Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Polynomial patterns in the primes   Let $P_1,\dots,P_k \colon {\bf Z} \to {\bf Z}$ be polynomials of degree at
most $d$ for some $d \geq 1$, with the degree $d$ coefficients all distinct,
and admissible in the sense that for every prime $p$, there exists integers
$n,m$ such that $n+P_1(m),\dots,n+P_k(m)$ are all not divisible by $p$. We show
that there exist infinitely many natural numbers $n,m$ such that
$n+P_1(m),\dots,n+P_k(m)$ are simultaneously prime, generalizing a previous
result of the authors, which was restricted to the special case
$P_1(0)=\dots=P_k(0)=0$ (though it allowed for the top degree coefficients to
coincide). Furthermore, we obtain an asymptotic for the number of such prime
pairs $n,m$ with $n \leq N$ and $m \leq M$ with $M$ slightly less than
$N^{1/d}$.
  Our arguments rely on four ingredients. The first is a (slightly modified)
generalized von Neumann theorem of the authors, reducing matters to controlling
certain averaged local Gowers norms of (suitable normalizations of) the von
Mangoldt function. The second is a more recent concatenation theorem of the
authors, controlling these averaged local Gowers norms by global Gowers norms.
The third ingredient is the work of Green and the authors on linear equations
in primes, allowing one to compute these global Gowers norms for the normalized
von Mangoldt functions. Finally, we use the Conlon-Fox-Zhao densification
approach to the transference principle to combine the preceding three
ingredients together.
  In the special case $P_1(0)=\dots=P_k(0)=0$, our methods also give infinitely
many $n,m$ with $n+P_1(m),\dots,n+P_k(m)$ in a specified set primes of positive
relative density $\delta$, with $m$ bounded by $\log^L n$ for some $L$
independent of the density $\delta$. This improves slightly on a result from
our previous paper, in which $L$ was allowed to depend on $\delta$.

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The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: mtrl-th
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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: astro-ph