I am asked to find when, for some nonzero complex numbers $\alpha,\beta,\gamma$, the set defined by $\{l\alpha+m\beta+n\gamma|l,m,n\in \mathbb{Z}\}$ is a lattice. Assuming it is a lattice, I constructed a homomorphism between $\mathbb{Z}$-modules, and I think I successfully deduced that thus would imply that $a\alpha+b\beta+c\gamma=0$ where not all of $a,b,c$ are zero (and at most one pair is dependent).
I believe the converse is true: if $a\alpha+b\beta+c\gamma=0$ then they generate a lattice. I proved that if two are dependent this is true. However, I am having a very hard time proving that if all three are dependent this is true.
My intuition is that if $a\alpha+b\beta=-c\gamma$, then the set $\{l\alpha+m\beta+n\gamma|l,m,n\in \mathbb{Z}\}$ is simply the lattice generated by $\alpha$ and $\beta$ contracted by a factor of $1/c$ in the direction parallel to $\gamma$. I'm not completely sure how to put this into math, let alone prove it. I feel like my method is leading to some kind of trigonometry, but I feel like there must be a better way to prove this. Is the statement even true?