Let $j:V_{\lambda}\rightarrow V_{\lambda}$ be an elementary embedding. Then $\{\mathrm{crit}(k)\mid k\in\langle j\rangle\}$ has order type $\omega$, so let $\mathrm{crit}_{n}(j)$ denote the $n$-th element of the set $\{\mathrm{crit}(k)\mid k\in\langle j\rangle\}$. Let $X$ be the set of all elementary embeddings $k:V_{\lambda}\rightarrow V_{\lambda}$ such that for all $\alpha<\lambda$, there is an $\ell\in\langle j\rangle$ where $\ell|_{V_{\alpha}}=k|_{V_{\alpha}}$.
Suppose that $j_{1},\dots,j_{r}\in X\setminus\{j\}$. Let $T=\{n\mid\mathrm{crit}_{n}(j)\in\langle j_{1},\dots,j_{r}\rangle\}$.
Define $$\overline{d}(T)=\limsup_{n\rightarrow\infty}\frac{|T\cap n|}{n}.$$ Let $f_{T}:\omega\rightarrow\omega$ be unique strictly increasing function where $f_{T}[\omega]=T$.
Is it possible for $\overline{d}(T)>0$?
What are the possible growth rates of the function $f_{T}$?
I am also interested in the above two questions in the special case when $j_{1}=t_{1}(j),\dots,j_{r}=t_{r}(j)$ for terms $t_{1},\dots,t_{r}$ in the language with function symbols $*,\circ.$
These question also makes sense when $k_{1},\dots,k_{g}:V_{\lambda}\rightarrow V_{\lambda}$ are elementary and $X$ is replaced with the collection of all elementary embeddings $\ell:V_{\lambda}\rightarrow V_{\lambda}$ such that for all $\alpha<\lambda$, there is some $k\in\langle k_{1},\dots,k_{g}\rangle$ where $k|_{V_{\alpha}}=\ell|_{V_{\alpha}}.$
