On exceptional sets in Erd\H{o}s-R\'{e}nyi limit theorem revisited

Abstract

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 $x.$ Erd\H{o}s and R\'enyi proved that $\lim\limits_{n\to\infty}\frac{r_n(x)}{\log_2n}=1$ for Lebesgue almost all $x\in[0,1]$. Let $H$ denote the set of monotonically increasing functions $\varphi:\mathbb{N}\to (0,+\infty)$ with $\lim\limits_{n\to\infty}\varphi(n)=+\infty$. For any $\varphi\in H$, we prove that the set \[ E_{\max}^\varphi=\left\{x\in [0,1]:\liminf\limits_{n\to\infty}\frac{r_n(x)}{\varphi(n)}=0, \limsup\limits_{n\to\infty}\frac{r_n(x)}{\varphi(n)}=+\infty\right\} \] either has Hausdorff dimension one and is residual in $[0,1]$ or empty. The result solves a conjecture posed in \cite{LW5} affirmatively.