Abstract
We study aperiodic balanced sequences over finite alphabets. A sequence \(\mathbf {v}\) of this type is fully characterised by a Sturmian sequence \(\mathbf {u}\) and two constant gap sequences \(\mathbf {y}\) and \(\mathbf {y}'\). We study the language of \(\mathbf {v}\), with focus on return words to its factors. We provide a uniform lower bound on the asymptotic critical exponent of all sequences \(\mathbf {v}\) arising by \(\mathbf {y}\) and \(\mathbf {y}'\). It is a counterpart to the upper bound on the least critical exponent of \(\mathbf {v}\) conjectured and partially proved recently in works of Baranwal, Rampersad, Shallit and Vandomme. We deduce a method computing the exact value of the asymptotic critical exponent of \(\mathbf {v}\) provided the associated Sturmian sequence \(\mathbf {u}\) has a quadratic slope. The method is used to compare the critical and the asymptotic critical exponent of balanced sequences over an alphabet of size \(d\le 10\) which are conjectured by Rampersad et al. to have the least critical exponent.
The research received funding from the project CZ.02.1.01/0.0/0.0/16_019/0000778. We would like to thank Daniela Opočenská for her careful and readily usable implementation of our program computing the asymptotic critical exponent.
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Notes
- 1.
More precisely, the minimality in the case \(d = 4\) was proved by Peltomäki in a private communication to Rampersad.
References
Balková, L., Bucci, M., De Luca, A., Hladký, J., Puzynina, S.: Aperiodic pseudorandom number generators based on infinite words. Theoret. Comput. Sci. 647, 85–100 (2016)
Balková, L., Pelantová, E., Starosta, Š.: Sturmian jungle (or garden?) on multiliteral alphabets. RAIRO-Theoret. Inf. Appl. 44, 443–470 (2010)
Baranwal, A.R.: Decision algorithms for Ostrowski-automatic sequences, the master thesis, University of Waterloo (2020). http://hdl.handle.net/10012/15845
Baranwal, A.R., Shallit, J.: Critical exponent of infinite balanced words via the Pell number system. In: Mercaş, R., Reidenbach, D. (eds.) WORDS 2019. LNCS, vol. 11682, pp. 80–92. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-28796-2_6
Berthé, V., et al.: Acyclic, connected and tree sets. Monatshefte für Mathematik 176(4), 521–550 (2014). https://doi.org/10.1007/s00605-014-0721-4
Carpi, A., de Luca, A.: Special factors, periodicity, and an application to Sturmian words. Acta Inform. 36, 983–1006 (2000). https://doi.org/10.1007/PL00013299
Currie, J.D., Mol, L., Rampersad, N.: The repetition threshold for binary rich words. Discrete Math. Theoret. Comput. Sci. 22(1) (2020). https://doi.org/10.23638/DMTCS-22-1-6. no. 6
Dejean, F.: Sur un théorème de Thue. J. Combin. Theory Ser. A 13, 90–99 (1972)
Damanik, D., Lenz, D.: The index of Sturmian sequences. Eur. J. Comb. 23, 23–29 (2002)
Dolce, F., Perrin, D.: Eventually dendric shift spaces. In: Ergodic Theory and Dynamical Systems, pp. 1–26 (2020)
Durand, F.: A characterization of substitutive sequences using return words. Discrete Math. 179, 89–101 (1998)
Dvořáková, L., Medková, K., Pelantová, E.: Complementary symmetric Rote sequences: the critical exponent and the recurrence function. Discrete Math. Theoret. Comput. Sci. 20(1) (2020). https://doi.org/10.23638/DMTCS-22-1-20. \(\#20\).
Hedlund, G.A., Morse, M.: Symbolic dynamics II - Sturmian trajectories. Am. J. Math. 62, 1–42 (1940)
Hubert, P.: Suites équilibrées. Theoret. Comput. Sci. 242, 91–108 (2000)
Justin, J., Pirillo, G.: Episturmian words and episturmian morphisms. Theoret. Comput. Sci. 276, 281–313 (2002)
Justin, J., Pirillo, G.: Fractional powers in Sturmian words. Theoret. Comput. Sci. 223, 363–376 (2001)
Pytheas Fogg, N.: Substitutions in Dynamics, Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol. 313. Springer, Heidelberg (2002). Edited by Berthé, V., Ferenczi, S., Mauduit, C., Siegel, A
Rampersad, N., Shallit, J., Vandomme, É.: Critical exponents of infinite balanced words. Theoret. Comput. Sci. 777, 454–463 (2019)
Shallit, J., Shur, A.: Subword complexity and power avoidance. Theoret. Comput. Sci. 792, 96–116 (2019)
Vuillon, L.: A characterization of Sturmian words by return words. Eur. J. Comb. 22, 263–275 (2001)
Vuillon, L.: Balanced words. Bull. Belgian Math. Soc. 10, 787–805 (2003)
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Dolce, F., Dvořáková, L., Pelantová, E. (2021). On Balanced Sequences and Their Asymptotic Critical Exponent. In: Leporati, A., Martín-Vide, C., Shapira, D., Zandron, C. (eds) Language and Automata Theory and Applications. LATA 2021. Lecture Notes in Computer Science(), vol 12638. Springer, Cham. https://doi.org/10.1007/978-3-030-68195-1_23
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