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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 $n$.

Laplacian
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    The summands must form an equilateral triangle, which gives conditions on the differences $\theta_i - \theta_j$. – WimC Jan 04 '22 at 13:11
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    $(\theta_2-\theta_1)T=2\pi n \pm 2\pi/3$, $(\theta_3-\theta_1)T=2\pi m\pm 4\pi/3$ $\Rightarrow$ $\frac{\theta_2-\theta_1}{\theta_3-\theta_1}=\frac{3n\pm 1}{3m\pm 2}$, $n,m\in\mathbb{Z}$. – Ivan Kaznacheyeu Jan 04 '22 at 14:30
  • There is a solution $T$, but the solution need not be real. – GEdgar Jan 04 '22 at 16:06

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