Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long lines in subsets of large measure in high dimension   We show that for any set $A\subseteq [0,1]^n$ with $\text{Vol}(A)\ge 1/2$
there exists a line $\ell $ such that the one-dimensional Lebesgue measure of
$\ell \cap A$ is at least $\Omega ( n^{1/4} )$. The exponent $1/4$ is tight.
More generally, for a probability measure $\mu $ on $\mathbb R ^n$ and $0<a<1$
define \begin{equation*} L(\mu ,a):= \inf_{A ; \mu(A) = a} \sup _{\ell \text{
line}} \big| \ell \cap A\big| \end{equation*} where $|\cdot | $ stands for the
one-dimensional Lebesgue measure. We study the asymptotic behavior of $L(\mu
,a)$ when $\mu $ is a product measure and when $\mu $ is the uniform measure on
the $\ell _p$ ball. We observe a rather unified behavior in a large class of
product measures. On the other hand, for $\ell_p$ balls with $1 \leq p \leq
\infty$ we find that there are phase transitions of different types.

9
The categories of this paper are: nlin
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
3
The categories of this paper are: physics