On the existence of minimal \(\beta\)-powers
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.
scientific article; zbMATH DE number 6073873
scientific article