— 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:
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.
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 —
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:
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.
- 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?