Falconer distance problem on Riemannian manifolds

TL;DR

This paper extends the Falconer distance problem to Riemannian manifolds, establishing conditions under which the distance set of a Borel set has positive measure based on its Hausdorff dimension.

Contribution

It provides a new threshold for the Hausdorff dimension ensuring positive Lebesgue measure of the distance set on Riemannian manifolds, generalizing prior Euclidean results.

Findings

Distance set has positive measure if Hausdorff dimension exceeds a specific threshold.

Derived a dimension threshold depending on the manifold's dimension.

Extended Falconer distance problem to 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 $$\dim_{\mathcal{H}}(E)>\frac d2+\frac14+\frac{1-(-1)^d}{8d}.$$