Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

13
The categories of this paper are: quant-ph
29
The categories of this paper are: funct-an
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
30
The categories of this paper are: q-alg