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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]

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 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 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]

We prove a number of conjectures recently stated by P. Barry, related to the paperfolding sequence and the Rueppel sequence. Furthermore, we study the regularity of sequences involved in the paper, and prove that for all q ≥ 2 , the sequence consisting of the positive integers whose odd part is of the form 4 k + 1 is not q -regular. Finally we establish the 2-regularity of two sequences of Hankel determinants. [ABSTRACT FROM AUTHOR]

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]

The main subject of this paper are binary pattern sequences, that is, sequences of the form (− 1) # (n , A) where A is a set of strings of 0s and 1s, and # (n , A) denotes the total number of times patterns from A appear in the binary expansion of n. A sequence is said to be noncorrelated if the corresponding spectral measure is equal to the Lebesgue measure. We show that it is possible to algorithmically verify if a given binary pattern sequence is noncorrelated. As an application, we compute that there are exactly 2272 noncorrelated binary pattern sequences of length ≤4. If we restrict our attention to patterns that do not end with 0, we put forward a sufficient condition for a pattern sequence to be noncorrelated. We conjecture that this condition is also necessary, and verify this conjecture for lengths ≤5. [ABSTRACT FROM AUTHOR]

We extend the notion of k -automatic sequences to Ostrowski-automatic sequences, and develop a procedure to computationally decide certain combinatorial and enumeration questions about such sequences that can be expressed as predicates in first-order logic. Our primary contribution is the design and implementation of an adder recognizing addition in a generalized Ostrowski numeration system. We also provide applications of our work to several topics in combinatorics on words, including repetitions and pattern avoidance. We partially resolve a previous conjecture about balanced words by Rampersad et al., and make the first progress on an open problem on rich words by Vesti. We also prove some known results about Lucas words using only machine computation. [ABSTRACT FROM AUTHOR]

We present a short proof of Cobham's theorem without using Kronecker's approximation theorem, making it suitable for generalization beyond automatic sequences. [ABSTRACT FROM AUTHOR]

Continued fraction expansions of automatic numbers have been extensively studied during the last few decades. The research interests are, on one hand, in the degree or automaticity of the partial quotients following the seminal paper of Baum and Sweet in 1976, and on the other hand, in calculating the Hankel determinants and irrationality exponents, as one can find in the works of Allouche–Peyrière–Wen–Wen, Bugeaud, and the first author. This paper is motivated by the converse problem: to study Stieltjes continued fractions whose coefficients form an automatic sequence. We consider two such continued fractions defined by the Thue–Morse and period-doubling sequences, respectively, and prove that they are congruent to algebraic series in ℤ [ [ x ] ] modulo 4. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo 4 are 2 -automatic. [ABSTRACT FROM AUTHOR]

In this article, we prove that all completely multiplicative automatic sequences (a n) n ∈ N defined on C , vanishing or not, can be written in the form a n = b n χ n , where (b n) n ∈ N is an almost constant sequence, and (χ n) n ∈ N is a Dirichlet character. [ABSTRACT FROM AUTHOR]

A formal inverse of a given automatic sequence (the sequence of coefficients of the composition inverse of its associated formal power series) is also automatic. The comparison of properties of the original sequence and its formal inverse is an interesting problem. Such an analysis has been done before for the Thue-Morse sequence. In this paper, we describe arithmetic properties of formal inverses of the generalized Thue-Morse sequences and formal inverses of two modifications of the Rudin-Shapiro sequence. In each case, we give the recurrence relations and the automaton, then we analyze the lengths of factors consisting of identical letters as well as the frequencies of letters. We also compare the obtained results with the original sequences. [ABSTRACT FROM AUTHOR]

In a recent paper, one of us posed three open problems concerning squarefree arithmetic progressions in infinite words. In this paper we solve these problems and prove some additional results. For instance, among other things, we show that there exists a squarefree word w over a ternary alphabet such that for every p ≥ 3 , the subsequence of w indexed by the multiples of p contains a square. [ABSTRACT FROM AUTHOR]

Hankel determinants and automatic sequences are two classical subjects widely studied in Mathematics and Theoretical Computer Science. However, these two topics were considered totally independently, until in 1998, when Allouche, Peyrière, Wen and Wen proved that all the Hankel determinants of the Thue-Morse sequence are nonzero. This property allowed Bugeaud to prove that the irrationality exponents of the Thue-Morse-Mahler numbers are exactly 2. Since then, the Hankel determinants of several other automatic sequences, in particular, the paperfolding sequence, the Stern sequence, the period-doubling sequence, are studied by Coons, Vrbik, Guo, Wu, Wen, Bugeaud, Fu, Han, Fokkink, Kraaikamp, and Shallit. On the other hand, it is known that the Hankel determinants of a rational power series are ultimately zero, and the Hankel determinants of a quadratic power series over finite fields are ultimately periodic. It is therefore natural to ask if we can obtain similar results about the Hankel determinants of algebraic series. In the present paper, we provide a partial answer to this question by establishing the automaticity of the reduced Hankel determinants modulo 2 of a family of automatic sequences. As an application of our result, we give upper bounds for the irrationality exponent of a family of automatic numbers. [ABSTRACT FROM AUTHOR]

Studying and comparing arithmetic properties of a given automatic sequence and the sequence of coefficients of the composition inverse of the associated formal power series (the formal inverse of that sequence) is an interesting problem. This problem was studied before for the Thue–Morse sequence. In this paper, we study arithmetic properties of the formal inverses of two sequences closely related to the well-known Baum–Sweet sequence. We give the recurrence relations for their formal inverses and we determine whether the sequences of indices at which these formal inverses take value 0 and 1 are regular. We also show an unexpected connection between one of the obtained sequences and the formal inverse of the Thue–Morse sequence. [ABSTRACT FROM AUTHOR]

In this paper, we study the abelian complexity ρ n a b (t (k)) of generalized Thue-Morse sequences t (k) for every integer k ≥ 2. We obtain the exact value of ρ n a b (t (k)) for every integer n ≥ k. Consequently, ρ n a b (t (k)) is ultimately periodic with period k. Moreover, we show that the abelian complexities of a class of infinite sequences are automatic sequences. [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]

In this paper, we study the abelian complexity of the Rudin–Shapiro sequence and a related sequence. We show that these two sequences share the same complexity function ρ ( n ) , which satisfies certain recurrence relations. As a consequence, the abelian complexity function is 2-regular. Further, we prove that the box dimension of the graph of the asymptotic function λ ( x ) is 3/2, where λ ( x ) = lim k → ∞ ⁡ ρ ( 4 k x ) / 4 k x and ρ ( x ) = ρ ( ⌊ x ⌋ ) for every x > 0 . [ABSTRACT FROM AUTHOR]

Allouche and Shallit introduced the notion of a regular power series as a generalisation of automatic sequences. Becker showed that all regular power series satisfy Mahler equations and conjectured equivalent conditions for the converse to be true. We prove a stronger form of Becker's conjecture for a subclass of Mahler power series. [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]

We use a decision procedure for the “Fibonacci-automatic” words to solve problems about a number of different sequences. In particular, we prove that there exists an aperiodic infinite binary word avoiding the pattern x x x R . This is the first avoidability result concerning a nonuniform morphism proven purely mechanically. [ABSTRACT FROM AUTHOR]

Spiking neural P systems (in short, SN P systems) are membrane computing models inspired by the pulse coding of information in biological neurons. SN P systems with standard rules have neurons that emit at most one spike (the pulse) each step, and have either an input or output neuron connected to the environment. A variant known as SN P modules generalize SN P systems by using extended rules (more than one spike can be emitted each step) and a set of input and output neurons. In this work we continue relating SN P modules and finite automata. In particular, we amend and improve previous constructions for the simulatons of deterministic finite automata and state transducers. Our improvements reduce the number of neurons from three down to one, so our results are optimal. We also simulate finite automata with output, and we use these simulations to generate automatic sequences. [ABSTRACT FROM AUTHOR]

The Hankel determinants of a given power series f can be evaluated by using the Jacobi continued fraction expansion of f . However the existence of the Jacobi continued fraction needs that all Hankel determinants of f are nonzero. We introduce Hankel continued fraction , whose existence and uniqueness are guaranteed without any condition for the power series f . The Hankel determinants can also be evaluated by using the Hankel continued fraction. It is well known that the continued fraction expansion of a quadratic irrational number is ultimately periodic. We prove a similar result for power series. If a power series f over a finite field satisfies a quadratic equation, then the Hankel continued fraction is ultimately periodic. As an application, we derive the Hankel determinants of several automatic sequences, in particular, the regular paperfolding sequence. Thus we provide an automatic proof of a result obtained by Guo, Wu and Wen, which was conjectured by Coons–Vrbik. [ABSTRACT FROM AUTHOR]

Let s( n) be the sum of digits of the expansion of the integer n in base k. Motivated by certain identities of Nieto, and Bateman and Bradley involving sums of the form $${\sum_{i=0}^{2^{n}-1}{(-1)}^{s_{2}(i)}{(x+i)}^{m}}$$ for m = n and m = n + 1, we consider the sequence of polynomials Here, the sequence $${{\mathbf{u}}}$$ satisfies the recurrence relation and $${q \in V}$$ , where V is a finitely dimensional vector space over the field K and $${P\: V \to V}$$ is a linear endomorphism. Moreover, $${\zeta_{k} \neq 1}$$ is a k-th root of unity. We prove that computing the polynomials $${f_{m,n}^{\mathbf{u}}}$$ is essentially equivalent with computing its constant term and we find an explicit formula for this number. This allows us to prove several interesting identities involving the sum of (binary) digits function. We also prove some related results which are of independent interests and can be seen as a further generalization of certain sums involving the Prouhet-Thue-Morse sequence. [ABSTRACT FROM AUTHOR]

We obtain simple relations between the Hankel determinants of the formal power series ∏k≥0(1 + Jx3k) where , and prove that the sequence of Hankel determinants is an aperiodic automatic sequence taking value in {±1, ±J, ±J2}. This research is essentially inspired by the works about Hankel determinants of Thue-Morse-like sequences by Allouche, Peyrière, Wen and Wen (1998), Bacher (2006) and the first author (2013). It is worth mentioning that we do not have a unified result, even though Bacher's result and ours are very similar. [ABSTRACT FROM AUTHOR]

In this article, we provide a coalgebraic account of parts of the mathematical theory underlying context-free languages. We characterize context-free languages, and power series and streams generalizing or corresponding to the context-free languages, by means of systems of behavioural differential equations; and prove a number of results, some of which are new, and some of which are new proofs of existing theorems, using the techniques of bisimulation and bisimulation up to linear combinations. Furthermore, we establish a link between automatic sequences and these systems of equations, allowing us to, given an automaton generating an automatic sequence, easily construct a system of behavioural differential equations yielding this sequence as a context-free stream. [ABSTRACT FROM AUTHOR]

The notion of a -automatic set of integers is well-studied. We develop a new notion - the -automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability. [ABSTRACT FROM AUTHOR]

We introduce the sequence ( i n ) n ≥ 0 defined by i n = ( − 1 ) inv 2 ( n ) , where inv 2 ( n ) denotes the number of inversions (i.e., occurrences of 10 as a scattered subsequence) in the binary representation of n . We show that this sequence has many similarities to the classical Rudin–Shapiro sequence. In particular, if S ( N ) denotes the N -th partial sum of the sequence ( i n ) n ≥ 0 , we show that S ( N ) = G ( log 4 ⁡ N ) N , where G is a certain function that oscillates periodically between 3 / 3 and 2 . [ABSTRACT FROM AUTHOR]

The Hankel determinants of certain automatic sequences f are evaluated modulo a prime number. In most cases, the Hankel determinants of automatic sequences do not have any closed-form expressions; the traditional methods, such as LU -decomposition and Jacobi continued fraction, cannot be applied directly. Our method is based on a simple idea: the Hankel determinants of each sequence g equal to f modulo p are equal to the Hankel determinants of f modulo p . The clue then consists of finding a nice sequence g , whose Hankel determinants have closed-form expressions. Several examples are presented, including a result saying that the Hankel determinants of the Thue–Morse sequence are nonzero, first proved by Allouche, Peyrière, Wen and Wen using determinant manipulation. [ABSTRACT FROM AUTHOR]

Extending the result of Allouche et al., we prove that for any k , the two-dimensional sequence (modulo 2) { | Δ k ( t ) n p | } n , p ≥ 0 is 2-automatic. Moreover, we prove that the three-dimensional sequence (modulo 2) { | Δ k ( t ) n p | } n , p , k ≥ 0 is 2-automatic. [ABSTRACT FROM AUTHOR]

In this paper, which may be considered a sequel to a recent article by Eric Rowland and Reem Yassawi, we present yet another approach for the automatic generation of automata (and an extension that we call congruence linear schemes) for the fast (log-time) determination of congruence properties, modulo small (and not so small!) prime powers, for a wide class of combinatorial sequences. Even more interesting than the new results that could be obtained is the illustratedmethodology, that of designing ‘meta-algorithms’ that enable the computer to develop algorithms, that it (or another computer) can then proceed to use to actually prove (potentially!) infinitely many new results. This paper is accompanied by a Maple package,AutoSquared, and numerous sample input and output files, that readers can use as templates for generating their own, thereby proving many new ‘theorems’ about congruence properties of many famous (and, of course, obscure) combinatorial sequences. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, improving the method of Allouche et al., we calculate the Hankel determinant of the regular paperfolding sequence, and prove that the Hankel determinant sequence modulo 2 is periodic with period 10 which answers Coonʼs conjecture. Then we extend Bugeaudʼs method to obatin the exact value of the irrationality exponent for some general transcendental numbers. Using the results above, we prove that the irrationality exponents of the regular paperfolding numbers are exactly 2. [Copyright &y& Elsevier]

We describe a technique for mechanically proving certain kinds of theorems in combinatorics on words, using finite automata and a software package for manipulating them. We illustrate our technique by applying it to (a) solve an open problem of Currie and Saari on the lengths of unbordered factors in the Thue-Morse sequence; (b) verify an old result of Prodinger and Urbanek on the regular paperfolding sequence; (c) find an explicit expression for the recurrence function for the Rudin-Shapiro sequence; and (d) improve the avoidance bound in Leech's squarefree sequence. We also introduce a new measure of infinite words called condensation and compute it for some famous sequences. We follow up on the study of Currie and Saari of least periods of infinite words. We show that the characteristic sequence of least periods of a k-automatic sequence is (effectively) k-automatic. We compute the least periods for several famous sequences. Many of our results were obtained by machine computations. [ABSTRACT FROM AUTHOR]

The critical exponent of an infinite word is defined to be the supremum of the exponent of each of its factors. For k-automatic sequences, we show that this critical exponent is always either a rational number or infinite, and its value is computable. Our results also apply to variants of the critical exponent, such as the initial critical exponent of Berthé, Holton, and Zamboni and the Diophantine exponent of Adamczewski and Bugeaud. Our work generalizes or recovers previous results of Krieger and others, and is applicable to other situations; e.g., the computation of the optimal recurrence constant for a linearly recurrent k-automatic sequence. [ABSTRACT FROM AUTHOR]

Following up on a paper of Balamohan et al. [‘On the behavior of a variant of Hofstadter’s $q$-sequence’, J. Integer Seq. 10 (2007)], we analyze a variant of Hofstadter’s $Q$-sequence and show that its frequency sequence is 2-automatic. An automaton computing the sequence is explicitly given. [ABSTRACT FROM AUTHOR]

Abstract: Let be an integer. A sequence of natural numbers is -synchronized if its graph is represented, in base , by a right-synchronized rational relation. We show that the factor complexity and the palindromic complexity of a -synchronized sequence are -regular sequences. We derive that the palindromic complexity of a -automatic sequence is -automatic. [ABSTRACT FROM AUTHOR]

Abstract: An infinite word is -automatic if, for all , its th letter is the output of a deterministic automaton fed with the representation of in the numeration system . In this paper, we consider an analogous definition in a multidimensional setting and study its relation to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for , we show that a multidimensional infinite word over a finite alphabet is -automatic for some abstract numeration system built on a regular language containing the empty word if and only if is the image by a coding of a shape-symmetric infinite word. [Copyright &y& Elsevier]

A pure morphic sequence is a right-infinite, symbolic sequence obtained by iterating a letter-to-word substitution. For instance, the Fibonacci sequence and the Thue–Morse sequence, which play an important role in theoretical computer science, are pure morphic. Define a coding as a letter-to-letter substitution. The image of a pure morphic sequence under a coding is called a morphic sequence. A sequence x is called uniformly recurrent if for each finite subword u of x there exists an integer l such that u occurs in every l-length subword of x. The paper mainly focuses on the problem of deciding whether a given morphic sequence is uniformly recurrent. Although the status of the problem remains open, we show some evidence for its decidability: in particular, we prove that it can be solved in polynomial time on pure morphic sequences and on automatic sequences. In addition, we prove that the complexity of every uniformly recurrent, morphic sequence has at most linear growth: here, complexity is understood as the function that maps each positive integer n to the number of distinct n-length subwords occurring in the sequence. [ABSTRACT FROM AUTHOR]

Abstract: We revisit a technique of S. Lehr on automata and use it to prove old and new results in a simple way. We give a very simple proof of the 1986 theorem of Honkala that it is decidable whether a given -automatic sequence is ultimately periodic. We prove that it is decidable whether a given -automatic sequence is overlap-free (or squarefree, or cubefree, etc.). We prove that the lexicographically least sequence in the orbit closure of a -automatic sequence is -automatic, and use this last result to show that several related quantities, such as the critical exponent, irrationality measure, and recurrence quotient for Sturmian words with slope , have automatic continued fraction expansions if does. [Copyright &y& Elsevier]

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]

Let β be a fixed element of 픽q((X-1)) with polynomial part of degree ≥ 1, then any formal power series can be represented in base β, using the transformation Tβ : f ↦ {βf} of the unit disk $\mathcal{M}_0$. Any formal power series in $\mathcal{M}_0$ is expanded in this way into dβ(f) = (ai(X))i≥1, where $a_i(X)=[\beta T^{i-1}_\beta(f)]\in \mathbb{F}_q[X]$. The main aim of this paper is to characterize the formal power series β(|β| > 1), such that dβ(1) is finite, eventually periodic or automatic (such characterizations do not exist in the real case). [ABSTRACT FROM AUTHOR]

Abstract: We give some sufficient criteria for the existence of certain averages (mean, correlation functions) of generalized higher-dimensional automatic sequences and show how to calculate these averages. Then follows an exploration of the nature of necessary and sufficient conditions for the existence of averages. Some of these criteria are applied to averages which play a central role in the determination of the correlation function of an automatic sequence. [Copyright &y& Elsevier]

In this paper, we prove a conjecture of Chen and Wen that the boundary sequence of an automatic sequence is also automatic. In particular, we study the boundary sequences of the generalized Cantor sequences, and give a complete characterization of the periodic boundary sequences. As an application, for a class of automatic sequences, we prove that their abelian complexities are also automatic. [ABSTRACT FROM AUTHOR]

The article presents a function regularity relevant to the 2-adic logarithm. It provides mathematical formulas defining the regularity of the function contained in a finitely generated module. It mentions topics including dimension of linear span, liner independent and linear relation. Authors note of choosing a suitable subset to give a lower bound for the dimension.