In 1970th, Magidor proved the following important results:
(1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$ is strong limit and $2^{\aleph_\omega}=\aleph_{\omega+k+1},$ where $0< k \leq \omega$.
(2) Assuming the existence of a supercompact cardinal and a huge cardinal above it, it is consistent that $GCH$ holds below $\aleph_\omega$ and $2^{\aleph_\omega}=\aleph_{\omega+2}$.
On the other hand Shelah has proved the following strengthening of (1).
(3) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$ is strong limit and $2^{\aleph_\omega}=\aleph_{\omega+\alpha+1},$ where $0< \alpha < \omega_1$.
There are some references stating that motivated by Shelah's result, Magidor has proved the following:
Theorem. Assuming the existence of very large cardinals, it is consistent that $GCH$ holds below $\aleph_\omega$ and $2^{\aleph_\omega}=\aleph_{\omega+\alpha+1},$ where $0< \alpha < \omega_1$.
The proof seems to me uses a supercompact and $\alpha$-many huge cardinals above it.
I wonder to know if any one knows the basic idea of Magidor's result, or if there is any reference I can find the proof.
Remark. By work of Gitik-Magidor on extender based Prikry forcing we can obtain the above theorem in a simpler way and just using a strong cardinal; but I am still interested to see Magidor's original proof.
