What kind of $f(n): \mathbb{N} \to \mathbb{N}$'s make the ff statement true? What kind don't?
$\limsup A_{f(n)} \subseteq \limsup A_n$ where $n \in \mathbb{N}$ (*)
Well obviously the answers to each are:
$(f(n) \ | \ \limsup A_{f(n)} \subseteq \limsup A_n)$
and
$(f(n) \ | \ \limsup A_{f(n)} \subsetneq \limsup A_n)$
respectively, but I am wondering about some possible subsets of each.
For instance
$3n^2 + 5n \in (f(n) \ | \ \limsup A_{f(n)} \subseteq \limsup A_n)$?
or in general
$(f(n) = \sum_{i=0}^{m} a_in^i\ | a_in^i \in \mathbb{N}, m \in \mathbb{N}) \subseteq (f(n) \ | \ \limsup A_{f(n)} \subseteq \limsup A_n)$? If this is right then I guess all polynomial functions whose range $\in \mathbb{N}$ make (*) true.
What if $m = \infty$?
What about continuous functions whose range $\in \mathbb{N}$?
What about some counterexamples to (*)? I think a simple piecewise or discontinuous function will do, but I can't think of any.
Not a full answer to your question, but maybe this can help.
For any $x$ define $A_{x}:=\left\{ n\in\mathbb{N}\mid x\in A_{n}\right\} $ and $B_{x}:=\left\{ n\in\mathbb{N}\mid x\in A_{f\left(n\right)}\right\} $.
Then $x\in\limsup A_{n}$ if and only if $A_{x}$ is infinite, and $x\in\limsup A_{f\left(n\right)}$ if and only if $B_{x}$ is infinite.
Note that $n\in B_{x}\iff x\in A_{f\left(n\right)}\iff f\left(n\right)\in A_{x}$ showing that $B_{x}=f^{-1}\left(A_{x}\right)$.
Now we can say that $\limsup A_{f\left(n\right)}\subseteq\limsup A_{n}$ is true iff the implication: $$A_{x}\text{ is finite}\Rightarrow f^{-1}\left(A_{x}\right)\text{ is finite}$$ is true for each $x$.
It is immedeately clear that $f$ will satisfy this condition if it is injective and this can be weakened.