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).

The Thue-Morse set T is the set of those non-negative integers whose binary expansions have an even number of 1. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word abbabaabbaababba · · ·, which is the fixed point starting with a of the word morphism a → ab, b → ba. The numbers in T are sometimes called the evil numbers. We obtain an exact formula for the state complexity (i.e. the number of states of its minimal automaton) of the multiplication by a constant of the Thue-Morse set with respect to any integer base b which is a power of 2. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all 2 p -expansions of the set mT for any positive integers m and p. The used method is general for any b-recognizable set of integers. As an application, we obtain a decision procedure running in quadratic time for the problem of deciding whether a given 2 p -recognizable set is equal to some multiple of the Thue-Morse set.

We address the following decision problem. Given a numeration system [Formula: see text] and a [Formula: see text]-recognizable set [Formula: see text], i.e. the set of its greedy [Formula: see text]-representations is recognized by a finite automaton, decide whether or not [Formula: see text] is ultimately periodic. We prove that this problem is decidable for a large class of numeration systems built on linear recurrence sequences. Based on arithmetical considerations about the recurrence equation and on [Formula: see text]-adic methods, the DFA given as input provides a bound on the admissible periods to test.

The Thue-Morse set $\mathcal{T}$ is the set of those non-negative integers whose binary expansions have an even number of $1$'s. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word $${\tt 0110100110010110\cdots},$$ which is the fixed point starting with ${\tt 0}$ of the word morphism ${\tt 0\mapsto 01}$, ${\tt 1\mapsto 10}$. The numbers in $\mathcal{T}$ are commonly called the evil numbers. We obtain an exact formula for the state complexity of the set $m\mathcal{T}+r$ (i.e. the number of states of its minimal automaton) with respect to any base $b$ which is a power of $2$. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all $2^p$-expansions of the set of integers $m\mathcal{T}+r$ for any positive integers $p$ and $m$ and any remainder $r\in\{0,\ldots,m{-}1\}$. The proposed method is general for any $b$-recognizable set of integers.

The Thue-Morse set T is the set of those non-negative integers whose binary expansions have an even number of 1. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word abbabaabbaababba • • •, which is the fixed point starting with a of the word morphism a → ab, b → ba. The numbers in T are commonly called the evil numbers. We obtain an exact formula for the state complexity of the set mT + r (i.e. the number of states of its minimal automaton) with respect to any base b which is a power of 2. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all 2 p -expansions of the set of integers mT + r for any positive integers p and m and any remainder r ∈ {0, . . . , m − 1}. The proposed method is general for any b-recognizable set of integers. As an application, we obtain a decision procedure running in quadratic time for the problem of deciding whether a given 2 p -recognizable set is equal to a set of the form mT + r.