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Automatic Sequences and Generalised Polynomials

Part of: Topological dynamics Diophantine approximation, transcendental number theory Ergodic theory Sequences and sets Graph theory

Published online by Cambridge University Press:  13 June 2019

Jakub Byszewski  and
Jakub Konieczny
Jakub Byszewski
Affiliation:
Department of Mathematics and Computer Science, Institute of Mathematics, Jagiellonian University, ul. prof. Stanisława Łojasiewicza 6, 30-348 Kraków Email: jakub.byszewski@gmail.com
Jakub Konieczny
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG
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Abstract

We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic.

Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial $p(n)$ with at least one irrational coefficient (except for the constant one) and integer $m\geqslant 2$, the sequence $\lfloor p(n)\rfloor \hspace{0.2em}{\rm mod}\hspace{0.2em}m$ is never automatic.

We also prove that the conjecture is equivalent to the claim that the set of powers of an integer $k\geqslant 2$ is not given by a generalised polynomial.

Keywords

generalised polynomialautomatic sequenceIP setnilmanifoldlinear recurrence sequenceregular sequence

MSC classification

Primary: 11B85: Automata sequences
Secondary: 37A45: Relations with number theory and harmonic analysis 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 37B10: Symbolic dynamics 11J71: Distribution modulo one 11B37: Recurrences 05C20: Directed graphs (digraphs), tournaments
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 2 , April 2020 , pp. 392 - 426
Copyright
© Canadian Mathematical Society 2019

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Footnotes

1

Current address: Einstein Institute of Mathematics, Edmond J. Safra Campus, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 9190401, Israel Email: jakub.konieczny@gmail.com

This research was supported by the National Science Centre, Poland (NCN) under grant no. DEC-2012/07/E/ST1/00185.

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