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.
Categories: math.CA math.CO math.MG