I am working with subsets of $[n]$ of the form $(A+B)\cap A$, where $A+B$ is a sumset. Namely, I am interested if there are nonempty sets $B$ such that whenever $A$ covers a positive proportion of $[n]$, the set $(A+B)\cap A$ also covers a positive proportion of $[n]$. In fact it would be nice if the proportion is the same, or tends to the same limit. (What I mean by "positive proportion" only really makes sense when we take $n$ to infinity; I suppose if $A$ and $B$ are regarded as (possibly infinite) subsets of ${\bf N}$, then this is their asymptotic density.)

For instance $B = \{1\}$ is not an example, since if $A$ is the set of even integers in $[n]$, then $A$ covers one half of $[n]$ but $(A+B)\cap A = \emptyset$. The specific example I'm considering at the moment is the set $B = \{1, 2, 6, 24, 120,\ldots,\}$ of factorials, so information particular to this example would also be appreciated, if more general information is not known. But this problem seems natural enough that I thought it might have a name, I just didn't know what to search online. As always, thank you all in advance for the help!

__Edit.__ I thought I should illustrate a nontrivial example where $B$ is the set of factorials and the densities of $A$ and $(A+B)\cap A$ are the same. Let $p$ be a prime and let $a$ be a residue modulo $p$. Then the set $A$ of all integers congruent to $a$ modulo $p$ has asymptotic density $1/p$. Now since only finitely many members of $B$ are not divisible by $p$, if I'm not wrong the set $(A+B)\cap A$ should have the same asymptotic density as $A$. Is this true for all $A$, not just this example?