Automatic sequences based on Parry or Bertrand numeration systems

Abstract

We study the factor complexity and closure properties of automatic sequences based on Parry or Bertrand numeration systems. These automatic sequences can be viewed as generalizations of the more typical k-automatic sequences and Pisot-automatic sequences. We show that, like k-automatic sequences, Parry-automatic sequences have sublinear factor complexity while there exist Bertrand-automatic sequences with superlinear factor complexity. We prove that the set of Parry-automatic sequences with respect to a fixed Parry numeration system is not closed under taking images by uniform substitutions or periodic deletion of letters. These closure properties hold for k-automatic sequences and Pisot-automatic sequences, so our result shows that these properties are lost when generalizing to Parry numeration systems and beyond. Moreover, we show that a multidimensional sequence is U -automatic with respect to a positional numeration system U with regular language of numeration if and only if its U -kernel is finite. IntroductionRoughly speaking, an automatic sequence is an infinite word over a finite alphabet such that its nth symbol is obtained as the output given by a deterministic finite automaton fed with the representation of n in a convenient numeration system. Precise definitions are given in Subsection 2.2.If we consider the usual base-k numeration systems, then we get the family of k-automatic sequences [1]. These words are images under a coding of a fixed point of a substitution of constant length. On a larger scale, if one considers abstract numeration systems based on a regular language (see for instance [3, Chap. 3] or [18]), then we get the family of morphic words. Between these two extremes, we have the automatic sequences based on Pisot, Parry, and Bertrand numeration systems (the definitions are given in Subsection 2.1), and we have the following hierarchy: Integer base system Pisot systems Parry systemsBertrand systems with a regular numeration language Abstract numeration systems.Abstract numeration systems are uniquely based on the genealogical ordering of the words belonging to a regular language. This is contrasting with the more restricted case, treated in this paper, of positional numeration systems based on an increasing sequence of integers: a digit occurring in nth position is multiplied by the nth element of the underlying sequence.The Pisot-automatic sequences behave in many respects like k-automatic sequences. Most importantly, in both cases automatic sets have a characterization in terms of first-order logic. E-mail addresses: a.massuir@uliege.be (A. Massuir), r@turambar.org (J. Peltomäki), m.rigo@uliege.be (M. Rigo).

A full characterization of Bertrand numeration systems

“…Then, let us show that the condition given in the original Bertrand-Mathis result characterizing the Bertrand numeration systems is not necessary. This fact was already observed in [11].…”

“…The numeration language is the set N U = 0 * rep U (N). Similarly, the literature about positional numeration systems is vast; see [2,3,4,9,11,13,16] for the most topic-related ones.…”

“…The usual integer base numeration systems are Bertrand, as well the Zeckendorf numeration system [18]. This form of the definition of Bertrand numeration systems, as well as their names after Bertrand-Mathis, was first given in [3], and then used in [4,11,13]. Bertrand numeration systems were also reconsidered in [9].…”

Automatic winning shifts

“…The subshift generated by x is transitive by definition. By [19,Thm. 3.4] all Parry-automatic words have sublinear factor complexity.…”

“…3.4] all Parry-automatic words have sublinear factor complexity. Since all Pisot numeration systems are Parry numeration systems [19,Remark 3], it follows that x has sublinear factor complexity. It is a well-know fact that Pisot numeration systems are addable [10], so the claim follows from Theorem 6.2.…”