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We conjecture that bounded generalised polynomial functions cannot begenerated by finite automata, except for the trivial case when they areperiodic away from a finite set. Using methods from ergodic theory, we are ableto partially resolve this conjecture, proving that any hypotheticalcounterexample is periodic away from a very sparse and structured set. Inparticular, we show that for a polynomial $p(n)$ with at least one irrationalcoefficient (except for the constant one) and integer $m$, the sequence$\lfloor p(n) \rfloor \bmod{m}$ is never automatic. We also obtain aconditional result, where we prove the conjecture under the assumption that thecharacteristic sequence of the set of powers of an integer $k\geq 2$ is notgiven by a generalised polynomial.

Automatic sequences, generalised polynomials, and nilmanifolds