I have some problems to understand the following situation:
Let $A=\{z\in\mathbb C : |z|\ge 1$ and $|\Re(z)|\le\frac{1}{2}\}$, then the inequality $|cz+d|\le 1$ with $c,d\in\mathbb Z$ doesn't have solutions in $A$ if $|c|\ge 2$.
I have tried to write $z=x+iy$ so $|cz+d|^2=c^2(x^2+y^2)+d^2+2cdx\ge 5+2cdx...$
but probably this is the wrong way because there is the obnoxious term $2cdx$.