Given a positive integer $n$, define $\zeta = e^{2\pi i/n}$ and define $s: \mathbb Z^n \to \mathbb C$$$s(\vec x) = \sum_{k=0}^{n-1} x_k \zeta^k$$
Let us consider the set $S = \{ |s(\vec x)| : \vec x \in \{0,1\}^n\} \setminus \{0\}$. Is it possible to find an explicit $\vec x \in \{0,1\}^n \setminus \{ \vec 0\}$ such that $|s(\vec x)|$ is minimal, that is $|s(\vec x)| = \min S$?
Or alternatively is there an known lower bound for $\min S$?
My thought was to find a more general result for $T = \{ |s(\vec x)| : \vec x \in \mathbb Z^n\} \setminus \{ 0 \}$ instead of $S$, but a lower bound of this set there would probably be worse than one for $S$.