Intersections of sets of distances

We isolate conditions on the relative size of sets of natural numbers $A,B$ that guarantee a nonempty intersection $\Delta(A)\cap\Delta(B)\ne\emptyset$ of the corresponding sets of distances. Such conditions apply to a large class of zero density sets. We also show that a variant of Khintchine's Recurrence Theorem holds for all infinite sets $A=\{a_1<a_2<...\}$ with $a_n\ll n^{3/2}$.