Two remarks on Merimovich's model of the total failure of GCH

Let $M$ denote the Merimovich’s model in which for each infinite cardinal $\lambda ,2^{\lambda }={\lambda }^{+3}$. We show that in $M$ the following hold: (1) Shelah’s strong hypothesis fails at all singular cardinals, indeed, $\mathrm{\forall }\lambda (\lambda$ is a singular cardinal $\Rightarrow pp(\lambda)=\lambda^{+3}).$(2)Foreachsingularcardinal$\lambda$thereisaninnermodel$N$of$M$suchthat$M$and$N$havethesameboundedsubsetsof$\lambda,$ $\lambda$ is a singular cardinal in $N$, $({\lambda }^{+i}{)}^N=({\lambda }^{+i}{)}^M$, for $i=1,2,3,$ and $N\models 2^{\lambda }={\lambda }^{+}$. Thus it is possible to add many new fresh subsets to $\lambda$ without adding any new bounded subsets to $\lambda$.