On the Mattila-Sjölin theorem for distance sets
Alex Iosevich, Mihalis Mourgoglou, Krystal Taylor · Annales Academiae Scientiarum Fennicae Mathematica · 2012
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 ∆(E) = {|x -y| : x, y ∈ E} contains an interval.We prove this result for distance sets ∆ 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.