On the Mattila-Sjölin theorem for distance sets

We extend a result, due to Mattila and Sjölin, which says that if the Hausdorff dimension of a compact set E⊂ R^d, d ≥2, is greater than d+1/2, then the distance set Δ(E) = {|x-y|:x,y ∈ E} contains an interval. We prove this result for distance sets Δ_B(E) = {||x-y||_B: x,y ∈ E}, where ||·||_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.