Binary patterns in the Prouhet-Thue-Morse sequence

We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all (finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can actually be found in that sequence as segments (up to exchange of letters in the infinite case). This result was previously attributed to unpublished work by D. Guaiana and may also be derived from publications of A. Shur only available in Russian. We also identify the (finitely many) finite binary patterns that appear non trivially, in the sense that they are obtained by applying an endomorphism that does not map the set of all segments of the sequence into itself

Max Dehn, Axel Thue, and the Undecidable

This is a short essay on the roles of Max Dehn and Axel Thue in the formulation of the word problem for (semi)groups, and the story of the proofs showing that the word problem is undecidable.Comment: Definition of undecidability and unsolvability improve

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

Words with the Maximum Number of Abelian Squares

An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain $\Theta(n^2)$ distinct factors that are abelian squares. We study infinite words such that the number of abelian square factors of length $n$ grows quadratically with $n$.Comment: To appear in the proceedings of WORDS 201

Generalized Thue-Morse words and palindromic richness

We prove that the generalized Thue-Morse word $\mathbf{t}_{b,m}$ defined for $b \geq 2$ and $m \geq 1$ as $\mathbf{t}_{b,m} = (s_b(n) \mod m)_{n=0}^{+\infty}$, where $s_b(n)$ denotes the sum of digits in the base-$b$ representation of the integer $n$, has its language closed under all elements of a group $D_m$ isomorphic to the dihedral group of order $2m$ consisting of morphisms and antimorphisms. Considering simultaneously antimorphisms $\Theta \in D_m$, we show that $\mathbf{t}_{b,m}$ is saturated by $\Theta$-palindromes up to the highest possible level. Using the terminology generalizing the notion of palindromic richness for more antimorphisms recently introduced by the author and E. Pelantov\'a, we show that $\mathbf{t}_{b,m}$ is $D_m$-rich. We also calculate the factor complexity of $\mathbf{t}_{b,m}$.Comment: 11 page

Anti-Powers in Infinite Words

In combinatorics of words, a concatenation of $k$ consecutive equal blocks is called a power of order $k$. In this paper we take a different point of view and define an anti-power of order $k$ as a concatenation of $k$ consecutive pairwise distinct blocks of the same length. As a main result, we show that every infinite word contains powers of any order or anti-powers of any order. That is, the existence of powers or anti-powers is an unavoidable regularity. Indeed, we prove a stronger result, which relates the density of anti-powers to the existence of a factor that occurs with arbitrary exponent. As a consequence, we show that in every aperiodic uniformly recurrent word, anti-powers of every order begin at every position. We further show that every infinite word avoiding anti-powers of order $3$ is ultimately periodic, while there exist aperiodic words avoiding anti-powers of order $4$. We also show that there exist aperiodic recurrent words avoiding anti-powers of order $6$.Comment: Revision submitted to Journal of Combinatorial Theory Series