Everywhere -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 α -repetitive sequences. Such a sequence is defined by the property that there exists an integer N ≥ 2 such that every length- N factor has a repetition of order α as a prefix. If each repetition is of order strictly larger than α , then the sequence is called everywhere α + -repetitive. In both cases, the number of distinct minimal α -repetitions (or α + -repetitions) occurring in the sequence is finite. ral question regarding global regularity is to determine the least number, denoted by M ( α ) , of distinct minimal α -repetitions such that an α -repetitive sequence is not necessarily ultimately periodic. We call the everywhere α -repetitive sequences witnessing this property optimal. In this paper, we study optimal 2-repetitive sequences and optimal 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 ) for 1 ≤ α ≤ 15 / 7 .