MathDB

We consider the set \begin{align*} \mathbb{C}^{\nu} = \{ (z_1,z_2, \ldots , z_{\nu}) \in \mathbb{C} \},\qquad \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) \qquad \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
[*] 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} \qquad \qquad 1 \le i \le 4 \end{align*}

Problem Statement