Narrow progressions in the primes

TL;DR

This paper improves bounds on polynomial progressions within prime subsets, reducing the size of the common difference from a subpolynomial to a polylogarithmic scale, using advanced densification techniques.

Contribution

It shortens the progression size bounds in prime polynomial progressions and introduces the densification method to avoid complex correlation estimates.

Findings

Progression size reduced to polylogarithmic scale in N

Applicable to polynomial and linear progressions with improved bounds

Introduces densification method for prime pattern analysis

Abstract

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.