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.

中文翻译：

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自动序列和广义多项式

我们推测有界广义多项式函数不能由有限自动机生成，除非它们最终是周期性的微不足道的情况。使用遍历理论的方法，我们能够部分解决这个猜想，证明任何假设的反例都是周期性的，远离非常稀疏和结构化的集合。特别地，我们证明了多项式 $p(n)$ 至少有一个无理系数（常数项除外）和整数 $m\geq 2$，序列 $\lfloor p(n) \rfloor \bmod {m}$ 永远不会自动。我们还证明了该猜想等价于整数 $k\geq 2$ 的幂集不是由广义多项式给出的断言。