Factors of generalised polynomials and automatic sequences

TL;DR

This paper extends a known result by showing that automatic sequences and sequences generated by non-periodic generalized polynomials with zero-density exceptions cannot share arbitrarily long common factors.

Contribution

It generalizes the previous theorem to include sequences defined by generalized polynomials, broadening the scope of the original result.

Findings

Automatic and generalized polynomial sequences have bounded common factors.

The result applies to sequences with non-periodic behavior outside a zero-density set.

The theorem confirms limitations on the common structure of these sequences.

Abstract

The aim of this short note is to generalise the result of Rampersad--Shallit saying that an automatic sequence and a Sturmian sequence cannot have arbitrarily long common factors. We show that the same result holds if a Sturmian sequence is replaced by an arbitrary sequence whose terms are given by a generalised polynomial (i.e., an expression involving algebraic operations and the floor function) that is not periodic except for a set of density zero.