## Everywhere $$\alpha$$-repetitive sequences and Sturmian words.(English)Zbl 1187.68369

Summary: 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 $$\alpha$$-repetitive sequences. Such a sequence is defined by the property that there exists an integer $$N\geq 2$$ such that every length-$$N$$ factor has a repetition of order $$\alpha$$ as a prefix. If each repetition is of order strictly larger than $$\alpha$$, then the sequence is called everywhere $$\alpha ^{+}$$-repetitive. In both cases, the number of distinct minimal $$\alpha$$-repetitions (or $$\alpha ^{+}$$-repetitions) occurring in the sequence is finite.
A natural question regarding global regularity is to determine the least number, denoted by $$M(\alpha )$$, of distinct minimal $$\alpha$$-repetitions such that an $$\alpha$$-repetitive sequence is not necessarily ultimately periodic. We call the everywhere $$\alpha$$-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 $$M(\alpha )$$ for $$1\leq \alpha \leq 15/7$$.

### MSC:

 68R15 Combinatorics on words 68Q45 Formal languages and automata
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### References:

 [1] Lothaire, M., () [2] Mignosi, F.; Restivo, A.; Salemi, S., Periodicity and the Golden ratio, Theoret. comput. sci., 204, 1-2, 153-167, (1998) · Zbl 0913.68162 [3] Karhumäki, J.; Lepistö, A.; Plandowski, W., Locally periodic versus globally periodic infinite words, J. combin. theory ser. A, 100, 250-264, (2002) · Zbl 1011.68070 [4] A. Lepistö, On relations between local and global periodicity, Ph.D. Dissertation, Turku Centre for Computer Science, 2003 [5] Saari, K., Everywhere $$\alpha$$-repetitive sequences and Sturmian words, (), 363-372 · Zbl 1188.68218 [6] Choffrut, C.; Karhumäki, J., Combinatorics on words, (), 329-438 [7] K. Saari, On the periodicity and frequency of infinite words, Ph.D. Dissertation, Turku Centre for Computer Science, 2008 [8] Allouche, J.-P.; Shallit, J., Automatic sequences. theory, applications, generalizations, (2003), Cambrige University Press Cambridge · Zbl 1086.11015 [9] Wen, Z.-X.; Wen, Z.-Y., Some properties of the singular words of the Fibonacci word, European J. combin., 15, 587-598, (1994) · Zbl 0823.68087 [10] Knuth, D.E.; Morris, J.H.; Pratt, V.R., Fast pattern matching in strings, SIAM J. comput., 6, 2, 323-350, (1977) · Zbl 0372.68005 [11] Berthé, V.; Holton, C.; Zamboni, L.Q., Initial powers of Sturmian sequences, Acta arith., 122, 4, 315-347, (2006) · Zbl 1117.37005 [12] Krieger, D.; Shallit, J., Every real number greater than 1 is a critical exponent, Theoret. comput. sci., 381, 1-3, 177-182, (2007) · Zbl 1188.68216 [13] Carpi, A.; de Luca, A., Special factors, periodicity, and an application to Sturmian words, Acta inform., 36, 983-1006, (2000) · Zbl 0956.68119 [14] Damanik, D.; Lenz, D., The index of Sturmian sequences, European J. combin., 23, 23-29, (2002) · Zbl 1002.11020 [15] Currie, J.D.; Visentin, T.I., On abelian 2-avoidable binary patterns, Acta inform., 43, 521-533, (2007) · Zbl 1111.68094 [16] Richomme, G.; Saari, K.; Zamboni, L.Q., Standard words and abelian powers in Sturmian words, () · Zbl 1184.68378
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