Let $(\C,+)$ be the additive group of complex numbers and $z\in\C\sm\R$ with $

|z|>1$.

For each $k\in N$, let $I_k'(z)$ be the set of all complex numbers

 of a form $\al_1z^{k_1}+\al_2z^{k_2}+\cdots+\al_nz^{k_n}$,

 where $\al_i\in\Z$, $k_i\in N$ $(i=1,2,\cdots,n)$, $k\leq k_1<k_2<\cdots<k_n$

 and $n\in N$.

We prove that $\inf\{|w|:w\in I'_k(z)\}\to\infty$

 $(k\to\infty)$ if and only if $z$ is an algebraic integer with degree 2.

In this case, we can easily define a metrizable group topology $\tau$ on

 $(\C,+)$ such that the sequence $\{z^n:n\in N\}$ converges to $0$ in the topo

logical group

 $(\C,+,\tau)$.