Sturmian words and a criterium by Michaux-Villemaire

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. In this paper we study those properties for characteristic Sturmian words. In particular we answer a question posed by Michaux and Villemaire in that context.