Why does it seem that the BMOSL G's are much much much easier than the A's and N's.

I don’t even understand what the question is asking half the time. I’m concerned this doesn’t have a single solution on AoPS because then my solution could be a fakesolve. Let $\omega=\operatorname{cis}\left(\frac{2\pi}{\nu}\right)$, and $\tau=\operatorname{cis}\left(\frac{\pi}{\nu}\right)$. Notice now that it is kinda clear that $\phi^{(k)}(\omega_0,\omega_1,\dots,\omega_{\nu-1})=\tau(1-\omega)^k(\omega^0,\omega^1,\dots,\omega^{\nu-1})$, so clearly we just need $|1-\omega|\le1$ for (a). Doing geometry, if $\theta=\frac{2\pi}{\nu}$ then $|1-\omega|^2=|1|^2+|\omega|^2-2\cos(\theta)|1||\omega|=2(1-\cos(\theta))$, so $\cos(\theta)\ge\frac{1}{2}$, and then we get that $\nu\ge6$ for (a). For (b), we just need $|1-\omega|^k=2^{\frac{k}{2}}$, so we just need $k\ge2\cdot100=200$ for (b) lol.

AlastorMoody wrote: 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*} such periphrasis is to be shunned