Find $(a_1,...,a_m) \in \mathbb C^{m}, ~ |a_k| = 1$ such that $\lim_{p} \sum_{k=1}^m a_k^p$ exists Let $m\in\mathbb N^*$ be a non zero integer. Let $\mathbb U := \{ z \in \mathbb C : |z|=1\}$ be the unit circle.
Find all the $(a_1,...,a_m) \in \mathbb U^{m}$ such that
$$\lim_{p\to \infty}~~ \sum_{k=1}^m ~a_k^p$$ exists in $\mathbb C$.

Let us re-write $a_k = e^{i\theta_k}$ for a certain $\theta_k\in \mathbb R$. We need some condition on the $\theta_k$.
I only managed to do some things for small $m$:

*

*$m=1$: only $\theta_1 \in 2\pi\mathbb Z $ does;


*$m=2$: it is more difficult here. If this quantity admits a limit, then the real part of it does. Write $\mathcal Re(e^{i\theta_1p} + e^{i\theta_2p}) = 2\cos(p\frac{\theta_1 + \theta_2}{2})\cos(p\frac{\theta_1 - \theta_2}{2})$ (where $\mathcal R e (z)$ denotes the real part of $z$). We can here deduce that it is true for some values that fix $\cos(p\frac{\theta_1 + \theta_2}{2})\cos(p\frac{\theta_1 - \theta_2}{2})$ (for example $\frac{\theta_1 - \theta_2}{2} \in 2\pi \mathbb Z$ and $\frac{\theta_1 + \theta_2}{2} \in 2\pi \mathbb Z$)
I did some other cases but I cannot find any general result.
 A: Let $a_k$ be the roots of $P(z)=z^m+e_1z^{m-1}+\dots +e_{m-1}z + e_m$, then:
$$
a_k^m+e_1a_k^{m-1}+\dots +e_{m-1}a_k + e_m \tag{1} = 0
$$
Let $s_p = \sum_{k=1}^m ~a_k^p$, then multiplying $(1)$ by $a_k^{p-m}$ with $p \gt m$ and summing for $k=1,\dots,m$:
$$
s_p + e_1 s_{p-1}+\dots+e_{m-1}s_{p-m+1} +e_ms_{p-m} = 0 \tag{2}
$$
Assuming the limit exists $\lim_{p\to \infty} s_p = S$, then $(2)$ can be passed to the limit:
$$
S \cdot\left(1 + e_1 + \dots + e_{m-1}+e_m\right) = 0
$$
$$
\iff\;\;\;\; S \cdot P(1) = 0 \tag{3}
$$
If $S \ne 0$ (as proved below) then $(3)$ implies that $P(1) = 0$, so one of the $a_j = 1$. Removing that $a_j^p=1$ from all sums reduces the problem to the case with $m-1$ variables, then it follows by induction that all $a_k=1$.
What's left to prove is that $S \ne 0$, in other words that the limit can never be zero.
In fact, a stronger result is proved at Powers of complex numbers property, which is that for any $\varepsilon \gt 0$ there exist arbitrarily large integers $p$ such that $|s_p| > m - \varepsilon$. The proof uses the simultaneous version of Dirichlet's approximation theorem which has as a consequence that:

given finitely many numbers $x_1,\dots,x_m$ and $\delta>0$, one can find a positive integer $q$ such that each of the numbers $qx_1, qx_2,\dots, qx_m$ differs from some integer by less than $\delta$.

For $x_k = \frac{\theta_k}{2\pi}$ that yields the result, which implies that $\limsup_{p\to\infty} |s_p| = m$, so if $S = \lim_{p\to\infty} s_p$ exists then it must satisfy $|S|=m$, which is consistent with the previous finding.
