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On exceptional sets in Erdős-Rényi limit theorem. (English) Zbl 1408.11077
Summary: For $$x \in [0, 1]$$, the run-length function $$r_n(x)$$ is defined as the length of the longest run of 1’s amongst the first $$n$$ dyadic digits in the dyadic expansion of $$x$$. P. Erdős and A. Rényi [ J. Anal. Math. 23, 103–111 (1970; Zbl 0225.60015)] proved that $$\lim_{n \to \infty} \frac{r_n(x)}{\log_2 n} = 1$$ for Lebesgue almost all $$x \in [0, 1]$$. In this paper, we study the Hausdorff dimensions of the exceptional sets in Erdős-Rényi limit theorem. Let $$\varphi : \mathbb{N} \to(0, + \infty)$$ be a monotonically increasing function satisfying $$\lim_{n \to \infty} \frac{n}{\varphi(n^{1 + \alpha})} = + \infty$$ with some $$0 < \alpha \leq 1$$. We prove that the set $E_{\max}^\varphi = \left\{x \in [0, 1] : \lim \inf_{n \to \infty} \frac{r_n(x)}{\varphi(n)} = 0, \lim \sup_{n \to \infty} \frac{r_n(x)}{\varphi(n)} = + \infty \right\}$ has Hausdorff dimension one and is residual in $$[0, 1]$$.

##### MSC:
 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 28A78 Hausdorff and packing measures 28A80 Fractals
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