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Thue–Morse Along Two Polynomial Subsequences

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Combinatorics on Words (WORDS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9304))

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Abstract

The aim of the present article is twofold. We first give a survey on recent developments on the distribution of symbols in polynomial subsequences of the Thue–Morse sequence \(\mathbf {t}=(t(n))_{n\ge 0}\) by highlighting effective results. Secondly, we give explicit bounds on

$$\min \{n: (t(pn), t(qn))=(\varepsilon _1, \varepsilon _2)\},$$

for odd integers p, q, and on

$$\min \{n: (t(n^{h_1}), t(n^{h_2}))=(\varepsilon _1, \varepsilon _2)\}$$

where \(h_1, h_2\ge 1\), and \((\varepsilon _1, \varepsilon _2)\) is one of (0, 0), (0, 1), (1, 0), (1, 1).

Work supported by the ANR-FWF bilateral project MuDeRa “Multiplicativity: Determinism and Randomness” (France-Austria) and the joint project “Systèmes de numération : Propriétés arithmétiques, dynamiques et probabilistes” of the Université de Lorraine and the Conseil Régional de Lorraine.

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Acknowledgements

I am pleased to thank Jeff Shallit for bringing to my attention the question on bounding Thue–Morse on two multiples. I also thank him and E. Rowland for several discussions.

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Authors and Affiliations

  1. Institut Elie Cartan de Lorraine, UMR 7502, Université de Lorraine / CNRS, 54506, Vandoeuvre-lès-Nancy Cedex, Nancy, France

    Thomas Stoll

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  1. Thomas Stoll
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Correspondence to Thomas Stoll .

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Editors and Affiliations

  1. Universität Kiel, Kiel, Germany

    Florin Manea

  2. Universität Kiel, Kiel, Germany

    Dirk Nowotka

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Stoll, T. (2015). Thue–Morse Along Two Polynomial Subsequences. In: Manea, F., Nowotka, D. (eds) Combinatorics on Words. WORDS 2015. Lecture Notes in Computer Science(), vol 9304. Springer, Cham. https://doi.org/10.1007/978-3-319-23660-5_5

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  • DOI: https://doi.org/10.1007/978-3-319-23660-5_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23659-9

  • Online ISBN: 978-3-319-23660-5

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