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

Denote by $\mathbb{N}$ and $\mathbb{P}$ the set of all positive integers and prime numbers, respectively. Let $\mathbb{P}=\{p_1<p_2<\cdots <p_n<\dots \}$, where $p_n$ is the $n$-th prime number. For $k\in \mathbb{N}$ we recursively define subsequences $(p_n^{(k)}{)}_{n=1}^{+\mathrm{\infty }}$ of the sequence $(p_n{)}_{n=1}^{+\mathrm{\infty }}$ 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 ${\mathbb{P}}_k=\{p_1^{(k)}<p_2^{(k)}<\cdots <p_n^{(k)}<\dots \}$, ${\mathbb{P}}_n^{\mathrm{T}}=\{p_n^{(1)}<p_n^{(2)}<\cdots <p_n^{(k)}<\dots \}$ and $\text{Diag}\mathbb{P}=\{p_1^{(1)}<p_2^{(2)}<\cdots <p_k^{(k)}<\dots \}$ 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.