Let $z_1,\ldots,z_n\in \mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence of $$s_k=\sum_{1\leq i\leq n}z_i^k$$ contains infinitely many $(n-1)$-tuples of consecutive values of $s_k$ which are all $0$. Then (essentially) $$z_j=e(j/n),$$ where $e(x)=e^{2\pi ix}$.

Let $z_1,\ldots,z_n\in \mathbb{C}$ be a sequence such that $z_1=1$. Suppose that the sequence of $$s_k=\sum_{1\leq i\leq n}z_i^k$$ contains infinitely many $(n-1)$-tuples of consecutive values of $s_k$ which are all $0$. Then (essentially) $$z_j=e(j/n),$$ where $e(x)=e^{2\pi ix}$.

A conjecture of Turán.

This is true (in the stronger form with only two such tuples) - in fact if $n$ is odd then the $z_i$ must be exactly the $n$th roots of unity, and if $n$ is even they must be the vertices of two regular $(n/2)$-gons with the same circumscribed circle centred at the origin. This was first proved by Tijdeman [Ti66]. An independent proof of this was given in the comments section by Hu, Tang, and Zhang.