On the Mattila-Sjolin theorem for distance sets

TL;DR

This paper generalizes the Mattila-Sjolin theorem to distance sets defined by convex body norms, establishing that sets with Hausdorff dimension greater than (d+1)/2 have distance sets containing an interval, with detailed measure estimates.

Contribution

It extends the Mattila-Sjolin theorem to convex body norms with smooth, curved boundaries, providing new results on the structure of distance sets under these metrics.

Findings

Distance sets contain an interval for sets with Hausdorff dimension > (d+1)/2.

Established estimates for the Radon-Nikodym derivative of the distance measure.

Generalized the theorem to metrics induced by convex bodies with smooth boundaries.

Abstract

We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E \subset {\Bbb R}^d$, $d \ge 2$, is greater than $\frac{d+1}{2}$, then the distance set $\Delta(E)=\{|x-y|: x,y \in E \}$ contains an interval. We prove this result for distance sets $\Delta_B(E)=\{{||x-y||}_B: x,y \in E \}$, where ${|| \cdot ||}_B$ is the metric induced by the norm defined by a symmetric bounded convex body $B$ with a smooth boundary and everywhere non-vanishing Gaussian curvature. We also obtain some detailed estimates pertaining to the Radon-Nikodym derivative of the distance measure.