Many classical limsup and liminf sets in Diophantine approximation are well understood from a dimensional perspective. But what happens when one Diophantine set is removed from another? How large is what remains?

In this talk, I will discuss recent results on differences and intersections of Diophantine sets. In particular, we consider intersections of inhomogeneously badly approximable sets with well-approximable and exact-approximation sets, and investigate their size. These results reveal the rich structure that survives when competing Diophantine approximation conditions are imposed.