  We consider shifts of a set $A\subseteq\mathbb{N}$ by elements from another
set $B\subseteq\mathbb{N}$, and prove intersection properties according to the
relative asymptotic size of $A$ and $B$. A consequence of our main theorem is
the following: If $A=\{a_n\}$ is such that $a_n=o(n^{k/k-1})$, then the
$k$-recurrence set $R_k(A)=\{x\mid |A\cap(A+x)|\ge k\}$ contains the distance
sets of arbitrarily large finite sets.
