## Poincaré recurrence theorem and its friends

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 $X$ in $\mathbb{R}^n$, if $T:X\rightarrow X$ is a volume preserving map (which means that for any $A\subseteq X$ with positive volume, $\text{Vol}(T^{-1}A) = \text{Vol}(A)$), then $\text{Vol}(A)>0$ implies that almost every $x\in A$ returns to $A$ under iteration of $T$. That is, there exists $n\in\mathbb{N}$ so that $T^nx\in A$. 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 $(X,\Sigma, \mu)$, we say that $T:X\rightarrow X$ is measure preserving if for any $A\in\Sigma$, $\mu(T^{-1}A) = \mu(A)$.

Theorem. (Poincaré recurrence theorem) Let $(X,\Sigma,\mu,T)$ be as above. Then for any $A\in\Sigma$ such that $\mu(A)>0$, almost every $x\in A$ returns to $A$ under iteration of $T$.

Proof. Consider the sets $A, T^{-1}A, T^{-2}A, \ldots$. Because their measures are the same (and positive) and $\mu$ is a probability measure, we know that there exists $i such that $\mu(T^{-i}A\cap T^{-j}A)>0$ (otherwise, every pair of such sets are “almost” disjoint, and you can derive a contradiction easily by using finite additivity of $\mu$). That is, $\mu(A\cap T^{i-j}A)>0$ .
Now, assume that the theorem is not true and let $A$ be a set of positive measure such that no $x\in A$ returns. But there exists $n$ ( $=j-i$ above) such that $\mu(A\cap T^{-n}A)>0$, contradiction. $\square$

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 $P(x,y)\in \mathbb{Z}[x,y]$ is irreducible, then there exists $x_0\in\mathbb{Z}$ such that $P(x_0,y)\in\mathbb{Z}[y]$ is irreducible.

His proof relied on the following lemma:

Lemma. For any finite partition $\mathbb{N}=\bigcup_{i=1}^r C_i$, one of the $C_i$ contains arbitrarily large cube.

For “cube”, it means that the set of the form

$\displaystyle Q(t;x_1,\ldots,x_k) = \left\{t+\sum_{i=1}^k \varepsilon_ix_i: \varepsilon_i\in\{0,1\}\right\}$.

For example, $t+\{0,x_1,x_2,x_1+x_2\}$ is a cube. “Arbitrarily large” means that for any $k\in\mathbb{N}$, you can always find some $Q(t;x_1,\ldots,x_k)\subseteq C_i$.

The original proof of lemma was quite difficult, yet there is an easy proof using the concept of Poincaré recurrence!

For $E\subseteq\mathbb{N}$, define the upper density of $E$ by

$\displaystyle \overline{d}(E) = \limsup_{N\rightarrow\infty} \frac{|E\cap\{1,\ldots,N\}|}{N}$.

We claim that if $\overline{d}(E)>0$ then $E$ contains large cubes. This claim clearly implies the Lemma.  As in the proof of Poincaré recurrence theorem, there exists $n_1$ such that $\overline{d}(E\cap (E-n_1))>0$. As $E\cap (E-n_1)$ is of positive upper density, using again that proof we can find $n_2$ such that $\overline{d}((E\cap(E-n_1)) \cap [(E\cap(E-n_1))-n_2])>0$. Let $t\in E\cap(E-n_1) \cap [(E\cap(E-n_1))-n_2]$. Then $t+\{0,n_1,n_2,n_1+n_2\}\subseteq E$. 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 $\overline{d}(E)>0$ then $E$ contains arbitrarily long arithmetic progressions.

This famous theorem is known to be equivalent to the multiple recurrence theorem:

Theorem. (Furstenberg) Let $(X,\Sigma, \mu, T)$ be as above. Then for any $A\in\Sigma$ with $\mu(A)>0$, and for any $k\in\mathbb{N}$, there exists $n\in\mathbb{N}$ such that
$\mu(A\cap T^{-n}A\cap T^{-2n}A\cap\cdots\cap T^{-kn}A)>0$.

Another example is

Theorem. (Sarkozy) If $E\subseteq \mathbb{N}$ with positive upper density, then there exists $n\in\mathbb{N}$ such that $n^2\in E-E$.

If we want to apply Poincaré recurrence-type theorem, we need to show something like $\mu(A\cap T^{-n^2}A)>0$, which is much harder than the original one as $n^2$ is much more sparse. Yet we still have

Polynomial Szemerédi theorem. $(X, \Sigma, \mu, T)$ and $A$ are as above. Then for any $p_1,\ldots, p_k\in\mathbb{Z}[x]$ with $p_i(0)=0$, there exists $n\in\mathbb{N}$ such that
$\mu(A\cap T^{p_1(n)}A\cap\cdots\cap T^{p_k(n)}A)>0$.

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.