Let be a probability space and let be a (measurable) map such that for any (measurable) set we have , where as usual . Such is called a measure preserving transformation. In this setting, all sets in the pre-orbit of some given set , i.e. the sets , have the same measure. If in addition then it is clear that two of those sets must intersect (since the total measure is ). This simple observation is (a consequence of) the Poincaré recurrence theorem.
Several improvements can be made to this theorem: we can ask for more than two sets in the orbit to intersect simultaneously, we can ask to have not one but many pairs with non-empty intersection, we can require the intersection to be not only non-empty but to have positive measure (how much measure are we guaranteed to find?). In this post I will list many such extensions that have been proved so far, as well as related results and some questions (that as far as I know are open). I will not provide the proof to most of the results, however I will always put a reference.
— 1. Single Recurrence theorems —
We start by stating properly the Poincaré recurrence theorem. In what follows (and as usual) to say that is a measure preserving system means that is a -algebra on , is a probability measure and is a measure preserving transformation. The first 5 results appear in the beginning of this survey by Bergelson.
Theorem 1 (Poincaré recurrence theorem/Pigeonhole Principle) Let be a measure preserving system and let have positive measure. Then for some , the intersection has positive measure. Moreover one can take .
The points in the intersection are called recurrent because after iterations of the map they are back to . This is where the name “recurrence theorem” comes from.
The first part of this theorem will follow immediately from the next result which is also called the Poincaré recurrence theorem. It says roughly that almost all points in are recurrent:
Theorem 2 (Poincaré recurrence theorem, again) Let be a measure preserving system and let . Then the set has measure .
This implies also that if then for some we have , because the countable union of the intersections has measure .
Applying the previous theorem to each one of the transformations we obtain the following stronger looking result:
Theorem 3 (Poincaré recurrence theorem, yet again) Let be a measure preserving system and let . Then the set has measure .
By adapting the proof of this results we can get a more general statement that applies to any sequence of sets with a stationary property (so far we have been considering ).
Proposition 4 Let be an algebra of subsets of and be a finitely additive probability measure. For each let be such that for any we have . If then for some , the intersection has positive measure. Moreover one can take .
This can be useful in applications when we don’t have a (countably additive) measure, for instance when we deal with ultrafilters.
— 1.1. Large intersections —
Still on this setting when we only have a finitely additive measure and the events are not necessarily of the form for some measure preserving transformation we can improve the previous result in another direction, this answers the question of how big can we make the measure of the intersection in the previous proposition. If the sequence consists of independent events then the intersection has measure for any pair , so the next result is in a way the best possible:
Proposition 5 Let be an algebra over and be a finitely additive probability measure on . For each let be such that for any we have . If , then for each there exists a positive integer such that . Moreover we can take .
By the (infinite) Ramsey’s theorem this implies that, for any fixed , there exists a sequence such that for any one has . An interesting question for which I could not find an answer is if the sequence can always be taken to have positive upper density. On the other hand we can always find a sequence of density such that the intersection has positive measure, see proposition 15 bellow.
We now return to the case when and we consider the set of recurrent times.
Applying the previous result to for each implies that the set is always infinite, and indeed it contains multiples of each . Furthermore, the sets satisfies other notions of largeness: A subset of is called syndetic if it has bounded gaps, i.e. if there is some such that in any interval of length there exists at least one element of the set.
It turns out that more can be said. I now define the notion of IP-set and IP set, as well as their finitistic versions:
Definition 8 (IP, IP, IP and IP sets) Given a subset of the positive integers (not necessarily infinite) one forms the set of their finite sums . A set is called an IP-set it it contains for some infinite set and it is called an IP-set if it contains for some set with elements.
A subset of is an IP if it has non-empty intersection with every IP set, and it is an IP set if it has non-empty intersection with every IP set.
For instance the set of numbers that use only the digits and (in base ) is the IP set formed by the finite sums of the sequence , and the subset of those numbers with at most digits is an IP set. From this example we see that while IP sets have some nice structure, they may be very sparse. Also each IP set is an IP set for each .
As an example for IP we have the trivial example of itself. It is also not difficult to see that is an IP-set for any as well. Furthermore, since each IP set is an IP set, we have that each IP set is a IP set.
Moreover, the intersection of any two IP-sets is again an IP-set (this is essentially Hindman’s theorem). From this we see that IP-sets are in a sense very large. A good exercise is to prove that every IP-set is syndetic. Thus the following result generalizes Khintchine’s theorem.
Proposition 9 For any measure preserving system and any set with positive measure, the set of recurrent times given by definition 6 (with ) is an IP-set.
Proof: Let and take large to be determined later. We will prove that is an IP-set. Let be an IP set, say it contains the finite sums of the set . Let and consider the sets . We claim that two of those sets have intersection with measure larger than . Assuming the claim for now, say with then and . On the other hand so is non-empty and since was an arbitrary IP set we conclude that is an IP set.
To prove the claim let be the indicator function of and let . Note that . By Jensen’s inequality applied to and the convex function we get
Thus for some we have
and for sufficiently large (we need ) that is greater than and this finishes the proof.
— 1.2. Large recurrent times —
In this subsection I focus on results about how “large” is the set (see definition 6) in several ways. For these results I didn’t find a correspondent formulation for . The first result in this section is Sarkozy’s theorem (for which I presented two different proofs in previous posts) and states that intersects the image on any divisible polynomial:
It is also true that intersects the sets ad , where is the set of prime numbers, this is proved for instance in this paper by Kamae and Mendes-France. Moreover for each value of other that it is easy to construct a set that does not intersect (take a finite cyclic system). The same example shows that the set is not for a polynomial which is not divisible.
Actually, a little more can be said: let be either for a divisible polynomial or or . Then for every measure preserving system and any set with there exists (which may depend on as well as on the system and the choice of ) such that intersects .
Many other examples of sets that intersect for every system and set with positive measure are know, those are called recurrent sets.
— 2. Multiple recurrence —
By using essentially the proof of proposition 10 one can generalize that result to multiple intersections:
Proposition 12 Let be a probability space. Let be a sequence of sets with the same measure and assume that . Then for each and each there exist positive integers such that . Moreover we can take smaller than a constant depending only on and .
This can be considered the simplest case of multiple recurrence, but in many applications it is important to have some relation between . The most well-know case is Furstenberg’s multiple recurrence theorem (see for instance Furstenberg’s original paper or this recent exposition of Zhao):
First note that when this is theorem 1. Note also that this result only guarantees that the measure of the intersection is positive, there are very few results on multiple recurrence that guarantee a large intersection. The following exception is in the end of the already mentioned Bergelson’s survey (corollary 5.4).
Moreover the set of pairs that satisfy that is syndetic.
— 2.1. Large recurrent times —
Furstenberg’s multiple recurrence theorem (theorem 13) provides an alternative proof of the celebrated Szemerédi’s theorem stating that any subset with positive upper density contains arbitrarily long arithmetic progressions. To deduce Szemerédi’s theorem from theorem 13 Furstenberg created the so called correspondence principle. Conversely, it is also possible do deduce theorem 13 from Szemerédi’s theorem, one way to do this is using the following recurrence theorem (which is similar to proposition 12 but not quite the same):
The original proof of this fact, found by Bergelson, is given in this paper.
To give an example why it is of interest to have IP instead of just IP, I now present a proof of the following result:
The same result holds for instead of . Moreover, as remarked before, the result is false for any other constant different from , even for . The following argument was taken from this note.
Proof: By theorem 17 it suffices to show that for any there exists an IP set inside . By recent results of Green, Tao and Ziegler we have that for any there exists a vector such that is a vector of primes, where and , as long as the affine linear forms are affinely independent and for each there exists such that no coordinate of is divisible by .
If we make be the vector with all coordinates and be a matrix where each line is the characteristic function of a different nonempty subset of then we are in the conditions of the theorem and we get that the IP set generated by the coordinates of is inside .
Also a generalization of Sárközy’s theorem holds. In fact, not only the set intersects any divisible polynomial, but the intersection is very large in the following sense:
Theorem 19 Given any measure preserving system, any set with positive measure, any and any polynomials with , the set
is an IP set.
This result is due to Bergelson and McCutcheon and is part of several refinements of what is called the polynomial Szemerédi theorem.
Unfortunately it is not known (yet) if the set described in the previous theorem is actually IP for some . However, a result that generalizes both this theorem and theorem 18 was obtained by Frantzikinakis, Host and Kra in this paper.
Theorem 20 Given any measure preserving system, any set with positive measure, any and any polynomials with , there exists a prime such that
Again the result with is also true.
— 3. Final Remarks —
In this post I restricted the attention to a single measure preserving transformation, in other words, a action. Most of the results presented have generalizations for actions of other groups (usually or a general countable abelian group). There are also many other sets that I didn’t mention which are known to intersect the set from definition 6 and the set from definition 16, although I don’t know any other (non-trivial) example for sets that intersect every . A notable example of sets that intersect are certain IP-polynomials, an example is the set where is a polynomial and is an IP set. To construct another example of an IP-polynomial, let denote the family of finite non-empty subsets of . Let and be strictly increasing sequences in and for each let and . Now the set is a degree two IP polynomial that intersects .
On a different direction, one could “join” both definitions and consider the set of times of multiple returns with high intersection, formally the set
I don’t remember any result of this nature, either on the positive or negative direction.