Hausdorff Dimension and Distance Sets

摘要

According to a result of K. Falconer, the set D(A) = ((verticle line)x-y(verticleline); x,y (epsilon A)) of distances for a subset A of R(sup n) has positive 1-dimensional measure provided the Hausdorff dimension of A is larger than n+1/2(dim A > n/2 would be the optimal result.) The authors give an improvement of this statement in dimension 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 they don't plan to describe here.