Vtyjkyuk

I'm currently reading about Cremer Points and the fact that a generic real number satisfy a Cremer condition of degree $dgeq2$, 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 $lambda=e^2pi iomega$ with $omega$ irrational, we say that $lambda$ satisfies a Cremer condition of degree $dgeq2$ ($Cr_d$) iff $$limsup_ntoinftyfracloglog(frac1lambda^n-1)n>log(d)$$ which is equivalent to $$limsup_ntoinftyfraclog(log(q_n+1))q_n>log(d)$$ 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 $|xi-fracpq|<Phi(q)$ for infinitely many $fracpq$ is a generic set of the real numbers and a subset of irrationals satisfying any of Cremer's condition of degree $dgeq2$ ($Cr_infty$), thus proving that a generic real number satisfies a Cremer condition for all $dgeq2$, but I don't even know how to start proving that.

And last, but not least, could anyone explain me if $Cr_inftycap$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 $$sum_n=0^inftyfraclog(log(q_n+1))q_n<infty$$ ?

Thank you very much!