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A NOTE ON THE INTERSECTIONS OF THE BESICOVITCH SETS AND ERDŐS–RÉNYI SETS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 108 / Issue 1 / February 2020
- Published online by Cambridge University Press:
- 01 February 2019, pp. 33-45
- Print publication:
- February 2020
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- Article
- Export citation
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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: where
$$\begin{eqnarray}\bigg\{x\in (0,1]:\liminf _{n\rightarrow \infty }\frac{S_{\!n}(x)}{n}=\unicode[STIX]{x1D6FC},\limsup _{n\rightarrow \infty }\frac{S_{\!n}(x)}{n}=\unicode[STIX]{x1D6FD},\lim _{n\rightarrow \infty }\frac{r_{n}(x)}{\log _{2}n}=\unicode[STIX]{x1D6FE}\bigg\},\end{eqnarray}$$
$0\leq \unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D6FD}\leq 1$, 
$0\leq \unicode[STIX]{x1D6FE}\leq +\infty$.
