Plünnecke inequalities for countable abelian groups

A Appendix I: A Correspondence Principle for product sets

The aim of this appendix is to outline a complete proof of the Correspondence Principle stated in Section 1.2.

Let G be a countable group and let ℳ⁢(G) denote the set of means on G, which is a weak*-closed and convex subset of the dual of ℓ∞⁢(G). Given λ in ℳ⁢(G), we can associate to it a finitely additive probability measure λ′ by

Fix a weak*-compact and convex subset 𝒞⊂ℳ⁢(G). If X is a compact Hausdorff space, equipped with an action of G by homeomorphisms of X, such that there exists a point xo in X with a dense G-orbit, then we have a natural unital, injective and left G-equivariant C*-algebraic morphism

given by Θxo⁢φ⁢(g)=φ⁢(g⁢xo) for all g in G and φ in C⁢(X). Hence, its transpose Θxo* maps ℳ⁢(G) into 𝒫⁢(X). In particular, if μ=Θxo*⁢λ and B⊂X is a clopen set (so that the indicator function χB is a continuous function on X), then

where

Note that every functional of form λ↦λ′⁢(C) for C⊂G is weak*-continuous, so by the weak*-compactness of 𝒞, both the supremum and the infimum in the definitions of the upper and lower Banach densities with respect to 𝒞 are attained. Furthermore, since 𝒞 is also assumed convex, Bauer’s Maximum Principle (see e.g. [1, Theorem 7.69]) guarantees that these extremal values are attained at extremal elements of 𝒞, i.e. elements of 𝒞 which cannot be written as non-trivial convex combinations of other elements in 𝒞. Since Θxo* is affine and weak*-continuous on ℳ⁢(G), we see that the image

is weak*-compact and convex. Note that if A⊂G and B⊂X is any set, then A⁢Bxo=(A⁢B)xo. Hence, if A′⊂A is a finite set and B is a clopen set in X, then A′⁢B is again clopen in X, so that if μ=Θxo*⁢λ for some λ in 𝒞, then

In particular, we have

where the last equality holds because μ is σ-additive. We conclude that for every A⊂G and clopen set B⊂X, there exists an extremal element μ in 𝒞xo such that

The same argument also gives the following inequality. Fix A⊂G and a clopen set B⊂X. We can find an extremal element λ in 𝒞 such that

If we let ν=Θxo*⁢λ, then ν is an extremal element in 𝒞xo and

Furthermore, we have

where the last equality holds because ν is σ-additive. We conclude that whenever A⊂G and B⊂X is a clopen set, then there exists an extremal element ν in 𝒞xo such that

So far, everything we have said works for every compact Hausdorff space X, equipped with an action of G by homeomorphisms, every clopen subset B⊂X and point xo with dense orbit. The triple (X,B,xo) gives rise to a set Bxo⊂G and what we have seen is that one can estimate product sets of Bxo with any set A⊂G in terms of the size of the union AB of translates of the set B under the elements in A with respect to certain extremal elements in 𝒞xo. We wish to show that every subset B′⊂G is of this form.

This undertaking is not hard. Let 2G denote the set of all subsets of G equipped with the product topology. Since G is countable, this space is metrizable. Note that G acts by homeomorphisms on 2G by right translations and the set

is clopen. Given any set B′⊂G, we shall view it as an element (suggestively denoted by xo) in 2G and we let X denote the closure of the G-orbit of xo. If we write B=U∩X, then B is a clopen set in X and Bxo=B′. We can summarize the entire discussion so far in the following Correspondence Principle, which essentially dates back to Furstenberg [4].

Up until now, the discussion has been very general and no assumptions have been made on either the group G or the set 𝒞 of means involved. In order for the correspondence principle to be useful, we need to be able to better understand the extremal elements in 𝒞.

We shall now describe a situation when such an understanding is indeed possible. Let G be a countable abelian group and denote by ℒG the set of all invariant means on G. By a classical theorem of Kakutani–Markov, this set is always non-empty and it is clearly weak*-compact and convex. Given any compact Hausdorff space X, equipped with an action of G by homeomorphisms of X and containing a point xo in X with a dense G-orbit, the map

defined above is injective and left G-equivariant. Hence, its transpose must map ℒGonto the space of all G-invariant probability measures on X, which we denote by 𝒫G⁢(X). It is well known that the extremal elements in 𝒫G⁢(X) can be alternatively described as the ergodic probability measures on X, i.e. those G-invariant measures which do not admit any G-invariant Borel sets with μ-measures strictly between zero and one.

In particular, applying the discussion above to the set 𝒞=ℒG and adopting the conventions

we have proved the following version of the Correspondence Principle stated in Section 1.2.

We stress that this version of the correspondence principle does not apply to the set 𝒮⊂ℳ⁢(ℤ) of Birkhoff means on ℤ as its extremal points are not all mapped to ergodic measures under the map Θxo* above.

B Appendix II: Proof of Proposition 1.4

We recall the following simple lemma for completeness.

Let Y be a compact G-space, i.e. a compact Hausdorff space Y equipped with an action of G by homeomorphisms. We say that a point yo is G-transitive if its G-orbit is dense in Y. Recall that if A⊂Y is any subset and y is a point in Y, then we define

Proposition 1.4 will follow immediately from the following result which is interesting in its own right.