A NOTE ON THE INTERSECTIONS OF THE BESICOVITCH SETS AND ERDŐS–RÉNYI SETS
2019 ◽
Vol 108
(1)
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pp. 33-45
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Keyword(s):
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 $0\leq \unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D6FD}\leq 1$, $0\leq \unicode[STIX]{x1D6FE}\leq +\infty$.
Keyword(s):
1961 ◽
Vol 5
(1)
◽
pp. 35-40
◽
1955 ◽
Vol 7
◽
pp. 347-357
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1968 ◽
Vol 9
(2)
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pp. 146-151
◽
1953 ◽
Vol 1
(3)
◽
pp. 119-120
◽
1963 ◽
Vol 6
(2)
◽
pp. 70-74
◽
1964 ◽
Vol 16
◽
pp. 94-97
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1949 ◽
Vol 1
(1)
◽
pp. 48-56
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1966 ◽
Vol 18
◽
pp. 621-628
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1962 ◽
Vol 14
◽
pp. 565-567
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