Factors of generalised polynomials and automatic 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.2010 Mathematics Subject Classification. Primary: 11B85, 37A45.

Cobham’s Theorem and Automaticity

“…This family contains the family of Sturmian sequences as a subset. In recent work [5], they have extended some of the results of this paper to this more general class.…”

How to prove that a sequence is not automatic

“…• It is proved in [78] that, if a sequence has arbitrarily long blocks in common with a Sturmian sequence, then it cannot be q-automatic for any q ≥ 2. This is generalized from Sturmian sequences to generalized polynomials in [36].…”