On the existence of minimal \(\beta\)-powers
type: Item
instance of: scholarly article
zbMATH Open document ID: 1260.68314
DOI: 10.1142/S0129054111008969
author: Arseny M. Shur
published in: International Journal of Foundations of Computer Science
publication date: 2012-08-29T00:00:00Z
Mathematics Subject Classification ID: 68R15
zbMATH DE Number: 6073873
zbMATH Keywords: power-free word
zbMATH Keywords: minimal power
zbMATH Keywords: circular word
MaRDI profile type: Publication
cites work: The Thue-Morse word contains circular \(5/2^+\) power free words of every length
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full work available at URL: s0129054111008969
OpenAlex ID: W2033684775
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title: On the existence of minimal \(\beta\)-powers
review text: Let \(w\) be a finite word of length \(n\) over a \(k\)-letter alphabet. We say that \(w\) has period \(p\) if \(w[i] = w[i+p]\) for \(0\leq i < n-p\); the smallest such \(p\) is called the period. For example, the period of the German word nennen is \(3\). If the period of \(w\) is \(p\) and the length of \(w\) is \(n\), we say that \(w\) is a \(\beta\)-power for all real numbers \(\beta\) satisfying the inequality \((n-1)/p < \beta \leq n/p\). Thus nennen is a \(2\)-power and also an (\(e-1\))-power, where \(e \doteq 2.718\). A \(\beta\)-power is minimal if it contains no other \(\beta\)-powers as a factor (contiguous subword).NEWLINENEWLINENEWLINEIn this paper the author's goal is to characterize all triples (\(k, \beta, p\)) for which there exists, over a \(k\)-letter alphabet, a minimal \(\beta\)-power with the period \(p\). He fully resolves the question for \(k = 2, 3\) and gives a conjectural description for \(k \geq 4\) and \(\beta < 2\). As a consequence he strengthens a result of \textit{A. Aberkane} and \textit{J. D. Currie} [Bull. Belg. Math. Soc. - Simon Stevin 12, No. 4, 525--534 (2005; Zbl 1137.68046)] on the description of \(\beta\)-power-free circular words.
reviewed by: Jeffrey Shallit
description: scientific article; zbMATH DE number 6073873