I recently learned of a promising technique in ergodic Ramsey theory that is useful to establish multiple recurrence and convergence of nonconventional ergodic averages. The trick is to reduce the general statement to certain systems, called sated systems.
A common trick in ergodic theory, and indeed in other areas in mathematics, is to decompose a function (or the space) as a sum of a structured component and a random component. The idea is then to show that averages of the orbit of the random component converge to , while the orbit of the structured component can be analyzed using the structure it has. What is the random and the structured component will depend on the particular problem at hands, but intuitively, the “more random” we choose the random component, the easier it becomes to establish convergence to , but the less structure we obtain for the structured component, and hence the harder it is to deal with it. In this vague level, sated systems are such that low structure implies higher structure (or, equivalently, low randomness implies higher randomness).
The key observation is that every measure preserving system has a sated extension. In other words, for any system there exists an extension (i.e. another more complex system which contains the original system in a certain way) which is sated.
These ideas were first developed by Tim Austin in order to give a proof of convergence for the nonconventional ergodic averages associated with commuting measure preserving transformations (the first proof of that result, due to Tao, was finitely in nature). Later Host managed to improve Austin’s proof in the sense that the sated extension used was more concretely constructed (to be precise, the extension in Host’s work is a finite power of the original system). This may be useful to gain some information about the limit. Eventually, Austin was able to derive the multidimensional Szemeredi theorem using sated extensions (together with an infinitary removal lemma) and then also the density Halles-Jewett theorem, where the word sated is used for the first time (that I am aware of. Before this, sated extensions have been called pleasant, magical or isotropized).
This trick came to my attention because it was recently used (among other things) by Austin to give a proof of a generalization of Szemeredi’s theorem for amenable groups. Also, at the same time, Qing Chu and Pavel Zorin-Kranich used this idea to extend a result on large recurrence times for large intersections of commuting actions of amenable groups (both pre-prints appeared on the arXiv in the same week).
In this post I will use the idea to pass to a sated extension to deduce the convergence of nonconventional ergodic averages that arise when studying the ergodic Roth theorem. This convergence was first establish by Furstenberg in the same paper where he gave an ergodic theoretical proof of Szemeredi’s theorem. The proof using sated extensions is a very particular case of the work of Austin. I will follow the approach in Austin’s thesis.
Theorem 1 Let be a measure preserving system and let . Then the limit
exists in .
— 1. Sated extensions —
Instead of defining a general notion of sated systems, I will restrict this post to the notions needed to prove Theorem 1. The purpose is to illustrate the main ideas behind how this technique can be applied.
Definition 2 (-system) By a -system we mean a probability space together with two commuting measure preserving transformations and . We represent this as .
Definition 3 (Factor and extension) Let and be two -systems and let be a map such that for every and for each .
Then we say that is a factor of or (equivalently) that is an extension of . To be precise, the factor (or the extension) is the system (or the system ) together with the map .
As is usual in the study of multiple recurrence and convergence of non-conventional averages, factors and extensions play an important role in this post. We can associate a factor of a system with the system . Thus we can think of factors of as -subalgebras of that are globally invariant under and .
Moreover, a factor induces an inclusion by associating a function with . If we can consider orthogonal projection onto the space using the conditional expectation operator , where is the -algebra in the system . We may also represent this by the (intuitive) notation . The trivial fact that the projection of a vector already in the image subspace is the vector itself can be stated as
We will frequently associate -algebras with the subspaces of they induce. Thus, for instance we can write to denote that , or to mean that for every we have .
If and are two -algebras in some space , we denote by the -algebra generated by the union .
Definition 4 We will denote by the class of -systems such that , where is the -algebra of -invariant sets and is the -algebra of -invariant sets.
Given a -system , we denote by the -algebra . In other words is the largest -subalgebra of such that the system is in the class .
We can now define what a sated system for our situation will be.
Definition 5 (Sated system) A system is sated for the class if for any extension (where ) and any function we have
A way to think about sated extensions is in terms of the spaces. As mentioned above, an extension induces the inclusion (which by abuse of language we denote by the same letter) . It is clear that . Thus we have the trivial inclusion:
The system is sated for the class if and only if for every extension we also have the reversed inclusion:
Going back to the philosophical discussion in the beginning of the post, if a function is in the space (which means that has a certain structure) and the system is sated, we can deduce that actually belongs to the apriori smaller (and hence more structured) subspace .
— 2. Proof of Theorem 1 —
The idea of the proof is to pass to a sated extension where most of the work is done. This is the content of the following lemma:
in the norm.
Proof: We will apply the van der Corput trick. Let . For each we have
Observe that for any . Also, for any . Thus we can pull out from the form terms inside the integral to get
Since preserves the measure, for any . Thus we get
Taking the Cesaro limit as and applying the mean ergodic theorem to the measure preserving transformation we get
Now define the -system where the underlying space is , the -algebra is , the measure is defined by
(this defines a measure in by the Hanh-Kolmogorov theorem) and the transformations are and . We need to check that both and are measure preserving. Since , it suffices to check that for every . Indeed we have for each :
To prove that preserves we will use the observation that for every function . We have
This concludes the proof that is a -system. We can now rewrite equation (1) as
Now taking the Cesaro limit as we obtain
where is the -algebra of sets left invariant under . Now observe that , where denotes the constant function on . Thus we can rewrite the previous equation as
We have, . Thus is invariant under .
Recall that denotes the largest factor of in the class . Thus both functions and (and hence their product) are in . We deduce that the right hand side of equation (2) becomes
We will show that , which will imply that the integral vanishes and hence by the van der Corput trick, we conclude that
in as desired.
To prove that we claim that defined by induces a factor map of -systems. Indeed, let . Then
and hence is measure preserving. Also, if we have and , hence both diagrams
commute. Now is the point where we use the fact that is sated for the class . Observe that and hence by satedness (and then the hypothesis on ) we conclude that
We can finally prove Theorem 1:
Proof: Let be a measure preserving transformation on the probability space . Let be the system defined by and let be an extension of the -system. Let be arbitrary. Observe that for every we have
Thus the result will follow if we prove that for every the limit
exists in .
so it suffices to show that the limit
Thus we have: