## A dynamical proof of some Diophantine approximations

Diophantine approximations is a field in number theory which asks how good a real number can be approximated by some rational number. Quite a number of classical results are proved by using combinatorics, which is not friendly to people who cannot count. Here I will give a proof of some Diophantine approximations using only point set topology/dynamical system, or more precisely topological dynamics. (Main reference is this book.)

Throughout whole post, $X$ will be a compact topological space, and $G$ is a group or semigroup acting on $X$ by continuous transformations (i.e. $gx\mapsto T_g(x)$ for some continuous $T_g$). In this post, most of the time $G=\mathbb{N}$. In this case, $G$ is generated by $1$, and we denote by $T$ the transformation on $X$ representing the action of the element $1$. We then denote the dynamical system by $(X,T)$ and call it a cyclic system.

Definition 1 Let $T$ be a continuous map of a topological space $X$ into itself. A point $x\in X$ is called a recurrent point for $T$ (or for $(X,T)$) if for any neighbourhood $V$ containing $x$, there exists $n\geq 1$ such that $T^nx\in V$.

It means that if $x$ is a recurrent point, then it will eventually go back to somewhere near $x$. Surprisingly, every continuous map on a compact space has at least one recurrent point:

Theorem 1 (Birkhoff’s recurrence theorem) If $T$ is a continuous map of a compact space $X$ to itself, then the set of recurrent points for $T$ in $X$ is non empty.

Proof. Let $\mathcal{F}=\{Y\subset X: Y\text{ is closed, } Y\neq\emptyset\text{ and }TY\subset Y\}$, ordered by inclusion. Let $Y_1\supset Y_2\supset \cdots$ be a totally ordered chain. Then $Y_n\supset \bigcap_{m\in\mathbb{N}}Y_m$ for all $n$. The intersection is non empty because of  compactness. By Zorn’s lemma, $\mathcal{F}$ has a minimal element. Let $Y_0$ be minimal. We claim that every point in $Y_0$ is recurrent. Let $x\in Y_0$, and let $Y=\overline{\{T^nx: N\geq1\}}$. Note that $Y\subset Y_0$ since $Y_0$ is closed and invariant under $T$. But then so is $Y$. By minimality, $Y=Y_0$. Let $V$ be a neighbourhood of $x$ containing $x$. If $V\cap Y\neq\emptyset$, then we are done. If not, then $V\cap Y=\emptyset$, and so $V\cap Y_0=\emptyset$. But $x\in V\cap Y_0$, contradiction. $\Box$

Although we used Zorn’s lemma, it is known that the above result is independent of Zorn’s lemma.

In some special cases, we can guarantee that there are more recurrent points.

Definition 2 Let $K$ be a compact group, let $a\in K$ and let $T:K\rightarrow K$ be defined by $Tx=ax$. We call the dynamical system $(K,T)$ a Kronecker system.

Theorem 2 Every point in the space of a Kronecker system is recurrent.

Remark: Kronecker system was not invented by Kronecker. Reason for this terminology is that one of his theorem is equivalent to Theorem 2, with $K$ is torus instead of general compact group.

Proof. By Theorem 1, there is a recurrent point $x_0\in K$. Let $x\in K$ be arbitrary. Then $x=x_0(x_0^{-1}x)$. Let $V$ be a neighbourhood of $x$. Then $Vx^{-1}x_0$ is a neighbourhood of $x_0$. So there exists $n\geq 1$ such that $a^nx_0\in Vx^{-1}x_0$, i.e. $a^nx\in V$. $\Box$

Now we talk about how to construct a new dynamical system by using some given dynamical system(s).

Definition 3 Let $(X,G)$ and $(Y,G)$ be two dynamical systems with the same semigroup $G$ of operators. A homomorphism from $(X,G)$ to $(Y,G)$ is given be a continuous map $\phi:X\rightarrow Y$ satisfying

$\phi(gx)=g\phi(x)\text{ for all } x\in X, g\in G$.

(Note that the products $gx$ and $g\phi(x)$ refer to the action of $G$ on two different spaces.) A dynamical system $(Y,G)$ is a factor of the dynamical system $(X,G)$ if there exists a homomorphism $\phi$ from $(X,G)$ onto $(Y,G)$. In this case we also say that $(X,G)$ is an extension of $(Y,G)$.

Phenomena taking place on $(X,G)$ generally carry over to factors of $(X,G)$. For example:

Proposition 3 If $\phi$ determines a homomorphism of a cyclic system $(X,T)$ to $(Y,T)$ and $x\in X$ is recurrent for $(X,T)$, then $\phi(x)$ is recurrent for $(Y,T)$.

Proof. Let $x\in X$ be recurrent. Let $V$ be a neighbourhood of $\phi(x)$. By continuity of $\phi$, $\phi^{-1}(V)$ is a neighbourhood of $x$. Since $x$ is recurrent, there exists $n\geq 1$ such that $T^nx\in\phi^{-1}(V)$. Hence $\phi(T^nx)\in V$, and so $T^n\phi(x)\in V$. Therefore $\phi(x)$ is recurrent for $(Y,T)$. $\Box$

We can also form some special product:

Definition 4 Let $(Y,T)$ be a dynamical system, $K$ be a compact group and $\psi:Y\rightarrow K$ be continuous. Let $X=Y\times K$. Define $T:X\rightarrow X$ by $T(y,k)=(Ty,\psi(y)k).$ Then $(X,T)$ is called a group extension of $(Y,T)$, or sometimes called a skew product of $(Y,T)$ with $K$.

Remark: If $(X,T)=(Y\times K,T)$ is a group extension, then the elements of $K$ act on $X$ by right translation, i.e. $R_{k'}(y,k)=(y,kk')$. It is easy to show that $R_{k'}\circ T=T\circ R_{k'}$. As a result $R_{k'}$ is an automorphism of the system $(X,T)$.

The reason of forming such a weird product because we want to carry the good property of Kronecker system to the extended dynamical system:

Theorem 4 Let $y_0\in Y$ be recurrent for $(Y,T)$, and let $(X,T)$ be a group extension of $(Y,T)$, $X=Y\times K$. Then each of the points $(y_0, k_0)$, where $k_0\in K$, is a recurrent point of $(X,T)$.

Proof. Let $e$ denote the identity of $K$. We claim that $(y_0, e)$ is a recurrent point of $(X,T)$. Let $x\in X$ and let $Q(x)=\overline{\{T^nx:n\geq 1\}}$. Note that $x$ is recurrent if and only if $x\in Q(x)$. Now, since $y_0$ is recurrent for $(Y,T)$, $y_0\in\overline{\{T^ny_0:n\geq 1\}}$. Notice that $T^n(y_0,e)=(T^ny_0,\psi(y_0)^n)$, and so $T^n(y_0,\psi(y_0)^{-n})=(T^ny_0,e)$. Therefore $(y_0,k_1)\in Q(y_0,e)$, where $k_1=\psi(y_0)^{-n}$. Now $R_{k_1}(y_0,k_1)\in R_{k_1}Q(y_0,e)=QR_{k_1}(y_0,e)=Q(y_0,k_1)$, where the first equality comes from the remark above, we have $(y_0,{k_1}^2)\in Q(y_0, k_1)$. Using the fact that the relation “$x'\in Q(x)$” is transitive, we obtain that $(y_0, {k_1}^2)\in Q(y_0,e)$, and thus by induction we have $(y_0, {k_1}^n)\in Q(y_0,e)$ for all $n\in\mathbb{N}$. By Theorem 2 and the fact that $Q(y_0,e)$ is closed, we have $(y_0,e)\in Q(y_0,e)$,  i.e., $(y_0,e)$ is recurrent. By Proposition 3, $R_{k_0}(y,e)$ is recurrent, i.e. $(y_0,k_0)$ is recurrent. $\Box$

Some examples of non-Kronecker dynamical systems for which every point is recurrent:

1. Let $\mathbb{T}^2=\{(\theta,\phi):\theta,\phi\in\mathbb{R}/\mathbb{Z}\}$, and let $T:\mathbb{T}^2\rightarrow\mathbb{T}^2$ be defined by $T(\theta,\phi)=(\theta+\alpha,\phi+2\theta+\alpha)$, where $\alpha\in\mathbb{R}$ is fixed.
Notice that $(\mathbb{T}^2,T)$ is a group extension of the Kronecker system on the circle $T\theta=\theta+\alpha$, with $\psi(\theta)=2\theta+\alpha$. By Theorem 2 and Theorem 4, every point of $\mathbb{T}^2$ is recurrent. In particular, $(0,0)$ is recurrent. Now consider the orbit
$(0,0)\mapsto (\alpha,\alpha)\mapsto (2\alpha,4\alpha)\mapsto\cdots\mapsto (n\alpha,n^2\alpha)\mapsto\cdots$.
As a by-product, since $(0,0)$ is recurrent, for any $\varepsilon>0$, we can solve the Diophantine inequality $|\alpha n^2-m|<\varepsilon$, or $|\alpha-\frac{m}{n^2}|<\frac{\varepsilon}{n^2}$, becasue the second entry can be arbitrarily close to $0$ in $\mathbb{T}$, i.e. an integer.
2. More generally, let $p(x)$ be a polynomial of degree $d$ with real coefficients. Write $p_d(x)=p(x)$ and form successively $p_{d-1}(x)=p_d(x+1)-p_d(x)$, $p_{d-2}(x)=p_{d-1}(x+1)-p_{d-1}(x)$, and so on. Then $p_i(x)$ is of degree $i$. Let $\alpha$ be the constant $p_0(x)$. Define $T:\mathbb{T}^d\rightarrow\mathbb{T}^d$ by $T(\theta_1,\theta_2,\theta_3,\cdots,\theta_d)=(\theta_1+\alpha,\theta_2+\theta_1,\theta_3+\theta_2,\cdots,\theta_d+\theta_{d-1})$. This defines a group extension of a dynamical system on the $(d-1)$-torus, which in turn is a group extension of a dynamical system om the $(d-2)$-torus, and so on. Consider the orbit of the point $(p_1(0),p_2(0),\cdots,p_d(0))$. Since $p_i(n)+p_{i-1}(n)=p_i(n+1)$, we obtain that $T(p_1(n),p_2(n),\cdots,p_d(n))=(p_1(n+1),p_2(n+1),\cdots,p_d(n+1))$, and so $T^n(p_10),p_2(0),\cdots,p_d(0))=(p_1(n),p_2(n),\cdots,p_d(n))$. Therefore $p_d(n)$ comes arbitrary close to $p(0)$ modulo 1. Thus, again as a by-product, if $p(x)$ is a real polynomial with $p(0)=0$, then for any $\varepsilon>0$, we can solve the Diophantine inequality $|p(n)-m|<\varepsilon$.
In the proof of Theorem 1, do you mean $Y_n \supset \bigcap _{m\in \mathbb N} Y_m$?