On transcendence of numbers related to Sturmian and Arnoux-Rauzy words
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...
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| Format: | Conference item |
| Language: | English |
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Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2024
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| author | Kebis, P Luca, F Ouaknine, J Scoones, A Worrell, J |
| author_facet | Kebis, P Luca, F Ouaknine, J Scoones, A Worrell, J |
| author_sort | Kebis, P |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-09-25T04:16:12Z |
| format | Conference item |
| id | oxford-uuid:e2fa381d-fc43-420a-885c-3f594e3301ba |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:16:12Z |
| publishDate | 2024 |
| publisher | Schloss Dagstuhl – Leibniz-Zentrum für Informatik |
| record_format | dspace |
| spelling | oxford-uuid:e2fa381d-fc43-420a-885c-3f594e3301ba2024-07-16T10:35:29ZOn transcendence of numbers related to Sturmian and Arnoux-Rauzy wordsConference itemhttp://purl.org/coar/resource_type/c_5794uuid:e2fa381d-fc43-420a-885c-3f594e3301baEnglishSymplectic ElementsSchloss Dagstuhl – Leibniz-Zentrum für Informatik2024Kebis, PLuca, FOuaknine, JScoones, AWorrell, JWe 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. |
| spellingShingle | Kebis, P Luca, F Ouaknine, J Scoones, A Worrell, J On transcendence of numbers related to Sturmian and Arnoux-Rauzy words |
| title | On transcendence of numbers related to Sturmian and Arnoux-Rauzy words |
| title_full | On transcendence of numbers related to Sturmian and Arnoux-Rauzy words |
| title_fullStr | On transcendence of numbers related to Sturmian and Arnoux-Rauzy words |
| title_full_unstemmed | On transcendence of numbers related to Sturmian and Arnoux-Rauzy words |
| title_short | On transcendence of numbers related to Sturmian and Arnoux-Rauzy words |
| title_sort | on transcendence of numbers related to sturmian and arnoux rauzy words |
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