On sets of directions determined by subsets of ${\Bbb R}^d$

TL;DR

This paper investigates the relationship between the Hausdorff dimension of subsets of Euclidean space and the measure of directions and angles they determine, establishing threshold conditions for positivity of measures and extending results to finite point sets.

Contribution

It provides new threshold results linking Hausdorff dimension to direction and angle measures, and applies these to improve bounds on finite point sets in discrete geometry.

Findings

If the Hausdorff dimension of E exceeds d-1, the set of directions has positive surface measure.

For sets with Hausdorff dimension greater than (d-1)/2 + 1/3, the set of angles has positive Lebesgue measure.

Finite point sets satisfying certain energy conditions determine many distinct directions and angles.

Abstract

Given $E \subset \mathbb{R}^d$, $d \ge 2$, define ${\mathcal D}(E) \equiv {(x-y)/|x-y|: x,y \in E} \subset S^{d-1},$ the set of directions determined by $E$. We prove that if the Hausdorff dimension of $E$ is greater than $d-1$, then $\sigma({\mathcal D}(E))>0$, where $\sigma$ denotes the surface measure on $S^{d-1}$. This result is sharp since the conclusion fails to hold if $E$ is a $(d-1)$-dimensional hyper-plane. This result can be viewed as a continuous analog of a recent result of Pach, Pinchasi, and Sharir (\cite{PPS04}, \cite{PPS07}) on angles determined by finite subsets of $\mathbb{R}^d$. Also define ${\mathcal A}(E)=\{\theta(x,y,z): x,y,z \in E\},$ where $\theta(x,y,z)$ is the angle between $x-y$ and $y-z$. We use the techniques developed to handle the problem of directions and results on distance sets previously obtained by Wolff and Erdogan to prove that if the Hasudorff dimension of $E$ is greater than $(d-1)/2+1/3$, then the Lebesgue measure of ${\mathcal A}(E)$ is positive. This result can be viewed as a continuous analog of a recent result of Apfelbaum and Sharir (\cite{AS05}). At the end of this paper we show that our continuous results can be used to recover and in some case improve the exponents for the corresponding results in the discrete setting for large classes of finite point sets. In particular, we prove that a finite point set $P \subset {\Bbb R}^d$, $d \ge 3$, satisfying a certain discrete energy condition (Definition \ref{adaptablemama}), determines $\gtrapprox # P$ distinct directions and $\gtrapprox {(# P)}^{6/(3d-1)}$ distinct angles. In two dimensions, the lower bound on the number of angles is $\gtrapprox # P$.