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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.

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  • $\begingroup$ That's not a saving, is it, if $\epsilon>0$? $\endgroup$ Oct 7, 2014 at 6:16
  • $\begingroup$ Yes it is positive. $\endgroup$
    – Kunnysan
    Oct 7, 2014 at 7:08
  • $\begingroup$ Then it's not a saving, right? $O(\alpha^{-1})$ is smaller than $O(\alpha^{-(1-\epsilon)}$. $\endgroup$ Oct 7, 2014 at 8:09
  • $\begingroup$ Well, I actually thought about $\alpha<1$ case. Then it would be a saving indeed. Sorry about that. $\endgroup$
    – Kunnysan
    Oct 7, 2014 at 17:09
  • $\begingroup$ If $\alpha<1$ then isn't $O(1)$ better than $O(\alpha^{-1+\epsilon})$? $\endgroup$ Oct 7, 2014 at 22:35

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