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On the Repetition Threshold for Large Alphabets

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Mathematical Foundations of Computer Science 2006 (MFCS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4162))

Abstract

The (maximal) exponent of a finite non-empty word is the ratio among its length and its period. Dejean (1972) conjectured that for any n ≥5 there exists an infinite word over n letters with no factor of exponent larger than n/(n–1). We prove that this conjecture is true for n ≥38.

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References

  1. Berstel, J.: Axel Thue’s papers on repetition in words: a translation, Publications du LaCIM, Université du Québec á Montréal 20 (1995)

    Google Scholar 

  2. Berstel, J., Karhumäki, J.: Combinatorics on words: a tutorial. Bulletin of the EATCS 79, 178–228 (2003)

    MATH  Google Scholar 

  3. Brandenburg, F.-J.: Uniformly growing k-th power-free homomorphisms. Theoret. Comput. Sci. 23, 69–82 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  4. Carpi, A.: On the repetition threshold for large alphabets, Dipartimento di Matematica e Informatica dell’Università di Perugia, Tech. rep. no. 5-2006 (2006)

    Google Scholar 

  5. Crochemore, M., Goralcik, P.: Mutually avoiding ternary words of small exponent. Int. J. Alg. Comp. 1, 407–410 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Dejean, F.: Sur un théorème de Thue. J. Combin. Th. A 13, 90–99 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  7. Ilie, L., Ochem, P., Shallit, J.: A generalization of repetition threshold. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 818–826. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  8. Karhumäki, J., Shallit, J.: Polynomial versus exponential growth in repetition-free binary words. J. Comb. Theory Ser. A 105, 335–347 (2004)

    Article  MATH  Google Scholar 

  9. Mignosi, F., Pirillo, G.: Repetitions in the Fibonacci infinite word. RAIRO Inf. Theor. and Appl. 26, 199–204 (1992)

    MATH  MathSciNet  Google Scholar 

  10. Mohammad-Noori, M., Currie, J.D.: Dejean’s conjecture and Sturmian words. European J. Comb. (to appear)

    Google Scholar 

  11. Moulin-Ollagnier, J.: Proof of Dejean’s conjecture for alphabets with 5, 6, 7, 8, 9, 10 and 11 letters. Theoret. Comput. Sci. 95, 187–205 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Pansiot, J.-J.: A propos d’une conjecture de F. Dejean sur les répétitions dans les mots. Discr. Appl. Math. 7, 297–311 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  13. Shallit, J.: Simultaneous avoidance of large squares and fractional powers in infinite binary words. Int. J. Found. Comput. Sci. 15, 317–327 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Thue, A.: Uber unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl., Christiania 7, 1–22 (1906); reprinted in: Nagell, T. (ed.) Selected Mathematical Papers of Axel Thue, pp. 139–158. Universitetsforlaget, Oslo (1977)

    Google Scholar 

  15. Thue, A.: Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. Mat. Nat. Kl., Christiania 1, 1–67 (1912); reprinted in: Nagell, T. (ed.) Selected Mathematical Papers of Axel Thue, pp. 413–478. Universitetsforlaget, Oslo (1977)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Perugia, Italy

    Arturo Carpi

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  1. Arturo Carpi
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Comenius University, Mlynská dolina, 84248, Bratislava, Slovakia

    Rastislav Královič

  2. Institute of Informatics, University of Warsaw, Poland

    Paweł Urzyczyn

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© 2006 Springer-Verlag Berlin Heidelberg

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Carpi, A. (2006). On the Repetition Threshold for Large Alphabets. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_20

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  • DOI: https://doi.org/10.1007/11821069_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37791-7

  • Online ISBN: 978-3-540-37793-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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