The Poincaré recurrence theorem states that a dynamical system (under suitable conditions) will eventually return to a condition that is very close to the original condition. Slightly more formally, for a compact set in , if is a volume preserving map (which means that for any with positive volume, ), then implies that almost every returns to under iteration of . That is, there exists so that . This theorem was proved when Poincaré was studying the stability of the solar system.
We can formulate the statement rigorously and the proof is indeed very simple. For a probability space , we say that is measure preserving if for any , .
Theorem. (Poincaré recurrence theorem) Let be as above. Then for any such that , almost every returns to under iteration of .
Proof. Consider the sets . Because their measures are the same (and positive) and is a probability measure, we know that there exists such that (otherwise, every pair of such sets are “almost” disjoint, and you can derive a contradiction easily by using finite additivity of ). That is, .
Now, assume that the theorem is not true and let be a set of positive measure such that no returns. But there exists ( above) such that , contradiction.
We did not actually use the countably additivity of a probability measure. It suggests that the Poincaré recurrence theorem should hold in a more general setting. Roughly speaking, if we can define a certain kind of density on a set appropriately, then sets with positive density should have rich structures (large set returns to large set).
Let us forget Poincaré recurrence theorem now and consider the following theorem by Hilbert:
Theorem. (Hilbert) If is irreducible, then there exists such that is irreducible.
His proof relied on the following lemma:
Lemma. For any finite partition , one of the contains arbitrarily large cube.
For “cube”, it means that the set of the form
For example, is a cube. “Arbitrarily large” means that for any , you can always find some .
The original proof of lemma was quite difficult, yet there is an easy proof using the concept of Poincaré recurrence!
For , define the upper density of by
We claim that if then contains large cubes. This claim clearly implies the Lemma. As in the proof of Poincaré recurrence theorem, there exists such that . As is of positive upper density, using again that proof we can find such that . Let . Then . Bigger cubes are constructed in exactly the same way.
For more complicated structures, Poincaré recurrence-type theorem may not be sufficient. For example,
Theorem. (Szemerédi) If then contains arbitrarily long arithmetic progressions.
This famous theorem is known to be equivalent to the multiple recurrence theorem:
Theorem. (Furstenberg) Let be as above. Then for any with , and for any , there exists such that
Another example is
Theorem. (Sarkozy) If with positive upper density, then there exists such that .
If we want to apply Poincaré recurrence-type theorem, we need to show something like , which is much harder than the original one as is much more sparse. Yet we still have
Polynomial Szemerédi theorem. and are as above. Then for any with , there exists such that
To me it is very amazing that many results in additive number theory comes from Poincaré recurrence theorem (it seems that this kind of approaches is called ergodic Ramsey theory). There are many more examples which are consequences of Poincaré recurrence theorem (and not restricted to number theory). If you want to know more, a good reference is “The multifarious Poincaré recurrence theorem” by Vitaly Bergelson.