Automatic sequences, generalised polynomials, and nilmanifolds

TL;DR

This paper explores the limitations of automatic sequences generated by finite automata, especially those derived from generalized polynomials, using ergodic theory to partially prove conjectures about their structure and periodicity.

Contribution

It provides partial proof that certain generalized polynomial sequences are not automatic, advancing understanding of their structure and relation to nilmanifolds.

Findings

Sequences from polynomials with irrational coefficients are not automatic.

Counterexamples to the conjecture are shown to be periodic outside sparse sets.

Conditional results relate to the structure of powers of integers and generalized polynomials.

Abstract

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.