Everywhere αα-repetitive sequences and Sturmian words

Local constraints on an infinite sequence that imply global regularity are of general interest in combinatorics on words. We consider this topic by studying everywhere  αα-repetitive sequences  . Such a sequence is defined by the property that there exists an integer N≥2N≥2 such that every length-NN factor has a repetition of order αα as a prefix. If each repetition is of order strictly larger than αα, then the sequence is called everywhere  α+α+-repetitive  . In both cases, the number of distinct minimal αα-repetitions (or α+α+-repetitions) occurring in the sequence is finite.A natural question regarding global regularity is to determine the least number, denoted by M(α)M(α), of distinct minimalαα-repetitions such that an αα-repetitive sequence is not necessarily ultimately periodic. We call the everywhere αα-repetitive sequences witnessing this property optimal  . In this paper, we study optimal 2-repetitive sequences and optimal 2+2+-repetitive sequences, and show that Sturmian words belong to both classes. We also give a characterization of 2-repetitive sequences and solve the values of M(α)M(α) for 1≤α≤15/71≤α≤15/7.