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 \(E\subset \mathbb {R}^d\) needs to be in order to ensure that the radii set of \((d-1)\) -dimensional spheres determined by \(E\) has positive Lebesgue measure. 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 \(\dim _{{\mathcal H}}(E)>d-1+\frac{1}{d}\) ; In \(\mathbb {R}^2\) , the dimensional threshold is sharp. Second, by proving an intersection theorem, the author proves that for a.e \(a\in \mathbb {R}^d\) , the radii set of \((d-1)\) -spheres with center \(a\) determined by \(E\) must have positive Lebesgue measure if \(\dim _{{\mathcal H}}(E)>d-1\) , which is a sharp bound for this problem.