Automatic Sequences and Generalised Polynomials

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 ≥ 2, the sequence ⌊p(n)⌋ mod m is never automatic.We also prove that the conjecture is equivalent to the claim that the set of powers of an integer k ≥ 2 is not given by a generalised polynomial.

Factors of generalised polynomials and automatic sequences

“…This is indeed the case. The following result generalises [BK17,Theorem B]. We remind the reader that a set A ⊂ N 0 has upper Banach density zero if for every ε > 0 any interval of sufficiently large length l contains at most εl elements of A.…”

“…(In the case of a Sturmian sequence the corresponding nilmanifold is the circle with the irrational rotation by θ.) In a recent work, we have shown that generalised polynomials cannot be automatic unless they are periodic outside of a set of density zero (for this and related results see [BK17] and [BK16]). It is therefore natural to ask whether the result on common factors generalises to this more general context.…”

On multiplicative automatic sequences

“…Our first step is to show that is, using the terminology borrowed from , an arid set. Definition A set is a basic arid set of rank if it takes the form for some and .…”

Cobham’s Theorem and Automaticity

“…Much of what we do in the first part of this paper is based on this proof of Cobham's theorem. We also note that Byszewski and Konieczny [4] generalized Cobham's theorem by showing that if f and g coincide on a set of positions of density 1, then they are periodic on a set of positions of density 1.…”

“…In the present paper we answer, in the negative, the question, "Can a Sturmian sequence and a b-automatic sequence have arbitrarily large finite factors in common?" Byszewski and Konieczny [4] examine these questions for the family of generalized polynomial functions (these are sequences defined by expressions involving algebraic operations along with the floor function). This family contains the family of Sturmian sequences as a subset.…”