \({0 \leq \alpha \leq 1}\), we prove that the set
$$\left\{x\in [0,1]: \lim_{n\to \infty} \frac{S_n(x)}{n}=\alpha, \liminf_{n\to\infty} \frac{r_n(x)}{\varphi(n)}=0, \limsup _{n\to\infty} \frac{r_n(x)}{\varphi(n)}=+\infty\right\}$$
either has Hausdorff dimension \({H(\alpha)/{\rm log} 2}\) or is empty. Here \({H(\alpha) = -{\alpha}{\rm log} \alpha-(1-\alpha){\rm log}(1-\alpha)}\) is the classical entropy function.

Key words and phrases

Besicovitch set run-length function Erdős–Rényi limit theorem Hausdorff dimension 

Mathematics Subject Classification

28A80 11K55 

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Notes

Acknowledgement

The authors would like to thank the referee for valuable comments and suggestions that led to the improvement of the manuscript.

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceGuangzhou UniversityGuangzhouP. R. China
  2. 2.Department of MathematicsSouth China University of TechnologyGuangzhouP. R. China

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