Intersections of shifted sets

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 $a_n=o(n^{k\mathrm{/}k-1})$, then the $k$-recurrence set $R_k(A)=\{x\mid \mathrm{\mid }A\cap (A+x)\mathrm{\mid }\ge k\}$ contains the distance sets of arbitrarily large finite sets.