Given an $n \times n$ integer grid I chose any two grid points $a,b$, draw a line $l$ through $a$ and $b$ and measure the angle between $l$ and a horizontal line. I can do this for any grid point pair and I'll get a set $A$ of angles.
I am interested in finding the largest angle $\alpha$ s.t. every element in $A$ is an integer multiple of $\alpha$.
My question is if it is possible to give a good lower bound on $\alpha$? Maybe this goes into the topic of approximating non-integer numbers using Lattices?