Central sets generated by uniformly recurrent words

TL;DR

This paper explores the generation of central sets from uniformly recurrent words, revealing their rich combinatorial structure and connections to various mathematical areas such as topological dynamics and algebraic properties of the Stone-ech compactification.

Contribution

It introduces methods to generate central sets using uniformly recurrent words like Sturmian and Thue-Morse, linking combinatorics on words with topological and algebraic frameworks.

Findings

Central sets derived from Sturmian and Thue-Morse words exhibit rich combinatorial properties.

The study establishes new links between combinatorics on words and topological dynamics.

Generated central sets contain arbitrarily long arithmetic progressions and solutions to linear equations.

Abstract

A subset $A$ of $\nats$ is called an IP-set if $A$ contains all finite sums of distinct terms of some infinite sequence $(x_n)_{n\in \nats} $ of natural numbers. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of IP-sets possessing rich 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 investigate central sets in the framework of combinatorics on words. Using various families of uniformly recurrent words, including Sturmian words, the Thue-Morse word and fixed points of weak mixing substitutions, we generate an assortment of central sets which reflect the rich combinatorial structure of the underlying words. The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked. They include the general theory of combinatorics on words, abstract numeration systems, and the beautiful theory, developed by Hindman, Strauss and others, linking IP-sets and central sets to the algebraic/topological properties of the Stone-\v{C}ech compactification of $\nats .$