On radii of spheres determined by subsets of Euclidean space

TL;DR

This paper investigates the Hausdorff dimension thresholds needed for subsets of Euclidean space to ensure their associated radii sets of spheres have positive measure, revealing sharp bounds and frequency limitations.

Contribution

It establishes sharp Hausdorff dimension bounds for radii sets of spheres and introduces bounds on how often neighborhoods of a radius can repeat, using new intersection theorems.

Findings

Threshold of Hausdorff dimension for positive measure is d-1 in general and sharp in R^2.

Neighborhood repetition of a radius is limited by a statistical bound if dimension exceeds d-1+1/d.

For almost every center, the radii set has positive measure if the set's dimension exceeds d-1.

Abstract

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.