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
.
Proposition 3 (Conditional Cauchy-Schwartz inequality) For any functions
we have
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
Corollary 4 (Conditional Triangular Inequality) Let
. Then
Proof:
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.
Definition 5 (Weak mixing extension)
- 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:
Definition 6 (Compact extension)
- 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).
Lemma 8 Let
be a probability space, let
be
-subalgebras of
totally ordered by inclusion and let
be the
-algebra generated by all
. Let
and
. Then there is some
such that
.
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
and also
and
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
.
We are now ready to prove Furstenberg’s multiple recurrence (Theorem 2 of the previous post), conditional on Theorem 7.
Proof:
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.
Let be such that
and
. We need to show that, for each
,
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.
Pingback: Szemerédi’s Theorem Part IV – Weak mixing extensions | I Can't Believe It's Not Random!
Pingback: Szemerédi’s Theorem Part V – Compact extensions | I Can't Believe It's Not Random!
Pingback: Szemerédi Theorem Part VI – Dichotomy of extensions | I Can't Believe It's Not Random!