I'm currently reading about Cremer Points and the fact that a generic real number satisfy a Cremer condition of degree $d\geq2$, but I fail to see some of the results and some help would be much appreciated.
Let me first introduce the notation I'm following. If we consider $\lambda=e^{2\pi i\omega}$ with $\omega$ irrational, we say that $\lambda$ satisfies a Cremer condition of degree $d\geq2$ ($Cr_{d}$) iff $$\lim\sup_{n\to\infty}\frac{\log\log(\frac{1}{|\lambda^{n}-1|})}{n}>\log(d)$$ which is equivalent to $$\lim\sup_{n\to\infty}\frac{\log(\log(q_{n+1}))}{q_{n}}>\log(d)$$ where $q_{n}$ are the denominators of $\omega$'s continued simple fraction convergents.
First of all, I find myself uncable of proving this very equivalence. Then I read that I could consider $\Phi(q)=2^{-q!}$ and prove that the set of irrational $\xi$ such that $|\xi-\frac{p}{q}|<\Phi(q)$ for infinitely many $\frac{p}{q}$ is a generic set of the real numbers and a subset of irrationals satisfying any of Cremer's condition of degree $d\geq2$ ($Cr_{\infty}$), thus proving that a generic real number satisfies a Cremer condition for all $d\geq2$, but I don't even know how to start proving that.
And last, but not least, could anyone explain me if $Cr_{\infty}\cap$PM is empty and, if so, does an irrational number either belong to $Cr_{\infty}$ or PM? Where by PM I mean the Pérez-Marco numbers, that is all irrational numbers such that the denominators of the convergents of their simple continued fraction $q_{n}$ satisfy $$\sum_{n=0}^{\infty}\frac{\log(\log(q_{n+1}))}{q_{n}}<\infty$$ ?
Thank you very much!