Exercise 6, § 3

Let $(a_j)_{1 \leq j \leq n}$ be a finite sequence of complex numbers such that $|a_j| > 1$ for every $j$. For each of the $2^n$ sequences $s = (\varepsilon_j)_{1 \leq j \leq n}$ where $\varepsilon_j = 1$ or $\varepsilon_j = -1$, put $x_s = \sum_{j=1}^n \varepsilon_j a_j$. Let K be a closed disk of arbitrary center and radius 1 in $\mathbf{C}$. Prove that the number of sequences $s$ such that $x_s \in K$ is $\leq \binom{n}{\lfloor n/2 \rfloor}$. (Let S be the set $\{1, 2, \ldots, n\}$, T the set of the $j \in S$ such that $\Re(a_j)$ and $\Im(a_j)$ are both $\geq 0$ or both $\leq 0$. For every $s = (\varepsilon_j)$, let $U_s \subset S$ be the set of the $j$ such that $\Re(\varepsilon_j a_j) > 0$, and let $X \subset \mathscr{P}(S)$ be the set of the $U_s$ such that $x_s \in K$. Prove that it is not possible that for two distinct sequences $s, s'$, one has $U_s \in X$, $U_{s'} \in X$, $U_s \subset U_{s'}$ and $U_{s'} - U_s \subset T$ or $U_{s'} - U_s \subset S - T$. Apply then Exercise 13 of E, III, p. 85.)