On the intersections of the Besicovitch sets and exceptional sets in the Erdős-Rényi limit theorem (Q2317444)

Any $x \in (0,1)$ has a dyadic expansion $x = \frac{x_1}{2} + \frac{x_2}{2^2} + \cdots + \frac{x_n}{2^n} + \cdots,$ with digits $x_k \in \{0,1\}$. The run-length function $r_n$ is the maximal length of the strings of $1$s in the first $n$ digits, and denotes the sum of the $1$s in the first $n$ digits by $S_n$, and level sets defined in terms of the asymptotic size of both functions have been studied widely. Here the set of $x\in[0,1)$ for which $\lim_{n\to\infty}\frac{S_n(x)}{n}=\alpha$, $\liminf_{n\to\infty}\frac{r_n(x)}{\varphi(n)}=0$, and $\limsup_{n\to\infty}\frac{r_n(x)}{\varphi(n)}=\infty$ for a monotonically increasing unbounded function $\varphi\colon\mathbb{N}\to(0,\infty)$ is shown to either be empty or to have Hausdorff dimension $(-\alpha\log\alpha-(1-\alpha)\log(1-\alpha))/(\log2)$. It may help the reader to be alert to the use of $H$ for the classical entropy function appearing in the Hausdroff dimension formula, for the class of monotone functions used to characterize growth rates in the run-length function, and as a subscript denoting ``Hausdorff''.