- Thread starter
- #1

consider the map

$$|\cdot|:\mathbb{Z}

*\longrightarrow \mathbb{N},\qquad |a+ib|:=a^2+b^2$$*

1) Prove that $|\alpha|<|\beta|$ iff $|\alpha|\leq |\beta|-1$ and $|\alpha|<1$ iff $\alpha=0$

2) Let $\alpha,\beta\in\mathbb{Z}

1) Prove that $|\alpha|<|\beta|$ iff $|\alpha|\leq |\beta|-1$ and $|\alpha|<1$ iff $\alpha=0$

2) Let $\alpha,\beta\in\mathbb{Z}

*,\beta\neq 0$. Prove that the map $f:\mathbb{Z}**\longrightarrow\mathbb{Z}**, f(\gamma):=\alpha-\gamma\beta$ is the composition of a dilatation by the factor $\sqrt{|\beta|}$, a rotation (angle?) and a translation.*

3) Deduce that there exists $\gamma\in\mathbb{Z}3) Deduce that there exists $\gamma\in\mathbb{Z}

*$ such that $|f(\gamma)|$ is strictly smaller than $|\beta|$.*

$\textbf{Hint:}$ compare the size of a cell of the lattice $f(\mathbb{Z}$\textbf{Hint:}$ compare the size of a cell of the lattice $f(\mathbb{Z}

*)$ and the size of the set of points whose distance to $0$ is $\leq\sqrt{|\beta|}$.*

What i did: point 1) is a trivial consequence of the fact that the norm takes integer non negative values. For point 2), I use complex multiplication of numbers which is: multiply absolute values and add angles. For point 3), i'm actually waiting for a miracle... I suppose i should prove that there exists a cell in $f(\mathbb{Z}What i did: point 1) is a trivial consequence of the fact that the norm takes integer non negative values. For point 2), I use complex multiplication of numbers which is: multiply absolute values and add angles. For point 3), i'm actually waiting for a miracle... I suppose i should prove that there exists a cell in $f(\mathbb{Z}

*)$ intersecting the open ball centered at the origin with radius $\sqrt{|\beta|}$, but i have no idea how to write down this. Only thing i noticed is that $f$ acts with a rotation, which does not affect distance from the origin, so that the only changes in $|\gamma|$ come from dilatation and by adding $\alpha$.*

Could someone put me on the right direction? Thanks in advanceCould someone put me on the right direction? Thanks in advance