Suppose that the identity $$e^{i \theta_1} + \ldots + e^{i \theta_n} = e^{i k \theta_1} + \ldots + e^{i k \theta_n}$$ holds true $\forall k \neq 0$.
Is then true that we must have $\theta_i = 0$, $\forall i$?
I'm having a hard time seeing why it has to be true.
The condition $\forall k > 0 $ seems wrong to me. Indeed, restricting to a simpler case:
$$e^{i\theta_1} + e^{i\theta_2} = e^{ik\theta_1} + e^{ik\theta_2}$$
Choosing $\theta_1 = 0$ and $\theta_2 = \pi/2$ you get
$$-1 = e^{ik\pi/2}$$
Which doesn't hold unless $k\in\Omega \subset \mathbb{Z}$, $k = \{\pm 2, \pm 6, \pm 10, \pm 14, \ldots\}$
Now reason in reverse mode: for this choice of $k$ which as you see is not zero, $\theta_1$ and $\theta_2$ are not both zero.
Note: I won't delete this answer for I wrote it whilst the question was asked in a lightly blurry way.