Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Density of sets of natural numbers and the Levy group   Let $\N$ denote the set of positive integers. The asymptotic density of the
set $A \subseteq \N$ is $d(A) = \lim_{n\to\infty} |A\cap [1,n]|/n$, if this
limit exists. Let $ \mathcal{AD}$ denote the set of all sets of positive
integers that have asymptotic density, and let $S_{\N}$ denote the set of all
permutations of the positive integers \N. The group $\mathcal{L}^{\sharp}$
consists of all permutations $f \in S_{\N}$ such that $A \in \mathcal{AD}$ if
and only if $f(A) \in \mathcal{AD}$, and the group $\mathcal{L}^{\ast}$
consists of all permutations $f \in \mathcal{L}^{\sharp}$ such that $d(f(A)) =
d(A)$ for all $A \in \mathcal{AD}$. Let $f:\N \to \N $ be a one-to-one function
such that $d(f(\N))=1$ and, if $A \in \mathcal{AD}$, then $f(A) \in
\mathcal{AD}$. It is proved that $f$ must also preserve density, that is,
$d(f(A)) = d(A)$ for all $A \in \mathcal{AD}$. Thus, the groups
$\mathcal{L}^{\sharp}$ and $\mathcal{L}^{\ast}$ coincide.

26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
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The categories of this paper are: eess
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: math-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: q-bio