Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A generalization of Marstrand's theorem for projections of cartesian
  products   We prove the following variant of Marstrand's theorem about projections of
cartesian products of sets:
  Let $K_1,...,K_n$ Borel subsets of $\mathbb R^{m_1},... ,\mathbb R^{m_n}$
respectively, and $\pi:\mathbb R^{m_1}\times...\times\mathbb R^{m_n}\to\mathbb
R^k$ be a surjective linear map. We set $$\mathfrak{m}:=\min\{\sum_{i\in
I}\dim_H(K_i) + \dim\pi(\bigoplus_{i\in I^c}\mathbb R^{m_i}),
I\subset\{1,...,n\}, I\ne\emptyset\}.$$ Consider the space $\Lambda_m=\{(t,O),
t\in\mathbb R, O\in SO(m)\}$ with the natural measure and set
$\Lambda=\Lambda_{m_1}\times...\times\Lambda_{m_n}$. For every
$\lambda=(t_1,O_1,...,t_n,O_n)\in\Lambda$ and every $x=(x^1,,x^n)\in\mathbb
R^{m_1}\times...\times\mathbb R^{m_n}$ we define
$\pi_\lambda(x)=\pi(t_1O_1x^1,...,t_nO_nx^n)$. Then we have $(i)$ If
$\mathfrak{m}>k$, then $\pi_\lambda(K_1\times...\times K_n)$ has positive
$k$-dimensional Lebesgue measure for almost every $\lambda\in\Lambda$. $(ii)$
If $\mathfrak{m}\leq k$ and $\dim_H(K_1\times...\times
K_n)=\dim_H(K_1)+...+\dim_H(K_n)$, then \dim_H(\pi_\lambda(K_1\times...\times
K_n))=\mathfrak{m}$ for almost every $\lambda\in\Lambda$.

12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas