WORDS 2019: Combinatorics on Words pp 93-105

# Repetitions in Infinite Palindrome-Rich Words

• Aseem R. Baranwal
• 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)
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)
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)
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)
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).
7. 7.
Crochemore, M., Ilie, L.: Computing longest previous factor in linear time and applications. Inform. Process. Lett. 106(2), 75–80 (2008)
8. 8.
Currie, J., Rampersad, N.: A proof of Dejean’s conjecture. Math. Comput. 80(274), 1063–1070 (2011)
9. 9.
de Luca, A., Glen, A., Zamboni, L.Q.: Rich, Sturmian, and trapezoidal words. Theoret. Comput. Sci. 407, 569–573 (2008)
10. 10.
Dejean, F.: Sur un théorème de Thue. J. Combin. Theory. Ser. A 13(1), 90–99 (1972)
11. 11.
Glen, A., Justin, J., Widmer, S., Zamboni, L.Q.: Palindromic richness. Eur. J. Comb. 30, 510–531 (2009)
12. 12.
Groult, R., Prieur, E., Richomme, G.: Counting distinct palindromes in a word in linear time. Inform. Process. Lett. 110, 908–912 (2010)
13. 13.
Guo, C., Shallit, J., Shur, A.M.: Palindromic rich words and run-length encodings. Inform. Process. Lett. 116, 735–738 (2016)
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)
17. 17.
Pelantová, E., Starosta, Š.: Languages invariant under more symmetries: overlapping factors versus palindromic richness. Discrete Math. 313, 2432–2445 (2013)
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)
20. 20.
Rao, M.: Last cases of Dejean’s conjecture. Theoret. Comput. Sci. 412(27), 3010–3018 (2011)
21. 21.
Rote, G.: Sequences with subword complexity $$2n$$. J. Number Theory 46, 196–213 (1994)
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).
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)
25. 25.
Vesti, J.: Rich square-free words. Theoret. Comput. Sci. 687, 48–61 (2017)