On exceptional sets in Erdős-Rényi limit theorem revisited
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 \textit{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)].
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