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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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Subject Classification

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

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Abstract

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.

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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) https://doi.org/10.4230/LIPIcs.ICALP.2024.144

Author Details

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

Funding

  • Ouaknine, Joël: Also affiliated with Keble College, Oxford as https://emmy.network Fellow, and supported by DFG grant 389792660 as part of TRR 248 (see https://perspicuous-computing.science).
  • Worrell, James: Work supported by UKRI Frontier Research Grant EP/X033813/1.

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