Additive Properties of Sets and Substitutive Dynamics

In the chapter we discuss a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of N possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone–Čech compactification

β

N. This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. 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. One crucial additive property of central sets is that each central set contains all finite sums of distinct terms for some infinite increasing sequence of natural numbers, i.e. is an IP-set. By a celebrated result of N. Hindman, 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. We introduce an hierarchy of additive combinatorial properties for subsets of N and study them in terms of partition regularity. The results introduced in the chapter rely on interactions between different areas of mathematics: They include the general theory of combinatorics on words, numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone–Čech compactification of N.