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Critical Exponent of Infinite Balanced Words via the Pell Number System

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Combinatorics on Words (WORDS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11682))

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Abstract

In a recent paper of Rampersad et al., the authors conjectured that the smallest possible critical exponent of an infinite balanced word over a 5-letter alphabet is 3/2. We prove this result, using a formulation of first-order logic, the Pell number system, and a machine computation based on finite-state automata.

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Notes

  1. 1.

    Corresponding Walnut code is available at https://github.com/aseemrb/walnut.

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Acknowledgments

We thank Narad Rampersad and Luke Schaeffer for their helpful comments. We are also grateful to the referees who read the paper and offered many useful suggestions.

Author information

Authors and Affiliations

  1. School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Aseem R. Baranwal & Jeffrey Shallit

Authors
  1. Aseem R. Baranwal
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  2. Jeffrey Shallit
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Corresponding author

Correspondence to Aseem R. Baranwal .

Editor information

Editors and Affiliations

  1. Loughborough University, Loughborough, UK

    Robert Mercaş

  2. Loughborough University, Loughborough, UK

    Daniel Reidenbach

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Baranwal, A.R., Shallit, J. (2019). Critical Exponent of Infinite Balanced Words via the Pell Number System. In: Mercaş, R., Reidenbach, D. (eds) Combinatorics on Words. WORDS 2019. Lecture Notes in Computer Science(), vol 11682. Springer, Cham. https://doi.org/10.1007/978-3-030-28796-2_6

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  • DOI: https://doi.org/10.1007/978-3-030-28796-2_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-28795-5

  • Online ISBN: 978-3-030-28796-2

  • eBook Packages: Computer ScienceComputer Science (R0)

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