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In 1977, Furstenberg established his multiple recurrence theorem:
Theorem 1 (Furstenberg multiple recurrence) Let be a measure-preserving system, thus is a probability space and is a measure-preserving bijection such that and are both measurable. Let be a measurable subset of of positive measure . Then for any , there exists such that
Equivalently, there exists and such that
The multiple recurrence theorem is proven, roughly speaking, by an induction on the “complexity” of the system . Indeed, for very simple systems, such as periodic systems (in which is the identity for some , which is for instance the case for the circle shift , with a rational shift ), the theorem is trivial; at a slightly more advanced level, almost periodic (or compact) systems (in which is a precompact subset of for every , which is for instance the case for irrational circle shifts), is also quite easy. One then shows that the multiple recurrence property is preserved under various extension operations (specifically, compact extensions, weakly mixing extensions, and limits of chains of extensions), which then gives the multiple recurrence theorem as a consequence of the Furstenberg-Zimmer structure theorem for measure-preserving systems. See these lecture notes for further discussion.
From a high-level perspective, this is still one of the most conceptual proofs known of Szemerédi’s theorem. However, the individual components of the proof are still somewhat intricate. Perhaps the most difficult step is the demonstration that the multiple recurrence property is preserved under compact extensions; see for instance these lecture notes, which is devoted entirely to this step. This step requires quite a bit of measure-theoretic and/or functional analytic machinery, such as the theory of disintegrations, relatively almost periodic functions, or Hilbert modules.
However, I recently realised that there is a special case of the compact extension step – namely that of finite extensions – which avoids almost all of these technical issues while still capturing the essence of the argument (and in particular, the key idea of using van der Waerden’s theorem). As such, this may serve as a pedagogical device for motivating this step of the proof of the multiple recurrence theorem.
Let us first explain what a finite extension is. Given a measure-preserving system , a finite set , and a measurable map from to the permutation group of , one can form the finite extension
One easily verifies that this is indeed a measure-preserving system. We refer to as the cocycle of the system.
An example of finite extensions comes from group theory. Suppose we have a short exact sequence
of finite groups. Let be a group element of , and let be its projection in . Then the shift map on (with the discrete -algebra and uniform probability measure) can be viewed as a finite extension of the shift map on (again with the discrete -algebra and uniform probability measure), by arbitrarily selecting a section that inverts the projection map, identifying with by identifying with for , and using the cocycle
Thus, for instance, the unit shift on can be thought of as a finite extension of the unit shift on whenever is a multiple of .
Another example comes from Riemannian geometry. If is a Riemannian manifold that is a finite cover of another Riemannian manifold (with the metric on being the pullback of that on ), then (unit time) geodesic flow on the cosphere bundle of is a finite extension of the corresponding flow on .
Here, then, is the finite extension special case of the compact extension step in the proof of the multiple recurrence theorem:
Before we prove this proposition, let us first give the combinatorial analogue.
Lemma 3 Let be a subset of the integers that contains arbitrarily long arithmetic progressions, and let be a colouring of by colours (or equivalently, a partition of into colour classes ). Then at least one of the contains arbitrarily long arithmetic progressions.
Proof: By the infinite pigeonhole principle, it suffices to show that for each , one of the colour classes contains an arithmetic progression of length .
Let be a large integer (depending on and ) to be chosen later. Then contains an arithmetic progression of length , which may be identified with . The colouring of then induces a colouring on into colour classes. Applying (the finitary form of) van der Waerden’s theorem, we conclude that if is sufficiently large depending on and , then one of these colouring classes contains an arithmetic progression of length ; undoing the identification, we conclude that one of the contains an arithmetic progression of length , as desired.
Of course, by specialising to the case , we see that the above Lemma is in fact equivalent to van der Waerden’s theorem.
Now we prove Proposition 2.
Proof: Fix . Let be a positive measure subset of . By Fubini’s theorem, we have
where and is the fibre of at . Since is positive, we conclude that the set
is a positive measure subset of . Note for each , we can find an element such that . While not strictly necessary for this argument, one can ensure if one wishes that the function is measurable by totally ordering , and then letting the minimal element of for which .
Let be a large integer (which will depend on and the cardinality of ) to be chosen later. Because obeys the multiple recurrence theorem, we can find a positive integer and such that
Now consider the sequence of points
for . From (1), we see that
for some sequence . This can be viewed as a colouring of by colours, where is the cardinality of . Applying van der Waerden’s theorem, we conclude (if is sufficiently large depending on and ) that there is an arithmetic progression in with such that
for some . If we then let , we see from (2) that
for all , and the claim follows.
can be partitioned into the classes
where is the graph of . The multiple recurrence property for ensures that contains arbitrarily long arithmetic progressions, and so therefore one of the must also, which gives the multiple recurrence property for .
Remark 2 Compact extensions can be viewed as a generalisation of finite extensions, in which the fibres are no longer finite sets, but are themselves measure spaces obeying an additional property, which roughly speaking asserts that for many functions on the extension, the shifts of behave in an almost periodic fashion on most fibres, so that the orbits become totally bounded on each fibre. This total boundedness allows one to obtain an analogue of the above colouring map to which van der Waerden’s theorem can be applied.
In Lecture 11, we studied compact measure-preserving systems – those systems in which every function was almost periodic, which meant that their orbit was precompact in the topology. Among other things, we were able to easily establish the Furstenberg recurrence theorem (Theorem 1 from Lecture 11) for such systems.
In this lecture, we generalise these results to a “relative” or “conditional” setting, in which we study systems which are compact relative to some factor of . Such systems are to compact systems as isometric extensions are to isometric systems in topological dynamics. The main result we establish here is that the Furstenberg recurrence theorem holds for such compact extensions whenever the theorem holds for the base. The proof is essentially the same as in the compact case; the main new trick is to not to work in the Hilbert spaces over the complex numbers, but rather in the Hilbert module over the (commutative) von Neumann algebra . (Modules are to rings as vector spaces are to fields.) Because of the compact nature of the extension, it turns out that results from topological dynamics (and in particular, van der Waerden’s theorem) can be exploited to good effect in this argument.
[Note: this operator-algebraic approach is not the only way to understand these extensions; one can also proceed by disintegrating into fibre measures for almost every and working fibre by fibre. We will discuss the connection between the two approaches below.]