Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Narrow progressions in the primes   In a previous paper of the authors, we showed that for any polynomials
$P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset
$A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least
$\delta>0$, one can find a "polynomial progression" $a+P_1(r),\dots,a+P_k(r)$
in $A$ with $0 < |r| \leq N^{o(1)}$, if $N$ is sufficiently large depending on
$k,P_1,\dots,P_k$ and $\delta$. In this paper we shorten the size of this
progression to $0 < |r| \leq \log^L N$, where $L$ depends on $k,P_1,\dots,P_k$
and $\delta$. In the linear case $P_i = (i-1)\mathbf{m}$, we can take $L$
independent of $\delta$. The main new ingredient is the use of the
densification method of Conlon, Fox, and Zhao to avoid having to directly
correlate the enveloping sieve with dual functions of unbounded functions.

25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
27
The categories of this paper are: comp-gas
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
24
The categories of this paper are: dg-ga