sum of power of norm 1 complex numbers is frequently big?

Question

Given $\alpha_1,\dots,\alpha_k$ complex numbers of norm 1, is it true that $$s_n = \alpha_1^n + \dots + \alpha_k^n $$ has a subsequence converging to $k$?

I remember having proved this fact long ago, but right now it seems impossible to me.

Answer 1

Yes, this is true. Let $x_n=(\alpha_1^n,\dots,\alpha_k^n)\in (S^1)^k$. Since $(S^1)^k$ is compact, some subsequence of $(x_n)$ converges. So for any $\epsilon>0$, we can choose $N$ such that $|\alpha_m^n-\alpha_m^{n'}|<\epsilon$ for all $n,n'\geq N$ in our subsequence and $m=1,\dots,k$. Letting $d=n'-n$ be as large as we want, we then find that there are arbitrarily large $d$ such that $|\alpha_m^{d}-1|<\epsilon$ for all $m$. Choosing an increasing sequence of such $d$ while $\epsilon$ ranges over a sequence going to $0$, we get a subsequence of $(s_n)$ which converges to $k$.