On Radii of Spheres Determined by Subsets of Euclidean Space

In this paper the author considers the problem of how large the Hausdorff dimension of E ⊂ ℝ needs to be in order to ensure that the radii set of (d-1)-dimensional spheres determined by E has positive Lebesgue measure. The author also studies the question of how often can a neighborhood of a given radius repeat. There are two results obtained in this paper. First, by applying a general mechanism developed in Grafakos et al. (2013) for studying Falconer-type problems, the author proves that a neighborhood of a given radius cannot repeat more often than the statistical bound if dim_H(E) > d-1+1/d; In ℝ², the dimensional threshold is sharp. Second, by proving an intersection theorem, the author proves that for a.e a ∈ ℝ^d, the radii set of (d-1)-spheres with center a determined by E must have positive Lebesgue measure if dim_H(E) > d-1, which is a sharp bound for this problem.