Exact Diophantine approximation of real numbers by \(\beta\)-expansions
type: Item
instance of: scholarly article
title: Exact Diophantine approximation of real numbers by \(\beta\)-expansions
author: Xinyun Zhang
author: Wenmin Zhong
published in: Discrete and Continuous Dynamical Systems
publication date: 2024-08-13T00:00:00Z
review text: In this paper, the problem about the exact approximation order in the system of \(\beta\)-expansions is introduced.\N\NLet \(\beta>1\) be a real number and \(\psi:\mathbb N\to\mathbb R_{+}\) be a positive and non-increasing function defined on \(\mathbb N\) with \(\psi(n)\to 0\) as \(n\to+\infty\). Let \(E_\beta(\psi):=W(\psi)\backslash\bigcup_{0<c<1}W(c\psi)\) be the set of \(x\) which is approximable to order \(\psi\) but to no better order, where \N\[\NW(\psi):=\{x\in[0,1):x-\omega_n(x)<\psi(n)/\beta^n\,\hbox{for infinitely many}\, n\in\mathbb N\}. \N\]\NThe main result of this paper is to obtain the Hausdorff dimension of \(E_\beta(\psi)\): \N\[\N\dim_H E_\beta(\psi)=\frac{1}{1+\alpha}, \N\]\Nwhere \N\[\N\alpha:=\liminf_{n\to+\infty}\frac{-\log_\beta\psi(n)}{n}. \N\]\NThis result strengthens that in [\textit{L. Fang} et al., Math. Z. 296, No. 1--2, 13--40 (2020; Zbl 1452.11097)].
reviewed by: Takao Komatsu
Mathematics Subject Classification ID: 11K55
Mathematics Subject Classification ID: 28A80
zbMATH DE Number: 7896606
zbMATH Keywords: Cantor set
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: \(\beta\)-expansions
zbMATH Keywords: exact approximation
MaRDI profile type: Publication
cites work: Exact approximation order and well-distributed sets
cites work: Measure theoretic laws for lim sup sets
cites work: A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures
cites work: Sets of exact approximation order by rational numbers
cites work: Q3391526
cites work: Sets of exact approximation order by rational numbers III
cites work: Distribution of full cylinders and the Diophantine properties of the orbits in \(\beta\)-expansions
cites work: On the lengths of basic intervals in beta expansions
cites work: Approximation orders of real numbers by \(\beta\)-expansions
cites work: Sets of exact approximation order by complex rational numbers
cites work: The ergodic theory of shrinking targets
cites work: On simultaneous Diophantine approximations
cites work: Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen.
cites work: On theβ-expansions of real numbers
cites work: Some metrical theorems in number theory
cites work: Representations for real numbers and their ergodic properties
cites work: Shrinking target problems for beta-dynamical system
cites work: Modified shrinking target problem in beta dynamical systems
DOI: 10.3934/DCDS.2024042
description: scientific article; zbMATH DE number 7896606