Let $N \in \mathbb{N}$. The nonempty subset $P \subseteq \{0,1,\cdots,N-1\}$ satisfies $$ \sum_{k \in P} e^{2\pi i \frac{k}{N}}=0. $$
Example:
If $N$ is prime, then the only possibility is that $P=\{0,1,\cdots,N-1\}$.
If $N=pq$, then the set $P$ can be $\{0,q,2q,\cdots,(p-1)q\}$.
If $N=2m$, then the set $P$ can be $\{j,m+j\}$ for $0\le j<m$.
Question: can we describe all possibilities of the set $P$?