Suppose that $j,k:V_{\lambda}\rightarrow V_{\lambda}$ are elementary embeddings. Let $\mathrm{crit}_{n}(j,k)$ denote the $n$-th element in $\{\mathrm{crit}(\ell)\mid\ell\in\langle j,k\rangle\}$. Define $\alpha=\mathrm{crit}(j),\beta=\mathrm{crit}(k)$. Suppose that $\alpha\neq\beta.$ Let $R=\{\alpha\}\cup\{\ell(\alpha):\ell\in\langle j,k\rangle\},S=\{\beta\}\cup\{\ell(\beta):\ell\in\langle j,k\rangle\}$, and define $R'=\{n\in\omega:\mathrm{crit}_{n}(j,k)\in R\},S'=\{n\in\omega:\mathrm{crit}_{n}(j,k)\in S\}$. Then $\omega=R'\cup S'$.
Define $$\overline{d}(T)=\limsup_{n\rightarrow\infty}\frac{|T\cap n|}{n},$$ $$\underline{d}(T)=\liminf_{n\rightarrow\infty}\frac{|T\cap n|}{n},$$ $$d(T)=\lim_{n\rightarrow\infty}\frac{|T\cap n|}{n}$$ if it exists.
I want to know what combinations of values $\underline{d}(R'),\overline{d}(R'),\underline{d}(S'),\overline{d}(S')$ are possible. Is it possible to have $0<\overline{d}(R')<1$? Is it possible to have $0<\underline{d}(R')<1$? I am especially interested in the case when $R\cap S=\emptyset$.
