Marstrand’s intersection theorem (which is complementary to Marstrand’s celebrated projection theorem) is one of the seminal results of geometric measure theory; it is the intuitive statement that an s-dimensional set in the plane will have an (s-1)-dimensional intersection with (almost) every line that intersects the set non-trivially. In the first part of this talk, we discuss a vast generalization of the intersection theorem that investigates when a family of measures \((\nu_{\omega})\) has well-defined intersection \(\mu\cap \nu_{\omega}\) for all \(\mu\) and for almost every \(\omega\), and when this intersection has large Hausdorff dimension. In the second part of the talk, we will discuss applications to the study of arithmetic progressions in sparse subsets of Euclidean space. This is joint work with Daniel Glasscock.