A generalization of the Erdős-Rényi limit theorem and the corresponding multifractal analysis
type: Item
instance of: scholarly article
title: A generalization of the Erdős-Rényi limit theorem and the corresponding multifractal analysis
zbMATH Open document ID: 1423.11135
author: Haibo Chen
author: Min Yu
published in: Journal of Number Theory
publication date: 2018-09-24T00:00:00Z
review text: It is known that the well-known Erdős-Rényi limit theorem characterizes the limit behaviour of the run-length functions showing especially that the run-length functions for almost all numbers in the unit interval are of the logarithmic growth rate $\log_2n$.\par The aim of the present paper is to give a generalization of the classic Erdős-Rényi limit theorem through the introduction of a more general concept of the run-length function, as well as to complement the above-mentioned generalized limit theorem to the case of $m$-adic expansions, and to study the sets in which the run-length functions are endowed with different growth rates from the viewpoint of dimensional theory.\par In particular, the authors managed to give a generalization of the classic Erdős-Rényi theorem by studying the limit behaviour of the run-length function $R^p_n(x)$ of $x\in[0,1]$ in $\overline\Sigma p$, where $1\le p\le m-1$, $m\ge 2$, are integers and $\overline\Sigma p\subset\{0,1,\dots, m-1\}$ the alphabet with $p$ elements, and to prove that $$\lim_{n\to \infty}\frac{R^p_n(x)}{\log_{m/p}n}= 1,\qquad\text{for almost all }x\in [0,1],$$ \par in the sense of the Lebesgue measure.\par Furthermore, the authors manage to complement the generalized Erdős-Rényi theorem by finding that the level set in which the above limit equals to $\alpha$, for any $0\le\alpha\le+\propto$, is of full Hausdorff dimension. (It is known that for the level set $$E_m(\phi)= \Biggl\{x\in [0,1]: \lim_{n\to\infty} \frac{R^p_n(x)}{\varphi(n)}= 1\Biggr\},$$ where $\varphi$ is a positive function defined on $N$, its Hausdorff dimension is determined when the function $\varphi$ is of some particular growth rates.)
reviewed by: Chryssoula Ganatsiou
Mathematics Subject Classification ID: 11K55
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 6943100
zbMATH Keywords: run-length function
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Erdős-Rényi limit theorem
zbMATH Keywords: regularly varying sequence
MaRDI profile type: Publication
full work available at URL: j.jnt.2018.04.019
OpenAlex ID: W2807443704
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DOI: 10.1016/J.JNT.2018.04.019
description: scientific article; zbMATH DE number 6943100