Yes, you are right, I should have said characteristic set. I have modified that.

Thanks, that means also that it is the most oscillatory situation. Is your guess grounded on some homogeneity property of some sort? Do you know if there are some examples such that there exists $\alpha$ as in my question Q1?

Kats Thanks for your interest in my question. If $1=\cos z$ and $w=e^{iz}$, you get $w+\frac{1}{w}=2$ (not $2i$).