This is the third in a series of six posts on Szemerédi’s theorem. In the previous post I outlined the ideas of the ergodic theoretical proof by Furstenberg. In this post I will set up the machinery and give the precise definitions that appear on that proof (namely the notions of weak mixing extension and compact extension), and then I will show how Furstenberg’s multiple recurrence theorem follows from the three key results stated in the end of the previous post. Recall that in the first post of this series we deduced Szemerédi’s theorem from the multiple recurrence theorem; thus in this post we will reduce Szemerédi’s theorem to those three key results, which will be proved in the last three posts of this series.
From now on I will assume that denotes a probability preserving system. Also, for a function and any integer , we denote by the function . Note that if is measurable with respect to any factor of , then so is . Moreover is an isometry of any space (for any and any factor). In this post we will fix an extension of factors of .
We will make extensive use of the notion of conditional expectation, also explored in a previous post in this blog. In short if and is a -subalgebra, then the conditional expectation is the orthogonal projection of in the subspace . When is just in we can approximate it by functions in .
We need to set up some terminology before defining weak mixing and compact extensions.
Definition 1 (Conditional inner product) Let . We define their conditional inner product by:
Note that if is the trivial -algebra, then this degenerates to the usual inner product.
An immediate property of this inner product is that for every and . Indeed
This conditional inner product gives rise to a conditional norm, and then we can define the conditional Hilbert space (or Hilbert module over ).
Definition 2 (Hilbert Module) We define to be the subspace of consisting of those functions for which the conditional norm
is in .
Note that the functions and have the same norm. Thus, in particular, if then .
The conditional Cauchy-Schwartz inequality assures us that for , the inner product , which is a priori only in , is actually in .
Proof: Most proofs of the usual Cauchy-Schwartz inequality can be relativized do this situation. Note that if then the function is also in . We now use the trivial inequality with this function:
After rearranging this gives the desired inequality.
Observe that this also implies a conditional Triangular inequality
We define the norm on by making . Corollary 4 implies that this is indeed a norm. This turns into a complete metric space.
— 2. Weak mixing and compact extensions —
We can now define weak mixing extensions. Recall from the previous post the notation of uniform Cesàro limits.
- A function is conditionally weak mixing if for each we have
- The extension is called weak mixing if every such that is conditionally weak mixing.
Example 1 If is the trivial -algebra, then the extension is weak mixing if and only if the m.p.s. is a weakly mixing system.
It takes a little more effort to define compact extensions:
- A subset is conditionally pre-compact if it is totally bounded with respect to the norm, i.e. if there are finitely many functions such that for any and each , we have for some .
- A function is conditionally compact if the orbit is conditionally pre-compact.
- The extension is called compact if for each and each there is a subset such that and is conditionally compact.
We stress the subtlety that, in the definition of conditionally pre-compact set, the choice of depends on each .
Example 2 The first example of a compact extension is when is the trivial -algebra and is (isomorphic to) a rotation on a compact group, i.e. is a compact metrizable group, is the Borel -algebra, is the Haar measure and for some .
Indeed, in this case, the conditional norm coincides with the norm, and hence the extension is compact if and only in for each function , the orbit is pre-compact in the norm. Since is compact, for every there exists a finite set such that for any power of with , there is some such that . It is not hard to see that a similar phenomenon happens with the translations of a function ; more precisely, for every and there exists a finite set such that for any the function is -close to some function with . Finally, since is dense in , we conclude that any function is almost periodic.
The next example gives a more general example of a skew-product.
Example 3 Let , let be the Borel -algebra in and let be the Borel -algebra on . Let be the Haar measure on , let and define by . Let be the vertical -algebra. Then is a factor of and the extension is a compact extension.
Proof: Since we deduce that is indeed a factor. To see that the extension is compact recall that has the following orthonormal basis formed by characters:
I will show that, for each , the character is conditionally compact. It follows from Fubini’s theorem that, for every and any point we have
Now, fix and let with each be an -net of the unit circle. Let . For every and , applying (1) and a simple computation we have
Thus, choosing the appropriate we conclude that and hence every character is conditionally compact. It is easy to see that finite linear combinations of conditionally compact functions are still conditionally compact, and hence we found a dense subset of formed by compact functions.
— 3. Proof of the multiple recurrence theorem —
With all the definitions in place I will recall Theorem 8 from the previous post:
- Let be an extension between factors of . If the extension is weak mixing and is a Sz factor, then also is a Sz factor.
- Let be an extension between factors of . If the extension is compact and is a Sz factor, then also is a Sz factor.
- Let be a factors of . If the extension is not weak mixing, then there exists a non-trivial extension of which is compact.
In this section I will deduce Furstenberg’s multiple recurrence (Theorem 2 of the previous post) from Theorem 7. In the previous post we saw how the multiple recurrence theorem implies Szemerédi’s theorem; thus we will have reduced Szemerédi’s theorem to Theorem 7. The final three posts of this series will be dedicated to prove this theorem, one point per post.
First we need a technical lemma (alternatively one could use some version of Doob Martingale convergence Theorem).
Proof: Let be the closure of the union of the , precisely , where we view as a subspace of . I claim that if , then also .
To see this assume without loss of generality that , let and choose be such that both and . Note that multiplying with the characteristic function of the set (which is in ) gives a function in closer to than , thus we can assume that . We now have
Putting the last three equations together we get that , since was arbitrary, this proves the claim.
Now let be the family of all sets such that . I claim that is a -algebra. Let , then and so . Now let be any sequence of sets in . Let , note that is the characteristic function of the intersection . By the previous claim, , and is the characteristic function of the intersection . Since is closed, we get that the intersection of all is still in and hence is indeed a -algebra as claimed.
Moreover, since any set in any has its characteristic function in we conclude that , so also . Finally since is a closed subspace we actually get that .
Let be the set of all Sz factors of the system , I want to show that . We can order partially by inclusion, and we will apply Zorn’s lemma to find a maximal element in . Let be a totally ordered family of . I claim that the -algebra generated by is also Sz.
There must exist some such that the set has positive measure, otherwise would be . Since
it suffices to assume that itself is the characteristic function of some set, say .
Fix and apply Lemma 8 to find a Sz factor such that
Call and . I claim now that in a set of positive measure.
Indeed, if that were not true, we would have for all , and hence
which is a contradiction. Thus the set has positive measure. Also , which is a Sz factor, so
and hence, the set is syndetic. For any denote . If then for all , and hence
Since each of the functions only takes values in we have
and taking conditional expectations we get, for ,
Finally integrating we obtain
Since this happens for every which is a syndetic set, say with gaps bounded by , we conclude that
This means that the factor is indeed a Sz factor.
We are now in conditions to apply Zorn’s lemma to find a maximal Sz factor . Assume, for the sake of a contradiction, that . If the extension is weak mixing, then, by the first point of Theorem 7 also is Sz. Otherwise, by the third point of Theorem 7, there exists an intermediate factor such that the extension is compact. But then, by the second point of Theorem 7, would also be Sz, contradicting the maximality of .
We have proved Szemeredi’s Theorem, assuming only Theorem 7.