The Hausdorff dimension of level sets described by Erdős-Rényi average
The authors study so called Erdős-Rényi averages of the digits of binary expansion of real numbers. Given a function $\phi$ such that $1 \leq \phi(n) \leq n$, one defines $I_{n, \phi (n)} (x)$ as the maximal sum of the digits of $x$ in a block of length $\phi (n)$ among the first $n$ digits. The Erdős-Rényi average is then the number $A_{n, \phi (n)} (x) = \frac{I_n \phi (n)}{\phi (n)}$ and the authors investigate the limits thereof when $n \to \infty$. More precisely, the authors are interested in the Hausdorff dimension of the set $ER_\phi (\alpha) = \{ x : A_{n, \phi(n)} (x) \to \alpha,\ n \to \infty \}$. They prove several results which either give the exact value of the Hausdorff dimension or at least provide its estimate, depending on $\alpha$ and $\phi$.
scientific article; zbMATH DE number 6802324
scientific article