On Radii of Spheres Determined by Subsets of Euclidean Space

Article

Abstract

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.

Keywords

Falconer-type problems Radii Intersection 

Mathematics Subject Classification

28A75 42B20 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of RochesterRochesterUSA

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