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 sets $B$ such that whenever $A$ covers a positive proportion of $[n]$, the set $(A+B)\cap A$ also covers a positive proportion of $[n]$. (What I mean by "positive proportion" onlly 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!