Abstract
We show that the minimal proportion of one letter in an n-th power-free binary word is asymptotically 1/n. We also consider a generalization of n-th power-free words defined through the notion of exponent: a word is χ-th power-free for a real χ, if it does not contain subwords of exponent χ or more. We study the minimal proportion of one letter in an χ-th power-free binary word as a function of χ and prove, in particular, that this function is discontinuous.
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A. И. 3иMиH. БлoKиpyющe MHOжeCTBa Teрмoв. Mameмamuvecxu6 C6opvcux, 119(3):363–375, 1982. English Translation: A.I.Zimin, Blocking sets of terms, Math. USSR Sbornik 47 (1984), 353–364.
A.A. EBIIOKHMOB. Iloambie MHoxceCTSa c.nos H rix-Hcaiosaie xapaKTepuCTHKH. In Memodbi ductcpemnoeo aacaAuda e uccaedoeanuu axcmpe.naAbHbax cmpyxmyp, volume 39, pages 7–19, Hosocri6xpcx, 1983.
K. Baker, G. McNulty, and W. Taylor. Growth problems for avoidable words. Theoret. Comp. Sci., 69:319–345, 1989.
D. Bean, A. Ehrenfeucht, and G. McNulty. Avoidable patterns in strings of symbols. Pacific J. Math., 85(2):261–294, 1979.
J. Berstel. Axel thue's work on repetitions in words. Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics, Montreal, 1992, June 1992. accessible at http://www-litp. ibp.fr:80/berstel/.
J. Berstel and D. Perrin. Theory of codes. Academic Press, 1985.
J. Cassaigne. Motifs évitables et régularités dans les mots. These de doctorat, Université Paris VI, 1994.
M. Crochemore and P. Goralcik. Mutually avoiding ternary words of small exponent. International Journal of Algebra and Computation, 1(4):407–410, 1991.
J. Currie. Open problems in pattern avoidance. American Mathematical Monthly, 100:790–793, 1993.
F. Dejean. Sur un théoréme de Thue. J. Combinatorial Th. (A), 13:90–99, 1972.
A. Kfoury. A linear time algorithm testing whether a word contains an overlap. RAIRO Inf. Th., 22:135–145, 1988.
M. Lothaire. Combinatorics on Words, volume 17 of Encyclopedia of Mathematics and Its Applications. Addison Wesley, 1983.
A. Restivo and S. Salemi. On weakly square free words. Bull. of the EATCS, 21:49–56, 1983.
P. Roth. Every binary pattern of length six is avoidable on the two-letter alphabet. Acta Informatica, 29:95–106, 1992.
A. Salomaa. Jewels of formal language theory. Computer Science Press, 1986.
M. Sapir. Combinatorics on words with applications, December 1993. accessible at http://www.math.unl.edu/-msapir/ftp/course.
A. Thue. Über unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, 7:1–22, 1906.
A. Thue. Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, 10:1–67, 1912.
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Kolpakov, R., Kucherov, G. (1997). Minimal letter frequency in n-th power-free binary words. In: Prívara, I., Ružička, P. (eds) Mathematical Foundations of Computer Science 1997. MFCS 1997. Lecture Notes in Computer Science, vol 1295. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0029978
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DOI: https://doi.org/10.1007/BFb0029978
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