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Sequences, Groups, and Number Theory pp 37–87Cite as

Birkhäuser

Number Theoretic Aspects of Regular Sequences

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Abstract

We present a survey of results concerning regular sequences and related objects. Regular sequences were defined in the early 1990s by Allouche and Shallit as a combinatorially, algebraically, and analytically interesting generalization of automatic sequences. In this chapter, after an historical introduction, we follow the development from automatic sequences to regular sequences, and their associated generating functions, to Mahler functions. We then examine size and growth properties of regular sequences. The last half of the chapter focuses on the algebraic, analytic, and Diophantine properties of Mahler functions. In particular, we survey the rational-transcendental dichotomies of Mahler functions, due to Bézivin, and of regular numbers, due to Bell, Bugeaud, and Coons.

The research of M. Coons was supported in part by Australian Research Council grant DE140100223. Lukas Spiegelhofer was supported by the Austrian Science Fund (FWF), projects F5502-N26 and F5505-N26, which are part of the Special Research Program “Quasi Monte Carlo Methods: Theory and Applications”, and also by the ANR–FWF joint project MuDeRa (Multiplicativity, Determinism and Randomness). The authors thank the Erwin Schrödinger Institute for Mathematics and Physics where part of this chapter was written during the workshop on “Normal Numbers: Arithmetic, Computational and Probabilistic Aspects.”

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Notes

  1. 1.

    For a detailed account of automatic sequences, see the monograph of Allouche and Shallit [14]. See also Section 1.6.1.

  2. 2.

    Two integers k and l are multiplicatively independent provided \(\log k/\log l\) is irrational.

  3. 3.

    This result is inherent in the work of Cobham. In the 1980s, Loxton and van der Poorten [389] claimed to have proved that an automatic number is either rational or transcendental, but a few unresolvable flaws were found in their argument. This is why their name is associated with the conjecture.

  4. 4.

    We make no comment on the randomness properties of integer sequences, but will be content with their generality as is.

  5. 5.

    Allouche and Shallit gave a more general treatment for sequences taking values in Noetherian rings. In our applications, the most important settings are those of the integers and complex numbers, depending on the type of result presented. For our purposes, for results on sequences and numbers, the integers will be the standard setting, and for results on power series those with complex coefficients will be the most important.

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Authors and Affiliations

  1. School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW, Australia

    Michael Coons

  2. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wien, Austria

    Lukas Spiegelhofer

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  1. Michael Coons
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  2. Lukas Spiegelhofer
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Correspondence to Michael Coons .

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Editors and Affiliations

  1. IRIF, Université Paris Diderot, Paris, France

    Valérie Berthé

  2. Department of Mathematics, University of Liège, Liège, Belgium

    Michel Rigo

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Coons, M., Spiegelhofer, L. (2018). Number Theoretic Aspects of Regular Sequences. In: Berthé, V., Rigo, M. (eds) Sequences, Groups, and Number Theory. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-69152-7_2

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