On the intersections of the Besicovitch sets and the Erdös–Rényi sets

Abstract

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 the n independent Bernoulli trials, that is the so called Erdos–Renyi sets. In 2013, Chen and Wen (J Math Anal Appl 401:29–37, 2013) considered the intersections of these two kinds of sets by determining the Hausdorff dimension of the sets $$\begin{aligned} \left\{ x\in [0,1):~\liminf \limits _{n\rightarrow \infty }\frac{S_{n}(x)}{n}\ge \alpha ,~ \lim \limits _{n\rightarrow \infty }\frac{R_{n}(x)}{\log _{2}n}=\beta \right\} ,~~0\le \alpha \le 1,~0\le \beta \le +\,\infty , \end{aligned}$$ where $$S_{n}(x)$$ denotes the summation of the first n digits and $$R_{n}(x)$$ is the maximal length of consecutive one digits in the first n terms of the dyadic expansion of $$x\in [0,1)$$ . In the present paper, we complement this result by computing the Hausdorff dimension of the following sets $$\begin{aligned} \left\{ x\in [0,1):~\lim \limits _{n\rightarrow \infty }\frac{S_{n}(x)}{n}=\alpha ,~\lim \limits _{n\rightarrow \infty }\frac{R_{n}(x)}{\log _{2}n}=\beta \right\} , ~~0\le \alpha \le 1,~0\le \beta \le +\,\infty . \end{aligned}$$