On the topological structure of univoque sets
type: Item
instance of: scholarly article
title: On the topological structure of univoque sets
zbMATH Open document ID: 1111.11005
author: P. Loreti
author: V. Komornik
published in: Journal of Number Theory
publication date: 2007-02-12T00:00:00Z
review text: The authors study topological properties of the set of real numbers in \((1,2)\) that are univoque (recall that a real number \(\beta\in(1,2)\) is called univoque if 1 has a unique \(\beta\)-expansion \(1=\sum_{i\geq 1} c_i\beta^{-i}\), with \(c_i\in\{0,1\})\). Paralleling the combinatorial characterization of the set \(U\) of univoque numbers [\textit{P. Erdős, I. Joó} and \textit{V. Komornik}, Bull. Soc. Math. Fr. 118, No. 3, 377--390 (1990; Zbl 0721.11005)], the authors give a combinatorial characterization of \(\overline u\), the topological closure of \(u\). In particular \((\overline u\setminus u)\) is a countable dense subset of \(\overline u\). Hence \(\overline u\) is perfect (thus a Cantor set as it is of measure zero). The proof uses a fine study of the combinatorics of words appearing in the expansions in non-integer (univoque) bases.
reviewed by: Jean-Paul Allouche
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 68R15
zbMATH DE Number: 5124513
zbMATH Keywords: golden number
zbMATH Keywords: non-integer bases
zbMATH Keywords: Cantor sets
zbMATH Keywords: Thue-Morse sequence
zbMATH Keywords: beta-expansion
MaRDI profile type: Publication
full work available at URL: j.jnt.2006.04.006
OpenAlex ID: W1972868132
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DOI: 10.1016/J.JNT.2006.04.006
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description: scientific article; zbMATH DE number 5124513