On volumes determined by subsets of Euclidean space

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

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