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Automatic Sequences and Generalised Polynomials
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 13 June 2019, pp. 392-426
- Print publication:
- April 2020
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- Article
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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.
