A NOTE ON THE INTERSECTIONS OF THE BESICOVITCH SETS AND ERDŐS–RÉNYI SETS
2019
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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