On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words

Authors Pavol Kebis, Florian Luca , Joël Ouaknine , Andrew Scoones , James Worrell

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Pavol Kebis
  • Department of Computer Science, University of Oxford, UK
Florian Luca
  • Mathematics Division, Stellenbosch University, Stellenbosch, South Africa
Joël Ouaknine
  • Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
Andrew Scoones
  • Department of Computer Science, University of Oxford, UK
James Worrell
  • Department of Computer Science, University of Oxford, UK

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Pavol Kebis, Florian Luca, Joël Ouaknine, Andrew Scoones, and James Worrell. On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 144:1-144:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We consider numbers of the form S_β(u): = ∑_{n=0}^∞ (u_n)/(βⁿ), where u = ⟨u_n⟩_{n=0}^∞ is an infinite word over a finite alphabet and β ∈ ℂ satisfies |β| > 1. Our main contribution is to present a combinatorial criterion on u, called echoing, that implies that S_β(u) is transcendental whenever β is algebraic. We show that every Sturmian word is echoing, as is the Tribonacci word, a leading example of an Arnoux-Rauzy word. We furthermore characterise ̅{ℚ}-linear independence of sets of the form {1, S_β(u₁),…,S_β(u_k)}, where u₁,…,u_k are Sturmian words having the same slope. Finally, we give an application of the above linear independence criterion to the theory of dynamical systems, showing that for a contracted rotation on the unit circle with algebraic slope, its limit set is either finite or consists exclusively of transcendental elements other than its endpoints 0 and 1. This confirms a conjecture of Bugeaud, Kim, Laurent, and Nogueira.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Computing methodologies → Algebraic algorithms
  • Transcendence
  • Subspace Theorem
  • Fibonacci Word
  • Tribonacci Word


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  1. B. Adamczewski and Y. Bugeaud. On the complexity of algebraic numbers i. expansions in integer bases. Annals of Mathematics, 165:547-565, 2005. Google Scholar
  2. B. Adamczewski and Y. Bugeaud. Dynamics for β-shifts and diophantine approximation. Ergodic Theory and Dynamical Systems, 27:1695-1711, 2007. Google Scholar
  3. B. Adamczewski, Y. Bugeaud, and F. Luca. Sur la complexité des nombres algébriques. Comptes Rendus Mathematique, 339:11-14, 2004. Google Scholar
  4. B. Adamczewski, J. Cassaigne, and M. Le Gonidec. On the computational complexity of algebraic numbers: the hartmanis-stearns problem revisited. Transactions of the American Mathematical Society, 373(5):3085-3115, 2020. Google Scholar
  5. B. Adamczewski and C. Faverjon. Mahler’s method in several variables and finite automata. arXiv preprint, 2020. URL: https://arxiv.org/abs/2012.08283.
  6. V. Berthe, S. Ferenczi, and L. Q. Zamboni. Interactions between dynamics, arithmetics and combinatorics: The good, the bad, and the ugly. Algebraic and Topological Dynamics, 385, 2005. Google Scholar
  7. J. Borwein and P. B. Borwein. On the complexity of familiar functions and numbers. SIAM Review, 30(4):589-601, 1988. Google Scholar
  8. Y. Bugeaud, D. H. Kim, M. Laurent, and A. Nogueira. On the diophantine nature of the elements of cantor sets arising in the dynamics of contracted rotations. Annali Scuola Normale Superiore di Pisa - Classe Di Scienze, XXII:1681-1704, 2021. Google Scholar
  9. A. Cobham. Uniform tag seqences. Math. Syst. Theory, 6(3):164-192, 1972. Google Scholar
  10. L. V. Danilov. Some classes of transcendental numbers. Mathematical notes of the Academy of Sciences of the USSR, 12(2):524-527, 1972. Google Scholar
  11. S. Ferenczi and C. Mauduit. Transcendence of numbers with a low complexity expansion. Journal of Number Theory, 67(2):146-161, 1997. Google Scholar
  12. G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, fifth edition, 1978. Google Scholar
  13. P. Kebis. Transcendence of numbers related to Episturmian words. PhD thesis, University of Oxford, 2023. Google Scholar
  14. M. Laurent and A. Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 12:175-191, 2018. Google Scholar
  15. Hendrik W Lenstra Jr. Finding small degree factors of lacunary polynomials. Number theory in progress, 1:267-276, 1999. Google Scholar
  16. A. N. Livshits. On the spectra of adic transformations of markov compacta. Russian Mathematical Surveys, 42(3):222, 1987. Google Scholar
  17. J. H. Loxton and A. J. Van der Poorten. Arithmetic properties of certain functions in several variables iii. Bulletin of the Australian Mathematical Society, 16(1):15-47, 1977. Google Scholar
  18. F. Luca, J. Ouaknine, and J. Worrell. On the transcendence of a series related to sturmian words, 2022. To appear in Annali della Scuola Normale Superiore di Pisa. URL: https://arxiv.org/abs/2204.08268.
  19. K. Mahler. Some suggestions for further research. Bulletin of the Australian Mathematical Society, 29:101-108, 1984. Google Scholar
  20. M. Morse and G. A. Hedlund. Symbolic dynamics: Sturmian trajectories. American Journal of Mathematics, 60:815-866, 1938. Google Scholar
  21. M. Morse and G. A. Hedlund. Symbolic dynamics ii: Sturmian trajectories. American Journal of Mathematics, 62:1-42, 1940. Google Scholar
  22. M. Queffélec. Substitution dynamical systems-spectral analysis, volume 1294. Springer, 2010. Google Scholar
  23. G. Rauzy. Nombres algébriques et substitutions. Bulletin de la Société Mathématique de France, 110:147-178, 1982. Google Scholar
  24. R. Risley and L. Zamboni. A generalization of sturmian sequences: Combinatorial structure and transcendence. Acta Arithmetica, 95, January 2000. Google Scholar