Let $z\_1,\ldots,z\_n$ be complex numbers of modulus one. Does it exist an increasing sequence $k\_j\in\mathbb{N}$ such that $\lim\_{j\to\infty}z\_i^{k\_j}=1$ for all i?



Simultaneous convergence of powers of unit complex numbers
Yes. Let $z=(z\_1,...,z\_n)$, a point on the torus $(S^1)^n$. Since the torus is compact, the sequence $z^m$ has a convergent subsequence $z^{m\_1}$, $z^{m\_2}$,..., and we may choose it so that $k\_j=m\_{j+1}-m\_j$ are nonnegative and increasing. Then ${\rm lim} z\_i^{k\_j}=1$ for all $i$. 


