Abstract
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.
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