On volumes determined by subsets of Euclidean space

Abstract

Given $E\subset {\mathbb{R}}^d$, define the \emph{volume set} of $E$, $\mathcal{V}(E)=\{det(x^1,x^2,...x^d):x^j\in E\}$. In ${\mathbb{R}}^3$, we prove that $\mathcal{V}(E)$ has positive Lebesgue measure if either the Hausdorff dimension of $E\subset {\mathbb{R}}^3$ is greater than 13/5, or $E$ is a product set of the form $E=B_1\times B_2\times B_3$ with $B_j\subset \mathbb{R},di{m}_{\mathcal{H}}(B_j)>2/3,j=1,2,3$. We show that the same conclusion holds for $\V(E)$ of Salem subsets $E\subset {\mathbb{R}}^d$ with $\hde>d-1$, and give applications to discrete combinatorial geometry.