Universal \(\beta\)-expansions
type: Item
instance of: scholarly article
title: Universal \(\beta\)-expansions
zbMATH Open document ID: 1049.11009
author: Nikita Sidorov
published in: Periodica Mathematica Hungarica
publication date: 2004-10-19T00:00:00Z
review text: For fixed \(\beta\in(1,2)\) and \(x\geq0\) the authors consider \(\beta\)-expansions, i.e., expansions of the form \(x=\sum_{n=1}^\infty\varepsilon_n\beta^{-n}\), where \(\epsilon=(\varepsilon_n)\) is a \(0-1\) sequence. Such an expansion is called universal if every finite \(0-1\) word occurs as an interval in the sequence \(\epsilon\). The author showed earlier [Am. Math. Mon. 110, 838--842 (2003; Zbl 1049.11085)] that almost every \(x\in I_\beta=[0,1/(\beta-1)]\) has a continuum of distinct \(\beta\)-expansions. Now he proves that almost every \(x\in I_\beta\) has a universal \(\beta\)-expansion and poses the question of whether every \(x\) which has a universal \(\beta\)-expansion must have a continuum of distinct \(\beta\)-expansions.
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 2108669
zbMATH Keywords: beta-expansion
zbMATH Keywords: complexity
zbMATH Keywords: universal expansion
zbMATH Keywords: normal expansion
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
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full work available at URL: b:mahu.0000010823.98226.4e
OpenAlex ID: W1974819605
DOI: 10.1023/B:MAHU.0000010823.98226.4E
description: scientific article; zbMATH DE number 2108669