cv.complex variables - Simultaneous convergence of powers of unit complex numbers

Yes.

A standard lemma is that some element of ${alpha, 2alpha, ..., malpha}$ is within $1/(m+1)$ of 0 mod 1, since otherwise, there would have to be two multiples of $alpha$ between some $k/(m+1)$ and $(k+1)/(m+1)$ which means their difference would be close to 1.

The same pigeonhole argument works on $(S^1)^n$. Consider the first $m^n+1$ multiples of $(alpha_1,...alpha_n)$. Two must be in the same $n$-dimensional box $[k_1/m,(k_1+1)/m]times...[k_n/m,(k_n+1)/m]$ which means their difference is a multiple $t_mtimes(alpha_1,..,alpha_n)$ within $1/m$ of 0 on each coordinate.

If you want more details and better approximations, then there are some multidimensional versions of simple continued fractions which might work, but this suffices to show that a sequence of integers ${t_m}$ exists so that $z_i^{t_m}$ is within $2pi/m$ of 1.

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