  We conjecture that bounded generalised polynomial functions cannot be
generated by finite automata, except for the trivial case when they are
periodic away from a finite set. 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 $m$, the sequence
$\lfloor p(n) \rfloor \bmod{m}$ is never automatic. We also obtain a
conditional result, where we prove the conjecture under the assumption that the
characteristic sequence of the set of powers of an integer $k\geq 2$ is not
given by a generalised polynomial.
