On sets of directions determined by subsets of ℝ^d

Abstract

Given E ⊂ ℝ^d, d ≥ 2, define

$$D(E) \equiv \left\{ {{{x - y} \over {\left| {x - y} \right|}}:x,y \in E} \right\} \subset {S^{d - 1}}$$

to be the set of directions determined by E. We prove that if the Hausdorff dimension of E is greater than d − 1, then σ(D(E)) > 0, where σ denotes the surface measure on S ^d−1. In the process, we prove some tight upper and lower bounds for the maximal function associated with the Radon-Nikodym derivative of the natural measure on D. 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 ([22, 23]) on directions determined by finite subsets of ℝ^d. We also discuss the case when the Hausdorff dimension of E is precisely d − 1, where some interesting counter-examples have been obtained by Simon and Solomyak ([25]) in the planar case. In response to the conjecture stated in this paper, T. Orponen and T. Sahlsten ([20]) have recently proved that if the Hausdorff dimension of E equals d − 1 and E is rectifiable and is not contained in a hyper-pane, the Lebesgue measure of the set of directions is still positive. Finally, we show that our continuous results can be used to recover and, in some cases, 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 ⊂ ℝ^d, d ≥ 3, satisfying a certain discrete energy condition (Definition 3.1) determines ≳ #P distinct directions.