From now on, instead of upper density, we will consider upper Banach density. This notion reflects better the the asymptotic behavior of the set. For example, the union of intervals , , has zero upper density, but has a rich combinatorics.
Definition 1 The upper Banach density of a set is
Note that , which will reveal to be good for our applications.
In the previous posts, we developed some machinery in Ergodic Theory and now we will relate them to combinatorics. The main tool is a correspondence principle due to Furstenberg. Before discussing it, let us restate a theorem of the last post.
This is the main result of Furstenberg’s 1977 paper and was used to prove Szemerédi Theorem.
1. Furstenberg Correspondence Principle
The question is: given such set , how to create a mps somehow related to ?
for any integers .
Theorems 2 and 4 prove Theorem 3: note that the arithmetic progression belongs to if and only if . So, positive denseness of the set , in particular, proves its non-emptyness. Apply Theorem 2 to the mps obtained by Theorem 4: given , there exists such that
As is arbitrary, Szemerédi Theorem is established.
The upper Banach density gives the same weight to all integers. It might happen (altough I still did not find any examples) that a set might have positive density if some subsets of are heavier than others. For example, if we give no weight to the even numbers, then the odd numbers have density one. In this post, I will give a general version of the correspondence principle. It follows the same ideas of the original proof of Theorem 4. I hope this extension, in addition of proving the classical correspondence principle, give rise to new applications of Theorem 2 for some classes of sets of zero upper Banach density that are big in some other sense.
2. A general correspondence principle
For example, every constant sequence satisfies this condition. Given a set , define the upper Banach density with respect to as:
where stands for the usual Dirac measure. This density is a well defined number between and . We will prove the following
for every .
Proof: We consider the most natural system: the set of characteristic functions of sets of integers.
The dynamics considered is the bilateral shift defined by
We apply a Krylov-Bogolubov argument to create a -invariant measure . Take , that is, with .
Let , be sequences of integers for which
We may assume (restricting , to subsequences, if necessary) that the probabilities defined by
converge in the weak- topology to a probability , that is,
for every continuous . Under the assumption (2), is -invariant. In fact,
which, by hypothesis, converges to zero as . Good: we have our probability space!
Continue the construction taking . Then
This is the connection between and we were looking for. It implies that
and, as is a clopen set (all cylinders are clopen),
It remains to verify (3), which actually follows from the last argument:
This concludes the proof.
It is established the connection between combinatorics of sets of integers and ergodic theory.