On radii of spheres determined by subsets of Euclidean space

In this paper we consider the problem that how large the Hausdorff dimension of E ⊂ Rd needs to be to ensure the radii set of (d − 1)-dimensional spheres determined by E has positive Lebesgue measure. We obtain two results. First, by extending a general mechanism for studying Falconer-type problems in [4], we prove that it holds when dimH(E) > d− 1+ 1 d and in R2, the index 3 2 is sharp for this method. Second, by proving an intersection theorem we prove for a.e a ∈ Rd, the radii set of (d− 1)-spheres with center a determined by E must have positive Lebesgue measure if dimH(E) > d− 1. It is obviously a sharp bound for this problem.