Sturmian words and a criterium by Michaux-Villemaire
From the authors' abstract: ``Michaux and Villemaire's proof of Cobham's theorem relies on the characterization of ultimately periodic words by means of the behaviour of certain repetitions in the word. Namely, they consider the length of the smallest shift between repetitions of a given length and the first position at which that smallest shift is observed.'' Their characterization needs that two sequences \(\delta\) and \(\alpha\) admit certain syndetic subsequences (an increasing sequence of integers is called syndetic if its first difference is bounded); and they ask whether it is possible to assume that only one of these two subsequences is syndetic. The authors of the present paper study in detail \(\delta\) and \(\alpha\) for characteristic Sturmian sequences. As a consequence of their study they prove that the answer to \textit{C. Michaux} and \textit{R. Villemaire}'s question is negative [Ann. Pure Appl. Logic 77, No. 3, 251--277 (1996; Zbl 0857.03003)].
scientific article; zbMATH DE number 2184113