Hausdorff dimension and distance sets

Abstract

According to a result of K. Falconer (1985), the set D(A)={|x-y|;x, y ∈A} of distances for a Souslin set A of ℝn has positive 1-dimensional measure provided the Hausdorff dimension of A is larger than (n+1)/2.* We give an improvement of this statement in dimensions n=2, n=3. The method is based on the fine theory of Fourier restriction phenomena to spheres. Variants of it permit further improvements which we don't plan to describe here. This research originated from some discussions with P. Mattila on the subject. © 1994 Hebrew University.