Falconer distance problem on Riemannian manifolds

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