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Repetitions in Infinite Palindrome-Rich Words

  • Aseem R. BaranwalEmail author
  • Jeffrey Shallit
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11682)

Abstract

Rich words are those containing the maximum possible number of distinct palindromes. Several characteristic properties of rich words have been studied; yet the analysis of repetitions in rich words still involves some interesting open problems. We consider lower bounds on the repetition threshold of infinite rich words over 2- and 3-letter alphabets, and construct a candidate infinite rich word over the alphabet \(\varSigma _2=\{0,1\}\) with a small critical exponent of \(2+\sqrt{2}/2\). This represents the first progress on an open problem of Vesti from 2017.

Keywords

Critical exponent Repetitions Rich words Palindrome 

Notes

Acknowledgments

We are grateful to the referees for their suggestions.

After our paper was submitted, we learned from Edita Pelantová that our word r is a complementary symmetric Rote word [21], and hence by [3, 18] it follows that \(\mathbf r\) is rich.

References

  1. 1.
    Angluin, D.: Learning regular sets from queries and counterexamples. Inf. Comput. 75(2), 87–106 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baranwal, A.R., Shallit, J.: Critical exponent of infinite balanced words via the Pell number system. Preprint: https://arxiv.org/abs/1902.00503 (2019)
  3. 3.
    Blondin Massé, A., Brlek, S., Labbé, S., Vuillon, L.: Palindromic complexity of codings of rotations. Theoret. Comput. Sci. 412, 6455–6463 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brlek, S., Hamel, S., Nivat, M., Reutenauer, C.: On the palindromic complexity of infinite words. Int. J. Found. Comput. Sci. 15, 293–306 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bucci, M., De Luca, A., Glen, A., Zamboni, L.Q.: A new characteristic property of rich words. Theoret. Comput. Sci. 410, 2860–2863 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, G., Puglisi, S.J., Smyth, W.F.: Fast & practical algorithms for computing all the runs in a string. In: Ma, B., Zhang, K. (eds.) CPM 2007, LNCS, vol. 4580, pp. 307–315. Springer, Heidelberg (2007).  https://doi.org/10.1007/978-3-540-73437-6_31
  7. 7.
    Crochemore, M., Ilie, L.: Computing longest previous factor in linear time and applications. Inform. Process. Lett. 106(2), 75–80 (2008)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Currie, J., Rampersad, N.: A proof of Dejean’s conjecture. Math. Comput. 80(274), 1063–1070 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    de Luca, A., Glen, A., Zamboni, L.Q.: Rich, Sturmian, and trapezoidal words. Theoret. Comput. Sci. 407, 569–573 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dejean, F.: Sur un théorème de Thue. J. Combin. Theory. Ser. A 13(1), 90–99 (1972)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Glen, A., Justin, J., Widmer, S., Zamboni, L.Q.: Palindromic richness. Eur. J. Comb. 30, 510–531 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Groult, R., Prieur, E., Richomme, G.: Counting distinct palindromes in a word in linear time. Inform. Process. Lett. 110, 908–912 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Guo, C., Shallit, J., Shur, A.M.: Palindromic rich words and run-length encodings. Inform. Process. Lett. 116, 735–738 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gusfield, D.: Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology. Cambridge University Press, Cambridge (1997)Google Scholar
  15. 15.
    Mousavi, H.: Automatic theorem proving in Walnut. Preprint: https://arxiv.org/abs/1603.06017 (2016)
  16. 16.
    Ostrowski, A.: Bemerkungen zur Theorie der diophantischen Approximationen. Abh. Math. Semin. Univ. Hamburg 1(1), 77–98 (1922)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pelantová, E., Starosta, Š.: Languages invariant under more symmetries: overlapping factors versus palindromic richness. Discrete Math. 313, 2432–2445 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pelantová, E., Starosta, Š.: Constructions of words rich in palindromes and pseudopalindromes. Discrete Math. Theoret. Comput. Sci. 18, Paper #16 (2016). https://dmtcs.episciences.org/2202
  19. 19.
    Rampersad, N., Shallit, J., Vandomme, E.: Critical exponents of infinite balanced words. Theoret. Comput, Sci. 777, 454–463 (2018)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rao, M.: Last cases of Dejean’s conjecture. Theoret. Comput. Sci. 412(27), 3010–3018 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rote, G.: Sequences with subword complexity \(2n\). J. Number Theory 46, 196–213 (1994)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Rubinchik, M., Shur, A.M.: EERTREE: an efficient data structure for processing palindromes in strings. In: Lipták, Z., Smyth, W.F. (eds.) IWOCA 2015. LNCS, vol. 9538, pp. 321–333. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-29516-9_27CrossRefGoogle Scholar
  23. 23.
    Schaeffer, L., Shallit, J.: Closed, palindromic, rich, privileged, trapezoidal, and balanced words in automatic sequences. Electronic J. Combinatorics 23, 1–25 (2016)Google Scholar
  24. 24.
    Vesti, J.: Extensions of rich words. Theoret. Comput. Sci. 548, 14–24 (2014)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Vesti, J.: Rich square-free words. Theoret. Comput. Sci. 687, 48–61 (2017)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Computer ScienceUniversity of WaterlooWaterlooCanada

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