On exceptional sets in Erdős-Rényi 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 $x.$ Erdős and Rényi proved that $\underset{n\to \infty }{\lim }\frac{r_n(x)}{{\log }_{2}n}=1$ for Lebesgue almost all $x\in [0,1]$. Let $H$ denote the set of monotonically increasing functions $\phi :\mathbb{N}\to (0,+\infty )$ with $\underset{n\to \infty }{\lim }\phi (n)=+\infty$. For any $\phi \in H$, we prove that the set \[ E_{\max}^φ=\left\{x\in [0,1]:\liminf\limits_{n\to\infty}\frac{r_n(x)}{φ(n)}=0, \limsup\limits_{n\to\infty}\frac{r_n(x)}{φ(n)}=+\infty\right\} \] either has Hausdorff dimension one and is residual in $[0,1]$ or empty. The result solves a conjecture posed in \cite{LW5} affirmatively.

9 pages