This is the fourth in a series of six posts I am writing about Szemerédi’s theorem. In the first three posts, besides setting up the notation and definitions necessary, I reduced Szemerédi’s theorem to three facts. Those three facts are proved in the last three posts in this series. In this post I will prove the fact that the Sz property lifts through weak mixing extensions.
To briefly recall our setup, is a measure preserving system, a -subalgebra is called a factor if for every also ; and we say a factor is Sz if for every nonnegative bounded function measurable in satisfying and every we have
The purpose of this post is to prove the following theorem:
— 1. Sz lifts through weak mixing extensions —
In this section we assume that is a weak mixing extension of a measure preserving system and is Sz. First we present a useful lemma that helps explain why we consider factors of instead of arbitrary -subalgebra:
Proof: Note that is a unitary operator on and the conditional expectation is the orthogonal projection onto . Since is a factor, it is easy to see that is also in . We will use the Riesz representation theorem, thus it suffices to show that for every we have . Indeed
The fact that is a factor, which implies that also , was used in the second equality.
The next lemma asserts that weak mixing extensions are also ergodic extensions:
Proof: By the ergodic theorem we know that this limit is the projection of onto the space of invariant functions. Thus it suffices to show that any -invariant function is measurable with respect to .
Let be a -invariant function and let . By Lemma 2, , so is also -invariant. Moreover is weak mixing and so
Thus and hence . Therefore every -invariant function is indeed measurable with respect to and this proves the result.
We need a technical lemma (which is, in some way or the other, behind all results in recurrence or convergence):
I have used the van der Corput trick several times before in this blog, I never quite proved this version here, but the proof from this post can easily be adapted to deal with uniform limits.
The following lemma is the key to show that weak mixing extensions of Sz systems are Sz. The proof is essentially the same proof that weak mixing systems satisfy the multiple recurrence property, properly relativized.
To get this lemma, we need to induct on some stronger hypothesis, and hence we will instead prove the following general case:
Lemma 6 Assume that the extension is weak mixing. Let and let be distinct and non-zero. Assume that, for some , we have . Then
Proof: We can (and will) assume, without loss of generality, that , since otherwise we can just permutate the values of . We proceed by induction on . For we can use Remark 1 to deduce that the extension is weak mixing for , and hence we can assume that . But now, this case reduces to Lemma 3.
Now let and let . We are going to apply the van der Corput trick, so it suffices to show that (2) is satisfied. We have
On the other hand we have the trivial bound
Taking the uniform Cesàro limit of the inner product and using the previous estimate and (3) (and the triangular inequality) we have
Finally we have that
where in the last line we used the definition of weak mixing extension (1).
We can now prove Theorem 1. Observe the following elementary identity:
The idea here is that the initial product was transformed into a sum of products, and the first product on this sum only has ‘s and each of the other products in this sum contain at one term .
Let be a non-negative function such that . Let and note that and a.e. (to see this, consider the set and use the definition of conditional expectation). Also let .
We will use the identity with and . Then, with exception of the first product, which consists only of ‘s, each other product in the sum contains some . Taking the uniform Cesàro limit and using Lemma 5 we get
Finally we can get multiple recurrence for :
where the inequality on the last line follows from the hypothesis that is Sz. Since was arbitrary (within the conditions and ) we showed that is Sz and this proves Theorem 1.