On the Mattila-Sjolin theorem for distance sets

Abstract

We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E\subset {\mathbb{R}}^d$, $d\ge 2$, 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.