— 1. Introduction —
Let be a probability space and
be a (measurable) map such that the set
has the same measure as the set
for all (measurable) sets
. We call the triple
a measure preserving system. All sets and maps from now on are measurable. Given a set
and some
we define
In a previous post I talked about recurrence theorems, and in particular about how large the set is in general. Note that by the mean ergodic theorem we have
and in particular this limit exists, so the set
is non-empty.
Definition 1 A set
is a set of nice recurrence if for every measure preserving system
, every
and every
the intersection
is non-empty.
We already observed that itself is a set of nice recurrence, but there are much more rarified sets of nice recurrence. For instance it follows from the result explored in this post that for any polynomial
with integer coefficients and
that the set
is a set of nice recurrence.
The property of nice recurrence is a strengthening of the property of recurrence. In this previous post (cf. Theorem 10 in there) I showed that recurrent sets are exactly the intersective sets. In this post I will prove an analog of this result regarding sets of nice recurrence.
— 2. A more general setup —
For the readers convenience, I will give first a proof that a set is recurrent if and only if it is an intersective set. There is a subtlety here regarding the notion of upper density, as for each Følner sequence there is a different notion of upper density. I find that, when playing with different Følner sequences, things become more clear by working with a general countable, abelian group, and so that’s what we will do. Actually this generalizes to any amenable group, but for the sake of presentation I will restrict to abelian groups.
The reader uncomfortable with Følner sequences and general abelian groups can use the model case and
.
Recall that in a group , a (left) Følner sequence is a sequence
of finite subsets of
such that for each
we have
A countable group is amenable if and only if it has a Følner sequence, and it is known that every abelian group is amenable and hence has a Følner sequence. For each Følner sequence we define the upper density of a set
by
Moreover we define the Banach upper density of a set
as the supremum over all Følner sequences of the upper density
. A measure preserving action of a group in a probability space
is an action of
on
such that each
induces a measure preserving transformation. We represent that measure preserving transformation by
. The action is ergodic if any set
invariant under
(in other words such that
for all
) has either measure
or measure
.
Definition 2 Let
be a countable abelian group, let
and let
be a Følner sequence on
.
is a set of recurrence if for every measure preserving action of
on
and every
with
there is some
such that
.
is
-intersective if for each
with
there is some
such that
.
is Banach intersective if it is
-intersective for all Følner sequences
on
.
is a set of nice recurrence if for every measure preserving action of
on
, every
and every
there is some
such that
.
is a set of ergodic nice recurrence if for every ergodic measure preserving action of
on
, every
and every
there is some
such that
.
is
-nicely intersective if for each
and each
there is some
such that
.
is Banach nicely intersective if it is
-nicely intersective for all Følner sequences
on
.
Note that in the definition of sets of recurrence and sets of nice recurrence there is no reference to Følner sequences.
— 3. Sets of recurrence —
For sets of recurrence we have the following characterization:
Theorem 3 Let
be a countable abelian group and let
. Let
be a Følner sequence on
. Then
is a set of recurrence if and only if
is
-intersective.
Proof: To prove one direction we can use the Furstenberg’s correspondence principle. Then assuming that is recurrent we get that
is
-intersective.
To prove the other direction, assume that is
-intersective and let
be a measure preserving action of
on a probability space
and let
have positive measure. For each
let
be the set
. Let
Note that for all
, so by Fatou’s lemma we get that
In particular we get that in a set of positive measure, call it
. For each
there is some
such that
is non-empty. Let
be in that intersection. Since there are only countably many choices for the pair
, we conclude that there is a subset
with positive measure and a pair
such that for each
we have
.
Thus if then both
(because
) and
(because
). Equivalently we have
and so
, so
.
As a corollary we get the following:
Corollary 4 Let
be a countable abelian group and let
. Then
is a set of ergodic recurrence if and only if
is Banach-intersective.
Proof: If is recurrent, then it is
-intersective for all Følner sequence
and hence it is Banach intersective.
Reciprocally, if it is Banach intersective, then it is -intersective for some (and actually all) Følner sequence and by the previous result we get that it is also recurrent.
Maybe even more surprising, at least a priori, is the following observation:
Corollary 5 Let
and
be two Følner sequence in
. Then a set
is
-intersective if and only if it is
-intersective.
— 4. Sets of nice recurrence —
We now prove a weak analogue of the Theorem 3 for sets of nice recurrence.
Theorem 6 Let
be a countable abelian group and let
be a Banach nicely intersective set. Then
is a set of ergodic nice recurrence.
Proof: Let be an ergodic measure preserving action of
on the probability space
and let
. For each
let
be the set
. Now fix an arbitrary Følner sequence
in
, fix
and let
Note that for and
we have
By the mean ergodic theorem and the previous equation we have
where the convergence holds in the norm. Since
are bounded functions we conclude that some subsequence of
converges almost everywhere, hence for almost every
we have
where the density is with respect to a subsequence of the original Følner sequence. Applying this procedure for each of the countably many elements
of
we have that for some Følner sequence
, for all
and all
in a set of full measure we have
The result now follows from the definitions.
Remark 1 Note that, from the proof, it follows that we don’t need
to be
-nicely intersective for all Følner sequences
in
, but just for some family of Følner sequences
such that any Følner sequence in
has a subsequence in
. If moreover we consider two Følner sequences to be the same if they give the same upper density to every set, is it true that there exists such a countable family
? Assuming that this is the case (although this seems rather unlikely), then we can even replace the condition of
to be Banach with the condition that
is
-nicely intersective for some Følner sequence.
Remark 2 Note that a partial reciprocate of the Theorem 6 (the fact that a set of ergodic nice recurrence is
-nicely intersective for some Følner sequence) follows from a strong form of Furstenberg’s correspondence principle, where we get an ergodic system. The precise statement we are invoking here is the proposition 3.1 of this paper by Bergelson, Host and Kra.
— 5. A negative result on recurrence —
In a previous post I stated the following theorem of Bergelson:
Theorem 7 Let
be a probability space, let
and let
be a countable family of subsets of
, all with
. Then there exists a set
with
(where the density is with respect to the Følner sequence
) and such that for each
we have
![]()
On a different direction, it is a consequence of (the proof of) the Poincaré recurrence theorem that:
Theorem 8 Let
be a probability space, let
and let
be a countable family of subsets of
, all with
and such that for
we have
. Then for each
there exists an infinite set
such that for each
we have
![]()
It is an interesting question whether we can make in this theorem to have positive density (even if not with density
). Unfortunately this is not the case in general:
Theorem 9 There exists a measure preserving system
and some
with
and some
such that for all set
with positive density there are some
such that
.
Proof: The trick is to use the fact that sets not all sets of recurrence are sets of nice recurrence, this was proved first by Forrest in 1990. Let be a set of recurrence which is not a set of nice recurrence. Let
be a measure preserving system,
and
such that
for all
.
Assume, for the sake of a contradiction that there was some set of positive density such that
for all
. Then there is some
such that
is non-empty. Let
. Then both
and so
contradicting the construction.
— 6. Some questions —
The proof of the Theorem 6 showed that for any ergodic measure preserving action of an abelian group on a probability space
and any
, the sequence
can be modeled as
for some Følner sequence
on
and some
(and moreover we can take
to be a subsequence of any arbitrary Følner sequence).
Some good questions are raised by this phenomenon, I don’t know if the answers are somewhere in the literature:
Questions Let
be a countable abelian group acting on a probability space
by measure preserving transformations (not necessarily ergodicaly). Let
.
- Is it true that there is some Følner sequence
on
and some set
such that for all
we have
?
- Is it true that for any Følner sequence
there is some set
such that for all
we have
? What if the action is ergodic?
- Is it true that there is a set
such that for all
we have
? What if the action is ergodic?
- Can we get more generally that for all
and
![]()
Other less ambitious questions are suggested by how weak the Theorem 6 is compared to the Theorem 3. We just showed that if is Banach nicely intersective (the strongest condition for nicely intersective) implies that
is a set of ergodic nice recurrence (the weakest condition for nice recurrence). By the correspondence principle we also have that being of nice recurrence implies being Banach nicely intersective.
Questions
- Do we have that if a set
is of ergodic nice recurrence, then it is of nice recurrence, or at least Banach nicely intersective?
- Does
-nicely intersective imply any of the other properties?
- Can one remove the ergodicity assumption, i.e. does Banach nicely intersective imply being a set of nice recurrence?
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