The Hausdorff dimension of level sets described by Erdős-Rényi average
type: Item
instance of: scholarly article
title: The Hausdorff dimension of level sets described by Erdős-Rényi average
zbMATH Open document ID: 1410.37006
author: Peng Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-11-02T00:00:00Z
review text: 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 $\phi$.
reviewed by: Tomas Persson
Mathematics Subject Classification ID: 37A45
Mathematics Subject Classification ID: 37C45
Mathematics Subject Classification ID: 11K55
zbMATH DE Number: 6802324
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Erdős-Rényi average
zbMATH Keywords: slowly varying sequence
zbMATH Keywords: Moran set
MaRDI profile type: Publication
full work available at URL: j.jmaa.2017.09.018
OpenAlex ID: W2754686508
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DOI: 10.1016/J.JMAA.2017.09.018
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description: scientific article; zbMATH DE number 6802324