# Automatic sequences and generalised polynomials

## TL;DR

This paper investigates the conjecture that bounded generalised polynomial functions are not generated by finite automata unless they are ultimately periodic, using ergodic theory to provide partial results and connections to automatic sequences.

## Contribution

It proves that certain sequences derived from polynomials with irrational coefficients are not automatic and relates the conjecture to the nature of powers of integers as generalised polynomials.

## Key findings

- Sequences from polynomials with irrational coefficients are not automatic.
- The conjecture is equivalent to powers of integers not being generalised polynomials.
- Partial resolution shows such sequences are periodic outside a sparse set.

## 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.