Suppose that $x_i \in [-1,1], i=0,n-1$ and consider the root of unity $\omega = \cos(2\pi/n)+i\sin(2\pi/n)$ for some $n \geq 2$. Consider complex numbers of the form $$ z = \sum_{i=0}^{n-1} x_i \omega^{i}$$

Are there any known upper bounds for $|z|, |\operatorname{Re} z|, |\operatorname{Im} z|$?

I came across this type of questions when evaluating some numerical errors. Some trivial estimations show that for $n=3$ one should have $\operatorname{Re} z \leq 2$, for example. I was wondering if the bounds are known for general $n$.

Suppose that $x_i \in [-1,1], i=0,n-1$ and consider the root of unity $\omega = \cos(2\pi/n)+i\sin(2\pi/n)$ for some $n \geq 2$. Consider complex numbers of the form $$ z = \sum_{i=0}^{n-1} x_i \omega^{i}$$

Are there any known lowest upper bounds for $|z|, |\operatorname{Re} z|, |\operatorname{Im} z|$?

I came across this type of questions when evaluating some numerical errors. Some trivial estimations show that for $n=3$ one should have $\operatorname{Re} z \leq 2$, for example. I was wondering if the bounds are known for general $n$.