Central sets defined by words of low factor complexity

Abstract

A subset $A$ of $\mathbb{N}$ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $(x_n)_{n\in \mathbb{N}} $ of natural numbers. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of IP-sets possessing additional nice combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. In this paper we show how certain families of aperiodic words of low factor complexity may be used to generate a wide assortment of central sets having additional nice properties inherited from the rich combinatorial structure of the underlying word. We consider Sturmian words and their extensions to higher alphabets (so-called Arnoux-Rauzy words), as well as words generated by substitution rules including the famous Thue-Morse word. We also describe a connection between central sets and the strong coincidence condition for fixed points of primitive substitutions which represents a new approach to the strong coincidence conjecture for irreducible Pisot substitutions. Our methods simultaneously exploit the general theory of combinatorics on words, the arithmetic properties of abstract numeration systems defined by substitution rules, notions from topological dynamics including proximality and equicontinuity, the spectral theory of symbolic dynamical systems, and the beautiful and elegant theory, developed by N. Hindman, D. Strauss and others, linking IP-sets to the algebraic/topological properties of the Stone-Čech compactification of $\mathbb{N}.$ Using the key notion of $p$-$\lim_n,$ regarded as a mapping from words to words, we apply ideas from combinatorics on words in the framework of ultrafilters.