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

Avoiding Abelian powers in binary words with bounded Abelian complexity

The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden's theorem, they proved that if a word avoids Abelian $k$-powers for some integer $k$, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian $k$-powers occur in a word having bounded Abelian complexity? In particular, does every uniformly recurrent word having bounded Abelian complexity begin in an Abelian $k$-power? While this is true for various classes of uniformly recurrent words, including for example the class of all Sturmian words, in this paper we show the existence of uniformly recurrent binary words, having bounded Abelian complexity, which admit an infinite number of suffixes which do not begin in an Abelian square. We also show that the shift orbit closure of any infinite binary overlap-free word contains a word which avoids Abelian cubes in the beginning. We also consider the effect of morphisms on Abelian complexity and show that the morphic image of a word having bounded Abelian complexity has bounded Abelian complexity. Finally, we give an open problem on avoidability of Abelian squares in infinite binary words and show that it is equivalent to a well-known open problem of Pirillo-Varricchio and Halbeisen-Hungerb\"uhler.Comment: 16 pages, submitte

Ten Conferences WORDS: Open Problems and Conjectures

In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding informations concerning advances in problems solving