Thue-Morse Word Research Articles

On monochromatic arithmetic progressions in binary words associated with pattern sequences

Let ev(n) denote the number of occurrences of a fixed pattern v in the binary expansion of n∈N. In this paper we study monochromatic arithmetic progressions in the class of binary words (ev(n)mod2)n≥0, which includes the famous Thue–Morse word t and Rudin–Shapiro word r. We prove that the length of a monochromatic arithmetic progression of difference d≥3 starting at 0 in r is at most (d+3)/2, with equality for infinitely many d. We also compute the maximal length of a monochromatic arithmetic progression in r of difference 2k−1 and 2k+1. For a general pattern v we show that the maximal length of a monochromatic arithmetic progression of difference d is at most linear in d. Moreover, we prove that a linear lower bound holds for suitable subsequences (dk)k≥0 of differences. We also offer a number of related problems and conjectures.

Repetition Factorization of Automatic Sequences

Following Inoue et al., we define a word to be a repetition if it is a (fractional) power of exponent at least [Formula: see text]. 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.

On the 2-binomial complexity of the generalized Thue–Morse words

In this paper, we study the 2-binomial complexity btm,2(n) of the generalized Thue–Morse words tm over the alphabet {0,1,…,m−1} for every integer m≥3. By using boundary words, we fully characterize when two factors of tm are 2-binomially equivalent. In particular, we obtain the exact value of btm,2(n) for every integer n≥m2. As a consequence, btm,2(n) is ultimately periodic with period m2. This result partially answers a question of Lejeune et al. (2020) [11].

ART-Owen Scrambling

We present a novel algorithm for implementing Owen-scrambling, combining the generation and distribution of the scrambling bits in a single self-contained compact process. We employ a context-free grammar to build a binary tree of symbols, and equip each symbol with a scrambling code that affects all descendant nodes. We nominate the grammar of adaptive regular tiles (ART) derived from the repetition-avoiding Thue-Morse word, and we discuss its potential advantages and shortcomings. Our algorithm has many advantages, including random access to samples, fixed time complexity, GPU friendliness, and scalability to any memory budget. Further, it provides two unique features over known methods: it admits optimization, and it is in-vertible, enabling screen-space scrambling of the high-dimensional Sobol sampler.

Monochromatic arithmetic progressions in binary Thue–Morse-like words

We study the length of monochromatic arithmetic progressions in the Thue–Morse word and in a class of generalised Thue–Morse words. In particular, we give exact values or upper bounds for the lengths of monochromatic arithmetic progressions of given fixed differences inside these words. Some arguments for these are inspired by van der Waerden's proof for the existence of arbitrary long monochromatic arithmetic progressions in any finite colouring of the (positive) integers. We also establish upper bounds for the length of monochromatic arithmetic progressions of certain differences in any fixed point of a primitive binary bijective substitution.

Automatic winning shifts

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.

On prefix palindromic length of automatic words

The prefix palindromic length PPLu(n) of an infinite word u is the minimal number of concatenated palindromes needed to express the prefix of length n of u. Since 2013, it is still unknown if PPLu(n) is unbounded for every aperiodic infinite word u, even though this has been proven for almost all aperiodic words. At the same time, the only well-known nontrivial infinite word for which the function PPLu(n) has been precisely computed is the Thue-Morse word t. This word is 2-automatic and, predictably, its function PPLt(n) is 2-regular, but is this the case for all automatic words?In this paper, we prove that this function is k-regular for every k-automatic word containing only a finite number of palindromes. For two such words, namely the paperfolding word and the Rudin-Shapiro word, we derive a formula for this function. Our computational experiments suggest that generally this is not true: for the period-doubling word, the prefix palindromic length does not look 2-regular, and for the Fibonacci word, it does not look Fibonacci-regular. If proven, these results would give rare (if not first) examples of a natural function of an automatic word which is not regular.

Minimal Automaton for Multiplying and Translating the Thue-Morse Set

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.

Anti-power $j$-fixes of the Thue-Morse word

Recently, Fici, Restivo, Silva, and Zamboni introduced the notion of a $k$-anti-power, which is defined as a word of the form $w^{(1)} w^{(2)} \cdots w^{(k)}$, where $w^{(1)}, w^{(2)}, \ldots, w^{(k)}$ are distinct words of the same length. For an infinite word $w$ and a positive integer $k$, define $AP_j(w,k)$ to be the set of all integers $m$ such that $w_{j+1} w_{j+2} \cdots w_{j+km}$ is a $k$-anti-power, where $w_i$ denotes the $i$-th letter of $w$. Define also $\mathcal{F}_j(k) = (2 \mathbb{Z}^+ - 1) \cap AP_j(\mathbf{t},k)$, where $\mathbf{t}$ denotes the Thue-Morse word. For all $k \in \mathbb{Z}^+$, $\gamma_j(k) = \min (AP_j(\mathbf{t},k))$ is a well-defined positive integer, and for $k \in \mathbb{Z}^+$ sufficiently large, $\Gamma_j(k) = \sup ((2 \mathbb{Z}^+ -1) \setminus \mathcal{F}_j(k))$ is a well-defined odd positive integer. In his 2018 paper, Defant shows that $\gamma_0(k)$ and $\Gamma_0(k)$ grow linearly in $k$. We generalize Defant's methods to prove that $\gamma_j(k)$ and $\Gamma_j(k)$ grow linearly in $k$ for any nonnegative integer $j$. In particular, we show that $\displaystyle 1/10 \leq \liminf_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 9/10$ and $\displaystyle 1/5 \leq \limsup_{k \rightarrow \infty} (\gamma_j(k)/k) \leq 3/2$. Additionally, we show that $\displaystyle \liminf_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3/2$ and $\displaystyle \limsup_{k \rightarrow \infty} (\Gamma_j(k)/k) = 3$.

On anti-powers in aperiodic recurrent words

Fici, Restivo, Silva, and Zamboni define a k-anti-power to be a concatenation of k consecutive words that are pairwise distinct and have the same length. They ask for the maximum k such that every aperiodic recurrent word must contain a k-anti-power, and they prove that this maximum must be 3, 4, or 5. We resolve this question by demonstrating that the maximum is 5. We also conjecture that if W is a reasonably nice aperiodic morphic word, then there is some constant C=C(W) such that for all i,k≥1, W contains a k-anti-power with blocks of length at most Ck beginning at its ith position. We settle this conjecture for binary words that are generated by a uniform morphism, characterizing the small exceptional set of words for which such a constant cannot be found. This generalizes recent results of the second author, Gaetz, and Narayanan that have been proven for the Thue-Morse word, which also show that such a linear bound is the best one can hope for in general.

On the longest common subsequence of Thue-Morse words

The length a(n) of the longest common subsequence of the nth Thue-Morse word and its bitwise complement is studied. An open problem suggested by Jean Berstel in 2006 is to find a formula for a(n). In this paper we prove new lower bounds on a(n) by explicitly constructing a common subsequence between the Thue-Morse words and their bitwise complement. We obtain the lower bound a(n)=2n(1−o(1)), saying that when n grows large, the fraction of omitted symbols in the longest common subsequence of the nth Thue-Morse word and its bitwise complement goes to 0. We further generalize to any prefix of the Thue-Morse sequence, where we prove similar lower bounds.

Computing the k-binomial complexity of the Thue–Morse word

Two words are k-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most k with the same multiplicities. This is a refinement of both the abelian equivalence and the Simon congruence. The k-binomial complexity of an infinite word x maps the integer n to the number of classes in the quotient, by this k-binomial equivalence relation, of the set of factors of length n occurring in x. This complexity measure has not been investigated very much. In this paper, we characterize the k-binomial complexity of the Thue–Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue–Morse word is aperiodic, its k-binomial complexity eventually takes only two values. In this paper, we first express the number of occurrences of subwords appearing in iterates of the form Ψℓ(w) for an arbitrary morphism Ψ. We also thoroughly describe the factors of the Thue–Morse word by introducing a relevant new equivalence relation.

Thue-Morse-Sturmian words and critical bases for ternary alphabets

The set of unique $\beta$-expansions over the alphabet $\{0,1\}$ is trivial for $\beta$ below the golden ratio and uncountable above the Komornik-Loreti constant. Generalisations of these thresholds for three-letter alphabets were studied by Komornik, Lai and Pedicini (2011, 2017). We use S-adic words including the Thue-Morse word (which defines the Komornik-Loreti constant) and Sturmian words (which characterise generalised golden ratios) to determine the value of a certain generalisation of the Komornik-Loreti constant to three-letter alphabets.

Functions on antipower prefix lengths of the Thue–Morse word

We say that a word w of length kn is a k-antipower if it can be written in the form w1⋯wk, where each wi is a distinct word of length n. We analyze prefixes of the Thue–Morse word ,t and lengths of antipowers occurring in them. Define Γ(k) to be the largest odd n such that the prefix of ,t of length kn is not a k-antipower, and γ(k) to be the smallest odd n such that the corresponding prefix is a k-antipower. We provide strong bounds on the asymptotic values of γ(k) and Γ(k)−γ(k). Our bounds on γ(k) affirmatively answer one conjecture of Defant and make substantial progress towards answering a second conjecture of Defant. It was previously known that Γ(k) and γ(k) grow linearly in k, but our bounds on Γ(k)−γ(k) prove that Γ(k)−γ(k) also grows linearly in k.

State Complexity of the Multiples of the Thue-Morse Set

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 sending a to ab and b to 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.

Palindromic length of words and morphisms in class [formula omitted

We study the palindromic length of factors of infinite words fixed by morphisms of the so-called class P introduced by Hof, Knill and Simon. We show that it grows at most logarithmically with the length of the factor. For the Fibonacci word and the Thue–Morse word we provide explicit bounds on their rate of growth. We also construct an infinite word rich in palindromes for which the palindromic length grows as n.

Circular critical exponents for Thue–Morse factors

We prove various results about the largest exponent of a repetition in a factor of the Thue–Morse word, when that factor is considered as a circular word. Our results confirm and generalize previous results of Fitzpatrick and Aberkane & Currie.

Subword complexity and power avoidance

We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that–the Thue–Morse word has the minimum possible subword complexity over all overlap-free binary words and all (73)-power-free binary words, but not over all (73)+-power-free binary words;–the twisted Thue–Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all (73)-power-free binary words;–if some word attains the minimum possible subword complexity over all square-free ternary words, then one such word is the ternary Thue word;–the recently constructed 1-2-bonacci word has the minimum possible subword complexity over all symmetric square-free ternary words.

Abelian-square-rich words

An abelian square is the concatenation of two words that are anagrams of one another. A word of length n can contain at most Θ(n2) distinct factors, and there exist words of length n containing Θ(n2) distinct abelian-square factors, that is, distinct factors that are abelian squares. This motivates us to study infinite words such that the number of distinct abelian-square factors of length n grows quadratically with n. More precisely, we say that an infinite word w is abelian-square-rich if, for every n, every factor of w of length n contains, on average, a number of distinct abelian-square factors that is quadratic in n; and uniformly abelian-square-rich if every factor of w contains a number of distinct abelian-square factors that is proportional to the square of its length. Of course, if a word is uniformly abelian-square-rich, then it is abelian-square-rich, but we show that the converse is not true in general. We prove that the Thue–Morse word is uniformly abelian-square-rich and that the function counting the number of distinct abelian-square factors of length 2n of the Thue–Morse word is 2-regular. As for Sturmian words, we prove that a Sturmian word sα of angle α is uniformly abelian-square-rich if and only if the irrational α has bounded partial quotients, that is, if and only if sα has bounded exponent.

Cyclic complexity of words

We introduce and study a complexity function on words cx(n), called cyclic complexity, which counts the number of conjugacy classes of factors of length n of an infinite word x. We extend the well-known Morse–Hedlund theorem to the setting of cyclic complexity by showing that a word is ultimately periodic if and only if it has bounded cyclic complexity. Unlike most complexity functions, cyclic complexity distinguishes between Sturmian words of different slopes. We prove that if x is a Sturmian word and y is a word having the same cyclic complexity of x, then up to renaming letters, x and y have the same set of factors. In particular, y is also Sturmian of slope equal to that of x. Since cx(n)=1 for some n≥1 implies x is periodic, it is natural to consider the quantity liminfn→∞cx(n). We show that if x is a Sturmian word, then liminfn→∞cx(n)=2. We prove however that this is not a characterization of Sturmian words by exhibiting a restricted class of Toeplitz words, including the period-doubling word, which also verify this same condition on the limit infimum. In contrast we show that, for the Thue–Morse word t, liminfn→∞ct(n)=+∞.