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

Alex Iosevich, Mihalis Mourgoglou, Steven Senger

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 H…