I came up with the following problem. Given mutually different $\theta_1,\ldots,\theta_3\in\mathbb R-\{0\}$, does there exist $T\in \mathbb R$ such that $$e^{i\theta_1 T} + e^{i\theta_2 T} + e^{i\theta_3 T}=0?$$
For simpler problem of determining the existence of the solution of $e^{i\theta_1 T} + e^{i\theta_2 T}=0$, the answer is positive by taking $T$ such that $(\theta_1-\theta_2)T \in 2\pi\mathbb Z + \pi$. However, I could not find the answer if more terms are added.
It would be more fun if the answer can address the general case where 3 is replaced by $n$.