SCHWEITZER 2003 Problem 3

Let $Z=\{ z_1,\ldots, z_{n-1}\}$ , $n\ge 2$ , be a set of different complex numbers such that $Z$ contains the conjugate of any its element. a) Show that there exists a constant $C$ , depending on $Z$ , such that for any $\varepsilon\in (0,1)$ there exists an algebraic integer $x_0$ of degree $n$ , whose algebraic conjugates $x_1, x_2, \ldots, x_{n-1}$ satisfy $|x_1-z_1|\le \varepsilon, \ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$ and $|x_0|\le \frac{C}{\varepsilon}$ . b) Show that there exists a set $Z=\{ z_1, \ldots, z_{n-1}\}$ and a positive number $c_n$ such that for any algebraic integer $x_0$ of degree $n$ , whose algebraic conjugates satisfy $|x_1-z_1|\le \varepsilon,\ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$ , it also holds that $|x_0|>\frac{c_n}{\varepsilon}$ .

(translated by L. Erdős)