  We study the generalization of the Falconer distance problem to the
Riemannian setting. In particular, we extend the result of
Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian
surfaces. Key new ingredients include a family of refined microlocal decoupling
inequalities, which are related to the work of Beltran-Hickman-Sogge on
Wolff-type inequalities, and an analog of Orponen's radial projection lemma
which has proved quite useful in recent work on distance sets.
