Falconer distance problem on Riemannian manifolds

Abstract

We prove that on a $d$-dimensional Riemannian manifold, the distance set of a Borel set $E$ has a positive Lebesgue measure if $${\dim }_{\mathcal{H}}(E)>\frac{d}{2}+\frac{1}{4}+\frac{1-(-1{)}^d}{8d}.$$

18 pages.We have removed the constant sectional curvature condition, which has led to a strengthened version of the main result