TY - JOUR

T1 - Central sets generated by uniformly recurrent words

AU - Bucci, Michelangelo

AU - Puzynina, Svetlana

AU - Zamboni, Luca Q.

PY - 2015/8/28

Y1 - 2015/8/28

N2 - A subset A of ℕ is called an IP-set if A contains all finite sums of distinct terms of some infinite sequence (xn)n∈ℕ 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 have 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-Čech compactification of ℕ.

AB - A subset A of ℕ is called an IP-set if A contains all finite sums of distinct terms of some infinite sequence (xn)n∈ℕ 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 have 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-Čech compactification of ℕ.

UR - http://www.scopus.com/inward/record.url?scp=84883345919&partnerID=8YFLogxK

U2 - 10.1017/etds.2013.69

DO - 10.1017/etds.2013.69

M3 - Article

AN - SCOPUS:84883345919

VL - 35

SP - 714

EP - 736

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 3

ER -