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A NOTE ON THE INTERSECTIONS OF THE BESICOVITCH SETS AND ERDŐS–RÉNYI SETS

  • JINJUN LI (a1) and MIN WU (a2)
Abstract

For $x\in (0,1]$ and a positive integer $n,$ let $S_{\!n}(x)$ denote the summation of the first $n$ digits in the dyadic expansion of $x$ and let $r_{n}(x)$ denote the run-length function. In this paper, we obtain the Hausdorff dimensions of the following sets:

$$\begin{eqnarray}\bigg\{x\in (0,1]:\liminf _{n\rightarrow \infty }\frac{S_{\!n}(x)}{n}=\unicode[STIX]{x1D6FC},\limsup _{n\rightarrow \infty }\frac{S_{\!n}(x)}{n}=\unicode[STIX]{x1D6FD},\lim _{n\rightarrow \infty }\frac{r_{n}(x)}{\log _{2}n}=\unicode[STIX]{x1D6FE}\bigg\},\end{eqnarray}$$
where $0\leq \unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D6FD}\leq 1$ , $0\leq \unicode[STIX]{x1D6FE}\leq +\infty$ .

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The first author was supported by the National Natural Science Foundation of China (11671189) and the Natural Science Foundation of Fujian Province (2017J01403). The second author was supported by the National Natural Science Foundation of China (11771153).

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