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 $E \subset {\Bbb R}^d$, $d \ge 2$, 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.