Hausdorff dimensions of some exceptional sets in Engel expansions
type: Item
instance of: scholarly article
title: Hausdorff dimensions of some exceptional sets in Engel expansions
zbMATH Open document ID: 1428.11140
author: Jia Liu
author: Meiying Lü
published in: Journal of Number Theory
publication date: 2017-12-21T00:00:00Z
review text: In this paper, it is noted that the condition $\lim_{n\to\infty}{\frac{\log {d_n(x)}}{n}}=1$ holds for almost all numbers $x\in (0,1]$ represented by the Engel expansion \[ x=\sum^{\infty} _{n=1}{\frac{1}{d_1(x)d_2(x)\cdots d_n(x)}}, \] where $(d_n(x))$ is a sequence of positive integers satisfying the conditions $d_1(x)\ge 2$ and $d_{n+1}(x)\ge d_{n}(x)$, where $n=1,2,\dots$. \par In the present article, the main attention is given to the Hausdorff dimension of the set \[ E(\phi)=\left\{x\in(0,1]: \lim_{n\to\infty}{ \frac{\log {d_n(x)}}{\phi(n)}}=1\right\}, \] where $x$ is represented by the Engel expansion and $\phi(n): \mathbb N \to \mathbb R^{+}$ is a function satisfying $\phi(n+1)-\phi(n)\to\infty$ as $n\to\infty$. Also, some partial cases of $E(\phi)$ are considered: \begin{itemize} \item the case when $\phi(n)=\gamma n^{\beta}$, where $\gamma>0$ and $\beta>1$; \item the case when $\phi(n)=\eta\tau^{n}$, where $\eta>0$ and $ \tau>1$. \end{itemize} The main statements on the Hausdorff dimension of these sets are proved. Certain well-known properties of the representation of real numbers by an Engel expansion, are described. A formula for computing the Hausdorff dimension for a class of Cantor sets related to Engel expansions is given.
reviewed by: Symon Serbenyuk
Mathematics Subject Classification ID: 11K55
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 11A67
zbMATH DE Number: 6820896
zbMATH Keywords: Engel expansions
zbMATH Keywords: exceptional sets
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Cantor sets
MaRDI profile type: Publication
full work available at URL: j.jnt.2017.09.015
OpenAlex ID: W2765593779
cites work: Q3995301
cites work: Representations of real numbers by infinite series
cites work: Hausdorff dimensions in Engel expansions
cites work: Some exceptional sets in Engel expansions
cites work: A problem of Galambos on Engel expansions
cites work: How many points have the same Engel and Sylvester expansions?
DOI: 10.1016/J.JNT.2017.09.015
description: scientific article; zbMATH DE number 6820896