Title: Automatic sequences and generalised polynomials,
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 $m\geq 2$, the
sequence $\lfloor p(n) \rfloor \bmod{m}$ is never automatic.
We also prove that the conjecture is equivalent to the claim that the set of
powers of an integer $k\geq 2$ is not given by a generalised polynomial.