Consider the set of real numbers $\Omega=\left\{\omega_1,\omega_2,\dots\omega_k\right\}$ such that:
$$\omega_1<\omega_2\dots<\omega_k,$$
and the family of intervals:
$$I_j=\left(\frac{2n\pi}{\omega_j},\frac{(2n+1)\pi}{\omega_j}\right),\text{ for $j$ even, }$$
$$I_j=\left(\frac{(2n+1)\pi}{\omega_j},\frac{(2n+2)\pi}{\omega_j}\right),\text{ for $j$ odd, }$$
where $n\in\mathbb{N}\cup\{0\}$. Compute the intersection:
$$I=\bigcap_{j=1}^kI_j.$$
Furthermore, the conditions over $\Omega$ such that $I\not=\emptyset$.
Note: Basically, a family of periodic sets which period decreases as $K$ increases. Also, it has this interlacing property for $j$ even or odd. The original problem comes from trying to solve for $\alpha$ the following family of inequalities: $$(-1)^j\sin(\alpha\omega_j)<0.$$
I will appreciate any insight or recommendation to solve the problem. Either (preferable) to solve it analytically or to develop an algorithm.
Thanks for your attention!