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DOI: 10.4230/LIPIcs.ICALP.2024.144

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

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

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

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)

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@InProceedings{kebis_et_al:LIPIcs.ICALP.2024.144,
  author =	{Kebis, Pavol and Luca, Florian and Ouaknine, Jo\"{e}l and Scoones, Andrew and Worrell, James},
  title =	{{On Transcendence of Numbers Related to Sturmian and Arnoux-Rauzy Words}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{144:1--144:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.144},
  URN =		{urn:nbn:de:0030-drops-202873},
  doi =		{10.4230/LIPIcs.ICALP.2024.144},
  annote =	{Keywords: Transcendence, Subspace Theorem, Fibonacci Word, Tribonacci Word}
}

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

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