A simple characterization of sets satisfying the Central Sets Theorem

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Held copy and library card.

Neil Hindman and Dona Strauss, "A simple characterization of sets satisfying the Central Sets Theorem," New York Journal of Mathematics 15 (2009), 405–413.

Summary

The paper separates the combinatorial conclusion of the Central Sets Theorem from the stronger algebraic condition traditionally used to obtain it. For a discrete semigroup $S$, Definition 1.2 calls $A\subseteq S$ central when $\overline A$ contains an idempotent from the smallest two-sided ideal $K(\beta S)$. In the commutative setting, Theorem 1.3 recalls the Central Sets Theorem in its finite-family form: one can choose translations $\alpha(F)$ and separated finite index sets $H(F)$ so that every expression (written there as a sum) selected along a chain of finite families of sequences lies in the central set. Definition 1.4 abstracts precisely this conclusion as the definition of a $C$-set. Definition 1.5 introduces the simpler one-step notion of a $J$-set: every finite family of sequences admits a common translate and a common finite index set whose associated sums all land in the set. Thus central sets are the algebraically convenient objects, while $C$-sets retain the combinatorial configurations supplied by the theorem.

Section 2 formulates both notions for an arbitrary, possibly noncommutative, semigroup. Definition 2.1 encodes a word

$$x(m,a,H,f)= \left(\prod_{j=1}^{m} \left(a(j)\prod_{t\in H(j)}f(t)\right)\right)a(m+1),$$

where the blocks $H(1),\ldots,H(m)$ are finite and successively separated; all products are taken in increasing order of indices. Definition 2.2(a) declares $A$ to be a $J$-set when a common choice of $m,a,H$ puts this word in $A$ for every sequence in any prescribed finite family. Definition 2.2(b) defines a $C$-set by choosing such data for every finite family, with separation between nested families and closure under products chosen along strictly increasing chains. Theorem 2.3 records that every central set has this property. The decisive algebraic reduction is Theorem 2.4: if $S$ is infinite, then $A\subseteq S$ is a $C$-set if and only if $\overline A$ contains an idempotent of

$$J(S)=\{p\in\beta S:(\forall B\in p)\ B\text{ is a }J\text{-set}\}.$$

This also identifies the precise weakening of centrality: $K(\beta S)$ is replaced by $J(S)$.

The proof of the promised intrinsic characterization is organized around the tree criterion in Lemma 2.6. For an infinite semigroup, an ultrafilter $p$ is idempotent exactly when every $A\in p$ supports a nonempty tree $T$ of finite $A$-valued functions whose successor sets $B_f(T)$ belong to $p$, with $B_{f\mathbin{\frown}x}(T)\subseteq x^{-1}B_f(T)$. Necessity is proved by iterating the refinement $B^*=\{x\in B:x^{-1}B\in p\}$; sufficiency reads the idempotence condition directly from the successor sets. Theorem 2.7 then gives the main characterization. It makes the following three conditions equivalent: $A$ is a $C$-set; $A$ supports such a tree with every finite intersection of successor sets a $J$-set; and $A$ contains a downward directed family $(C_F)_{F\in I}$ whose members have the corresponding left-translation absorption property and whose finite intersections are $J$-sets. A decreasing sequence $(C_n)$ of $J$-sets with the same absorption property always implies these conditions, and is equivalent to them when $S$ is countable. The implications pass from an idempotent in $J(S)$ to the tree, from the tree to finite intersections of successor sets, and from a directed family to a compact subsemigroup meeting the ideal $J(S)$, where an idempotent is recovered. Countability is used only to enumerate the tree and replace the directed family by a sequence.

The final results show both the reach and the limitation of the characterization. Theorem 2.8 constructs, in $(\mathbb N,+)$, an explicit decreasing sequence $C_n$ defined by omissions from blocks of binary support; each $C_n$ is a $J$-set and the sequence satisfies Theorem 2.7(d), so the resulting set is a $C$-set. The set has zero Banach density and is not central, showing that the converse to Theorem 2.3 fails. Theorem 2.9 proves that, under a surjective semigroup homomorphism $h:S\to T$, $J$-sets, $C$-sets, and central sets are preserved by both image and inverse image; at the ultrafilter level, the extension satisfies $\widetilde h[J(S)]=J(T)$. Corollary 2.10 consequently transports any $C$-set that is not central back along a surjective homomorphism.