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Parikh-collinear morphisms have the property that all the Parikh vectors of the images of letters are collinear, i.e., the associated adjacency matrix has rank 1. In the conference DLT–WORDS 2023 we showed that fixed points of Parikh-collinear morphisms are automatic. We also showed that the abelian complexity function of a binary fixed point of such a morphism is automatic under some assumptions. In this note, we fully generalize the latter result. Namely, we show that the abelian complexity function of a fixed point of an arbitrary, possibly erasing, Parikh-collinear morphism is automatic. Furthermore, a deterministic finite automaton with output generating this abelian complexity function is provided by an effective procedure. To that end, we discuss the constant of recognizability of a morphism and the related cutting set. [ABSTRACT FROM AUTHOR]

Following Inoue et al., we define a word to be a repetition if it is a (fractional) power of exponent at least 2. A word has a repetition factorization if it is the product of repetitions. We study repetition factorizations in several (generalized) automatic sequences, including the infinite Fibonacci word, the Thue-Morse word, paperfolding words, and the Rudin-Shapiro sequence. [ABSTRACT FROM AUTHOR]

Let t (n) denote the number of 1-bits in the base-2 representation of n , taken modulo 2. We show how to prove the classic conjecture of Leo Moser, on the rarefied sum ∑ 0 ≤ i < n (− 1) t (3 i) , using tools from automata theory and logic. The same technique can be used to prove results about analogous sums. [ABSTRACT FROM AUTHOR]

We explore the evaluation of infinite products involving the automatic sequence (d n : n ∈ N 0) known as the period-doubling sequence, inspired by the work of Allouche, Riasat, and Shallit on the evaluation of infinite products involving the Thue–Morse or Golay–Shapiro sequences. Our methods allow for the application of integral operators that result in new product expansions for expressions involving the dilogarithm function, resulting in new formulas involving Catalan's constant G, such as the formula ∏ n = 1 ∞ n + 2 n n + 1 4 n + 3 4 n + 5 4 n + 4 d n = e 2 G π 2 introduced in this article. More generally, the evaluation of infinite products of the form ∏ n = 1 ∞ e (n) d n for an elementary function e(n) is the main purpose of our article. Past work on infinite products involving automatic sequences has mainly concerned products of the form ∏ n = 1 ∞ R (n) a (n) for an automatic sequence a(n) and a rational function R(n), in contrast to our results as in above displayed product evaluation. Our methods also allow us to obtain new evaluations involving ζ (3) π 2 for infinite products involving the period-doubling sequence. [ABSTRACT FROM AUTHOR]

Motivated by near-identical graphs of two increasing continuous functions—one related to Zaremba's conjecture and the other due to Salem—we provide an explicit connection between fractals and regular sequences by showing that the graphs of ghost distributions, the distribution functions of measures associated to regular sequences, are sections of self-affine sets. Additionally, we provide a sufficient condition for such measures to be purely singular continuous. As a corollary, and analogous to Salem's strictly increasing singular continuous function, we show that the ghost distributions of the Zaremba sequences are singular continuous. [ABSTRACT FROM AUTHOR]

The Kronecker algebra K is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over K are said to be Kronecker modules. The classification of these modules can be obtained by solving a well-known tame matrix problem. Such a classification deals with solving systems of differential equations of the form A x = B x ′ , where A and B are m × n , F -matrices with F an algebraically closed field. On the other hand, researching the Yang–Baxter equation (YBE) is a topic of great interest in several science fields. It has allowed advances in physics, knot theory, quantum computing, cryptography, quantum groups, non-associative algebras, Hopf algebras, etc. It is worth noting that giving a complete classification of the YBE solutions is still an open problem. This paper proves that some indecomposable modules over K called pre-injective Kronecker modules give rise to some algebraic structures called skew braces which allow the solutions of the YBE. Since preprojective Kronecker modules categorize some integer sequences via some appropriated snake graphs, we prove that such modules are automatic and that they induce the automatic sequences of continued fractions. [ABSTRACT FROM AUTHOR]

The ring $\mathbb Z_{d}$ of d -adic integers has a natural interpretation as the boundary of a rooted d -ary tree $T_{d}$. Endomorphisms of this tree (that is, solenoidal maps) are in one-to-one correspondence with 1-Lipschitz mappings from $\mathbb Z_{d}$ to itself. In the case when $d=p$ is prime, Anashin ['Automata finiteness criterion in terms of van der Put series of automata functions', p-Adic Numbers Ultrametric Anal. Appl. 4 (2) (2012), 151–160] showed that $f\in \mathrm {Lip}^{1}(\mathbb Z_{p})$ is defined by a finite Mealy automaton if and only if the reduced coefficients of its van der Put series constitute a p -automatic sequence over a finite subset of $\mathbb Z_{p}\cap \mathbb Q$. We generalize this result to arbitrary integers $d\geq 2$ and describe the explicit connection between the Moore automaton producing such a sequence and the Mealy automaton inducing the corresponding endomorphism of a rooted tree. We also produce two algorithms converting one automaton to the other and vice versa. As a demonstration, we apply our algorithms to the Thue–Morse sequence and to one of the generators of the lamplighter group acting on the binary rooted tree. [ABSTRACT FROM AUTHOR]

We study the properties of the ternary infinite word p = 012102101021012101021012⋯, that is, the fixed point of the map h : 0 → 01, 1 → 21, 2 → 0. We determine its factor complexity, critical exponent, and prove that it is 2-balanced. We compute its abelian complexity and determine the lengths of its bispecial factors. Finally, we give a characterization of p in terms of avoided factors. [ABSTRACT FROM AUTHOR]

We discuss the use of negative bases in automatic sequences. Recently the theorem-prover Walnut has been extended to allow the use of base (—k) to express variables, thus permitting quantification over ℤ instead of ℕ. This enables us to prove results about two-sided (bi-infinite) automatic sequences. We first explain the theory behind negative bases in Walnut. Next, we use this new version of Walnut to give a very simple proof of a strengthened version of a theorem of Shevelev. We use our ideas to resolve two open problems of Shevelev from 2017. We also reprove a 2000 result of Shut involving bi-infinite binary words. [ABSTRACT FROM AUTHOR]

We pursue an exploration into the interdisciplinarity given by amalgamations of the study of mathematical artwork and the study of automatic sequences. [ABSTRACT FROM AUTHOR]

We study the properties of the uncountable set of Stewart words. These are Toeplitz words specified by infinite sequences of Toeplitz patterns of the form αβγ , where α , β , γ is any permutation of the symbols 0,1,?. We determine the critical exponent of the Stewart words, prove that they avoid the pattern x x y y x x , find all factors that are palindromes, and determine their subword complexity. An interesting aspect of our work is that we use automata-theoretic methods and a decision procedure for automata to carry out the proofs. [ABSTRACT FROM AUTHOR]

The Kronecker algebra K is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over K are said to be Kronecker modules. The classification of these modules can be obtained by solving a well-known tame matrix problem. Such a classification deals with solving systems of differential equations of the form Ax=Bx′, where A and B are m×n, F-matrices with F an algebraically closed field. On the other hand, researching the Yang–Baxter equation (YBE) is a topic of great interest in several science fields. It has allowed advances in physics, knot theory, quantum computing, cryptography, quantum groups, non-associative algebras, Hopf algebras, etc. It is worth noting that giving a complete classification of the YBE solutions is still an open problem. This paper proves that some indecomposable modules over K called pre-injective Kronecker modules give rise to some algebraic structures called skew braces which allow the solutions of the YBE. Since preprojective Kronecker modules categorize some integer sequences via some appropriated snake graphs, we prove that such modules are automatic and that they induce the automatic sequences of continued fractions.

Let (a (n) : n ∈ N 0) denote an automatic sequence. Previous research on infinite products involving automatic sequences has mainly dealt with identities for products as in ∏ n R (n) a (n) for rational functions R (n). This inspires the development of techniques for evaluating ∏ n f (n) a (n) more generally, for functions f (n) that are not rational functions. This leads us to apply Euler's product expansion for the Γ-function together with recursive properties of a (n) to obtain identities as in ∏ n f (n , z) a (n) = Γ (z + 1) , and this is motivated by how the equivalent series identity ∑ n a (n) ln ⁡ f (n , z) = ln ⁡ Γ (z + 1) could be applied in relation to the remarkable results due to Gosper on the integration of ln ⁡ Γ (z + 1). We succeed in applying this approach, using Gosper's integration identities, to obtain new infinite products that we evaluate in terms of the Glaisher–Kinkelin constant A and that involve the Thue–Morse sequence, the period-doubling sequence, and the regular paperfolding sequence. A byproduct of our method gives us a way of generalizing a Dirichlet series identity due to Allouche and Sondow, and we also explore applications related to a product evaluation due to Gosper involving A. [ABSTRACT FROM AUTHOR]

In a recent paper, Jason P. Bell and Jeffrey Shallit introduced the notion of Lie complexity and proved that the Lie complexity function of an automatic sequence is automatic. In this note, we give more facts concerning Lie complexity and define the extended Lie complexity and the prefix Lie complexity. Further, we prove that some proprieties of Lie complexity also hold for the extended Lie complexity. Particularly, we prove that the extended Lie complexity function and the first-order difference sequence of the prefix Lie complexity function of an automatic sequence are both automatic. This article is dedicated to Jean-Paul Allouche on the occasion of his 70th birthday. [ABSTRACT FROM AUTHOR]

We present family of automatic sequences that define algebraic continued fractions in characteristic 2. This family is constructed from ultimately period words and contains the period-doubling sequence. [ABSTRACT FROM AUTHOR]

To each one-dimensional subshift X , we may associate a winning shift W (X) which arises from a combinatorial game played on the language of X. Previously it has been studied what properties of X does W (X) inherit. For example, X and W (X) have the same factor complexity and if X is a sofic subshift, then W (X) is also sofic. In this paper, we develop a notion of automaticity for W (X) , that is, we propose what it means that a vector representation of W (X) is accepted by a finite automaton. Let S be an abstract numeration system such that addition with respect to S is a rational relation. Let X be a subshift generated by an S -automatic word. We prove that as long as there is a bound on the number of nonzero symbols in configurations of W (X) (which follows from X having sublinear factor complexity), then W (X) is accepted by a finite automaton, which can be effectively constructed from the description of X. We provide an explicit automaton when X is generated by certain automatic words such as the Thue-Morse word. [ABSTRACT FROM AUTHOR]