For each $n\in \mathbf{N}$, we place a set of complex numbers $E_{n}$ s.t.
$E_{n}:=\{E, E=F(exp(2\pi k i/n)),k\in\mathbf{N},1\leq k\leq n\}$.
For each $n$, elements of $E_{n}$ are pairwise distinct.
$F$ is some fixed function, and is holomorphic over the area we discuss.
Let $d(n)$ be the nearest distance among that of between each pair of $E$ in $E_{n}$. We know that it has an $O(\frac{1}{n})$ upper bound. Is it possible to state an order for its lower bound, without knowing the exact form of $F$? i.e. state the big O order of $\frac{1}{d(n)}$.
NB: I find cases considering $F$ self intersecting quite annoying. Hence answers only discussing adjacent $k$ are welcome.