Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Analytic sets of reals and the density function in the Cantor space   We study the density function of measurable subsets of the Cantor space.
Among other things, we identify a universal set $\mathcal{U}$ for
$\Sigma^{1}_{1}$ subsets of $( 0 ; 1 )$ in terms of the density function;
specifically $\mathcal{U}$ is the set of all pairs $( K , r )$ with $K$ compact
and $r \in ( 0 ; 1 )$ being the density of some point with respect to $K$. This
result yields that the set of all $K$ such that the range of its density
function is $S \cup \{ 0 , 1 \}$, for some fixed uncountable analytic set $S
\subseteq ( 0 ; 1 )$, is $\Pi^{1}_{2}$-complete.

29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
24
The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys