On distribution of subsequences of primes having prime indices with respect to the $(R)$-denseness and convergence exponent

Abstract: Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}={p_1<p_2<\dots <p_n<\dots}$, where $p_n$ is the $n$-th prime number. For $k\in\mathbb{N}$ we recursively define subsequences $(p^{(k)}n){n=1}^{+\infty}$ of the sequence $(p_n){n=1}^{+\infty}$ in the following way: let $p_n^{(1)}=p_n$ and $p_n^{(k+1)}=p{p_n^{(k)}}$. In this paper we study and describe some interesting properties of the sets $\mathbb{P}k={p_1^{(k)}<p_2^{(k)}<\dots<p_n^{(k)}<\dots}$, $\mathbb{P}_n^{\mathrm{T}}={p_n^{(1)}<p_n^{(2)}<\dots<p_n^{(k)}<\dots}$ and $\text{Diag}\mathbb{P}={p^{(1)}_1<p^{(2)}_2<\dots <p^{(k)}_k<\dots}$ and their elements, for $k,n\in\mathbb{N}$. Especially, we check whether these sets have dense sets of ratios in $\mathbb{R}+$. Moreover, we compute their exponents of convergence and asymptotics of their counting functions.