$\newcommand{\m}{\,\operatorname{mod}}$I cite this text, page $38$ - although they reference results from earlier on in the text:
Some definitions:
A topological dynamical system $(K;\phi)$ is a non-empty compact Hausdorff topological space $K$ equipped with a map $\phi:K\to K$ which is continuous in $K$'s topology.
Let $K=[0,1)$ be endowed with the metric $d(x,y)=|e^{2\pi i\cdot x}-e^{2\pi i\cdot y}|=4\sin^2\left(\frac{x-y}{2}\right)$; then $K$ is a compact Hausdorff metric space. For an $x\in\Bbb R$ we write $x\m1=x-\lfloor x\rfloor$. Given $\alpha\in K$, let $\phi(x)=x+\alpha\m1$ be the (continuous) dynamic, and let this topological dynamical system be denoted by $([0,1);\alpha)$.
Let $\Bbb T=\{z\in\Bbb C:|z|=1\}$, and take $a\in\Bbb T$, defining $\phi(z)=a\cdot z$ to obtain an invertible dynamic system, denoted $(\Bbb T;a)$.
$([0,1);\alpha)$ is isomorphic to $(\Bbb T;a)$ when $a=e^{2\pi i\cdot\alpha}$ under the isomorphism $\psi:[0,1)\to\Bbb T,\,x\mapsto e^{2\pi i\cdot x}$.
A group extension of a dynamic system $(K;\phi)$ by a compact topological group $G$, along $\Psi:K\to G$ continuous, is the product topological space $K\times G$ endowed with the dynamic $\psi(x,g)=(\phi(x),\Psi(x)\cdot g)$.
A point $x\in K$ is said to be recurrent if for every open neighbourhood $U$ of $x$, there is $m\in\Bbb N_1$ such that $\phi^m(x)\in U$.
Theorem proved in the text:
Let $(K;\phi)$ be a topological system, $G$ a compact group and $(H;\psi)$ the group extension of $K$ by $G$ along some $\Psi$. If a point $x_0\in K$ is recurrent, then $(x_0,g)$ is recurrent in $H$ for all $g\in G$.
They use this to show that:
Let $\alpha\in\Bbb R,\epsilon\gt0$ be given. Then $\exists n\in\Bbb N_1,m\in\Bbb Z$ such that: $$|n^2\alpha-m|\lt\epsilon$$
Their proof goes along the lines of:
By isomorphy with $(\Bbb T;a)$, the point $x=0$ is recurrent in $([0,1);\alpha)$ (proof is discussed earlier in the text, proposition $3.12$). Consider the group extension of this system, as a group with addition modulo $1$, with itself, along $\Psi:[0,1)\to[0,1),\,x\mapsto2x+\alpha\m1$, giving $H=[0,1)\times[0,1),\,\psi(x,y)=(x+\alpha,y+2x+\alpha)\m1$ as the extended system. As $0$ is recurrent in the original system, $(0,g)$ is recurrent in $H$ for any $g\in[0,1)$ - in particular, $(0,0)$ is recurrent in $(H;\psi)$. Consider the orbit of $(0,0)$ - by induction: $$\psi^n(0,0)=(n\alpha,n^2\alpha)$$
This following part is the part I don't understand:
The recurrence of $(0,0)$ implies that for any $\epsilon\gt0$, there exists $n\in\Bbb N_1$ with $d(0,n^2\alpha-\lfloor n^2\alpha\rfloor)\lt\epsilon\quad\blacksquare$
How exactly am I supposed to understand $d$ here? Recurrence of this point implies that if I pick an open neighbourhood, which in the product metric may be chosen as a ball $B=\{(x,y)\in[0,1)\times[0,1):d((x,y),(0,0))\lt\epsilon\}$, i.e. $\sqrt{d(x,0)^2+d(y,0)^2}\lt\epsilon$ since I am inferring a product metric here. The recurrence of $(0,0)$ implies there is $n$ so that $d((n\alpha,n^2\alpha)\m1,(0,0))\lt\epsilon$, or: $$\sqrt{d(n\alpha-\lfloor n\alpha\rfloor,0)^2+d(n^2\alpha-\lfloor n^2\alpha\rfloor,0)^2}\lt\epsilon$$
Where I am lazily implying by $d$ either the product metric or the metric given at the start, $d(x,y)=|e^{2\pi i\cdot x}-e^{2\pi i\cdot y}|$. The text tells me that this metric is continuous w.r.t. the usual metric on $\Bbb R$, so $d(\cdots)\lt\epsilon\implies|\cdots|\lt\epsilon$, I think.
I can't understand how their conclusion follows. What am I missing? I've been quite confused by this text so far, as someone who doesn't really study topology.
The answer (I think) is that the product topology on metric spaces may be induced by $d_p=\max\{d_1,d_2\}$, and so $d_p\lt\epsilon\implies d_1,d_2\lt\epsilon$, and therefore we have two results:
$$|n\alpha-\lfloor n\alpha\rfloor|\lt\epsilon,|n^2\alpha-\lfloor n^2\alpha\rfloor|\lt\epsilon$$
For some $n\in\Bbb N_1$.