Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multiple recurence and convergence for sequences related to the prime
  numbers   For any measure preserving system $(X,\mathcal{X},\mu,T)$ and
$A\in\mathcal{X}$ with $\mu(A)>0$, we show that there exist infinitely many
primes $p$ such that $\mu\bigl(A\cap T^{-(p-1)}A\cap T^{-2(p-1)}A\bigr) > 0$
(the same holds with $p-1$ replaced by $p+1$). Furthermore, we show the
existence of the limit in $L^2(\mu)$ of the associated ergodic average over the
primes. A key ingredient is a recent result of Green and Tao on the von
Mangoldt function. A combinatorial consequence is that every subset of the
integers with positive upper density contains an arithmetic progression of
length three and common difference of the form $p-1$ (or $p+1$) for some prime
$p$.

21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom