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Given a vector of complex number $\vec{z}=(z_1,\cdots, z_n)$ with $|z_i|=1$ and $z_i$ is not a root of unit, and a vector of complex numbers $\vec{r}=(r_1, \cdots, r_n)$ with $|r_i|=1$. Is it the case that

for any $\epsilon>0$, there exists some $k$ such that $|\vec{z}^k-\vec{r}|<\epsilon$, where $\vec{z}^k=(z_1^k, \cdots, z_n^k)$ and $|\cdot|$ denotes the module of vectors?

If $n=1$, this is obvious. But how about higher dimension, especially that it might be the case that, say, $\frac{z_1}{z_2}$ is a root of unit.

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If no pair of $z_i$ have angles whose multiples are ever rational multiples of one another modulo $\pi$ then it is true. I'm not sure what happens if two $z_i$ have angles that are rational multiplies of each other...certainly if a pair of $z_i$ are equal (have a rational multiplier of $1$ between them) then it's false.

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