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On exceptional sets in Erdős-Rényi limit theorem revisited. (English) Zbl 1371.28022
Let $$x\in [0,1]$$ , and $$r_n(x)$$ be the length of the longest run of 1’s amongst the first $$n$$ digits in the dyadic expansion of $$x$$. Such $$r_n$$ is called the run-length function. Erdős and Rényi proved that the rate of growth of $$r_n(x)$$ is $$\log_2 n$$ for almost all $$x\in [0,1]$$. The set of exceptional points is negligible from the measure-theoretical point of view, but J.-H. Ma et al. [Monatsh. Math. 151, No. 4, 287–292 (2007; Zbl 1170.28001)] showed that it has Hausdorff dimension one. In this paper the authors study the asymptotic behavior of the run-length function with respect to more general speeds than $$\log_2 n$$. Let $$\varphi :\mathbb{N}\to (0,+\infty )$$ be an increasing function such that $$\lim_{n\to\infty} \varphi {n}=+\infty$$, and $E_{\max}^\varphi =\{ x\in [0,1] : \liminf_{n\to\infty } \frac{r_n(x)}{\varphi (n)} =0, \, \limsup_{n\to\infty} \frac{r _n(x)}{\varphi (n)} =+\infty \} .$ The main result of the paper says that the set $$E_{\max}^\varphi$$ either has Hausdorff dimension one and is residual in $$[0,1]$$, or is empty. It solves the conjecture posed by the authors in [J. Math. Anal. Appl. 436, 355–365 (2016; Zbl 1408.11077)].

##### MSC:
 28A80 Fractals 54E52 Baire category, Baire spaces 60F99 Limit theorems in probability theory
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