Treffer: Central Sets and Their Combinatorial Characterization: Central sets and their combinatorial characterization

This paper makes another significant contribution to the relationship between combinatorial number theory and the algebraic theory of semigroup compactifications by putting the Central Sets Theorem [Proposition 8.21 of \textit{H. Furstenberg's} book ``Recurrence in ergodic theory and combinatorial number theory'' (Princeton, 1981; Zbl 0459.28023)] in an appropriate -- and probably definitive -- abstract context. As often in this field, the algebraic properties are relatively easy to state, but their combinatorial interpretations are quite complicated. Let \(\beta S\) be the Stone-Čech compactification of a discrete commutative semigroup \(S\) (the authors include an example to show that some of their main conclusions fail for non-commutative semigroups). A set \(A \subset S\) is called central (resp. quasi-central) if the closure \(\text{cl }A\) in \(\beta S\) contains an idempotent \(p \in K\), the smallest ideal in \(\beta S\) (resp. \(p \in \text{cl }K\)). The definition of rich subset is too complex to state in a brief review. However, if a set \(A\) is rich, given any countable collection \(Y\) of sequences in \(S\) another family of sequences can be generated for each of which all products of finitely many distinct terms of the sequence lie in \(A\). In particular, in the most important case when \(S = (\mathbb{N},+)\), a rich set \(A\) has the property that, for any sequence \(\langle x_i\rangle\) in \(\mathbb{N}\), \(S\) contains arbitrary long arithmetic progressions with increments in the set of finite sums of \(\langle x_i\rangle\). The authors associate a two-sided closed ideal \(J_Y\) with any \(Y\), and show that \(A\) is rich if and only if for each \(Y\) the intersection \(\text{cl }A \cap J_Y\) contains an idempotent. The Central Sets Theorem (that central sets are rich) is an immediate consequence. In addition, in any finite partition of a rich set into subsets, one of the subsets is rich. In the last part of the paper the authors show that the classes of central, quasi-central and rich sets are distinct in \((\mathbb{N},+)\). Part of the method for achieving this is the derivation of combinatorial characterizations of central and quasi-central sets.