On volumes determined by subsets of Euclidean space

TL;DR

This paper proves that the volume set of certain subsets of Euclidean space has positive Lebesgue measure under specific dimensional conditions, with applications to discrete geometry.

Contribution

It establishes new conditions under which the volume set of subsets in Euclidean space has positive measure, including for Salem sets and product sets.

Findings

Volume set has positive measure if Hausdorff dimension > 13/5 in R^3.

Product sets with dimension > 2/3 in each factor also have positive volume set.

Results extend to Salem sets with dimension > d-1.

Abstract

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.