Let, $A\subseteq\{z=(z_1,z_2)\in\mathbb{C}^2:|z|^2=|z_1|^2+|z_2|^2=1\}$ such that any two vectors in $A$ have angle between them $\ge\alpha$ for some $0<\alpha<1$. I want to prove that $$\#A=O(\alpha^{-1}) \ \ for \ \ \epsilon>0.$$ If the whole problem is in $\mathbb{R}^2$ instead of $\mathbb{C}^2$, then by a careful use of pigeon-hole principle I can prove that $\#A=O(\alpha^{-1})$, while I have no idea how to prove in complex case.
Thanks in advance.