Binary Patterns in Binary Cube-Free Words: Avoidability and Growth

The avoidability of binary patterns by binary cube-free words is investigated and the exact bound between unavoidable and avoidable patterns is found. All avoidable patterns are shown to be D0L-avoidable. For avoidable patterns, the growth rates of the avoiding languages are studied. All such languages, except for the overlap-free language, are proved to have exponential growth. The exact growth rates of languages avoiding minimal avoidable patterns are approximated through computer-assisted upper bounds. Finally, a new example of a pattern-avoiding language of polynomial growth is given.Comment: 18 pages, 2 tables; submitted to RAIRO TIA (Special issue of Mons Days 2012

Permutation Complexity Related to the Letter Doubling Map

Given a countable set X (usually taken to be the natural numbers or integers), an infinite permutation, \pi, of X is a linear ordering of X. This paper investigates the combinatorial complexity of infinite permutations on the natural numbers associated with the image of uniformly recurrent aperiodic binary words under the letter doubling map. An upper bound for the complexity is found for general words, and a formula for the complexity is established for the Sturmian words and the Thue-Morse word.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Morphic words and equidistributed sequences

The problem we consider is the following: Given an infinite word $w$ on an ordered alphabet, construct the sequence $\nu_w=(\nu[n])_n$, equidistributed on $[0,1]$ and such that $\nu[m]<\nu[n]$ if and only if $\sigma^m(w)<\sigma^n(w)$, where $\sigma$ is the shift operation, erasing the first symbol of $w$. The sequence $\nu_w$ exists and is unique for every word with well-defined positive uniform frequencies of every factor, or, in dynamical terms, for every element of a uniquely ergodic subshift. In this paper we describe the construction of $\nu_w$ for the case when the subshift of $w$ is generated by a morphism of a special kind; then we overcome some technical difficulties to extend the result to all binary morphisms. The sequence $\nu_w$ in this case is also constructed with a morphism. At last, we introduce a software tool which, given a binary morphism $\varphi$, computes the morphism on extended intervals and first elements of the equidistributed sequences associated with fixed points of $\varphi$

On additive properties of sets defined by the Thue-Morse word

In this paper we study some additive properties of subsets of the set \nats of positive integers: A subset $A$ of \nats is called {\it $k$-summable} (where k\in\ben) if $A$ contains \textstyle \big{\sum_{n\in F}x_n | \emp\neq F\subseteq {1,2,...,k\} \big} for some $k$-term sequence of natural numbers $x_1<x_2 < ... < x_k$. We say A \subseteq \nats is finite FS-big if $A$ is $k$-summable for each positive integer $k$. We say is A \subseteq \nats is infinite FS-big if for each positive integer $k,$ $A$ contains {\sum_{n\in F}x_n | \emp\neq F\subseteq \nats and #F\leq k} for some infinite sequence of natural numbers $x_1<x_2 < ...$. We say A\subseteq \nats is an IP-set if $A$ contains {\sum_{n\in F}x_n | \emp\neq F\subseteq \nats and #F<\infty} for some infinite sequence of natural numbers $x_1<x_2 < ...$. By the Finite Sums Theorem [5], the collection of all IP-sets is partition regular, i.e., if $A$ is an IP-set then for any finite partition of $A$, one cell of the partition is an IP-set. Here we prove that the collection of all finite FS-big sets is also partition regular. Let \TM =011010011001011010... denote the Thue-Morse word fixed by the morphism $0\mapsto 01$ and $1\mapsto 10$. For each factor $u$ of \TM we consider the set \TM\big|_u\subseteq \nats of all occurrences of $u$ in \TM. In this note we characterize the sets \TM\big|_u in terms of the additive properties defined above. Using the Thue-Morse word we show that the collection of all infinite FS-big sets is not partition regular

String attractors and combinatorics on words

The notion of string attractor has recently been introduced in [Prezza, 2017] and studied in [Kempa and Prezza, 2018] to provide a unifying framework for known dictionary-based compressors. A string attractor for a word w = w[1]w[2] · · · w[n] is a subset Γ of the positions 1, . . ., n, such that all distinct factors of w have an occurrence crossing at least one of the elements of Γ. While finding the smallest string attractor for a word is a NP-complete problem, it has been proved in [Kempa and Prezza, 2018] that dictionary compressors can be interpreted as algorithms approximating the smallest string attractor for a given word. In this paper we explore the notion of string attractor from a combinatorial point of view, by focusing on several families of finite words. The results presented in the paper suggest that the notion of string attractor can be used to define new tools to investigate combinatorial properties of the words

On the critical exponent of generalized Thue-Morse words

For certain generalized Thue-Morse words t, we compute the "critical exponent", i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.Comment: 13 pages; to appear in Discrete Mathematics and Theoretical Computer Science (accepted October 15, 2007

Permutation Complexity and the Letter Doubling Map

Given a countable set X (usually taken to be N or Z), an infinite permutation $\pi$ of X is a linear ordering $<_\pi$ of X. This paper investigates the combinatorial complexity of infinite permutations on N associated with the image of uniformly recurrent aperiodic binary words under the letter doubling map. An upper bound for the complexity is found for general words, and a formula for the complexity is established for the Sturmian words and the Thue-Morse word