BALKAN-SL 2008 A4

We consider the set \begin{align*} \mathbb{C}^{\nu} = { (z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C} }, \nu \geq 2 \end{align*} and the function $\phi : \mathbb{C}^{\nu} \longrightarrow \mathbb{C}^{\nu}$ mapping every element $(z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C}^{\nu}$ to \begin{align*}\phi ( z_1,z_2, \ldots , z_{\nu})= \left( z_1-z_2, z_2-z_3, \ldots, z_{\nu}-z_1 \right) \end{align*} We also consider the $\nu-$ tuple $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$ $\in \mathbb{C}^{\nu}$ of the $n-$ th roots of $-1$ , where \begin{align*} \omega_{\mu} = \cos \left( \frac{\pi + 2\mu \pi }{\nu} \right) + \iota \sin \left( \frac{\pi + 2\mu \pi}{\nu} \right) \mu =0,1, \ldots , \nu -1 \end{align*} Let after $\kappa$ (where $\kappa$ $\in$ $\mathbb{N}$ ), successive applications of $\phi$ to the element $(\omega_0, \omega_1, \ldots , \omega_{\nu-1} )$ , we obtain the element \begin{align*} \phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right) =\left( Z_{\kappa 1}, Z_{\kappa 2}, \ldots , Z_{\kappa \nu } \right) \end{align*} Determine [list=i] [] the values of $\nu$ for which all coordinates of $\phi ^{(\kappa)} \left( \omega_0, \omega_1, \ldots , \omega_{\nu-1} \right)$ have measures less than or equal to $1$ [] for $\nu =4$ , the minimal value of $\kappa \in \mathbb{N}$ , for which \begin{align*} \mid Z_{\kappa i} \mid \geq 2^{100} 1 \le i \le 4 \end{align*}