A characterization of fine words over a finite alphabet. (English) Zbl 1133.68065

Summary: To any infinite word \(\mathbf t\) over a finite alphabet \(\mathcal A\) we can associate two infinite words \(\min (\mathbf t)\) and \(\max (\mathbf t)\) such that any prefix of \(\min (\mathbf t)\) (resp. \(\max (\mathbf t)\)) is the lexicographically smallest (resp. greatest) amongst the factors of \(\mathbf t\) of the same length. We say that an infinite word \(\mathbf t\) over \(\mathcal A\) is fine if there exists an infinite word \(\mathbf s\) such that, for any lexicographic order, \(\min (\mathbf t) = a \mathbf s\) where \(a = \min (\mathcal A)\). In this paper, we characterize fine words; specifically, we prove that an infinite word \(\mathbf t\) is fine if and only if \(\mathbf t\) is either a strict episturmian word or a strict “skew episturmian word”. This characterization generalizes a recent result of G. Pirillo, who proved that a fine word over a 2-letter alphabet is either an (aperiodic) Sturmian word, or an ultimately periodic (but not periodic) infinite word, all of whose factors are (finite) Sturmian.


68R15 Combinatorics on words
68Q45 Formal languages and automata
Full Text: DOI arXiv


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