On radii of spheres determined by subsets of Euclidean space

In this paper we consider the problem of how large the Hausdorff dimension of $E\subset\R^d$ needs to be in order to ensure that the radii set of $(d-1)$-dimensional spheres determined by $E$ has positive Lebesgue measure. We also study the question of how often can a neighborhood of a given radius repeat. We obtain two results. First, by applying a general mechanism developed in \cite{mul} for studying Falconer-type problems, we prove that a neighborhood of a given radius cannot repeat more often than the statistical bound if $\dH(E)>d-1+\frac{1}{d}$; In $\R^2$, the dimensional threshold is sharp. Second, by proving an intersection theorem, we prove for a.e $a\in\R^d$, the radii set of $(d-1)$-spheres with center $a$ determined by $E$ must have positive Lebesgue measure if $\dH(E)>d-1$, which is a sharp bound for this problem.