  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.
