Everywhere α-repetitive sequences and Sturmian words

  • Authors:

  • Kalle Saari

  • Affiliations:

  • Department of Mathematics, University of Turku, 20014 Turku, Finland and Turku Centre for Computer Science, University of Turku, 20014 Turku, Finland

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2010

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Abstract

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@a-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 @a as a prefix. If each repetition is of order strictly larger than @a, then the sequence is called everywhere@a^+-repetitive. In both cases, the number of distinct minimal @a-repetitions (or @a^+-repetitions) occurring in the sequence is finite. A natural question regarding global regularity is to determine the least number, denoted by M(@a), of distinct minimal@a-repetitions such that an @a-repetitive sequence is not necessarily ultimately periodic. We call the everywhere @a-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(@a) for 1@?@a@?15/7.