Polynomial progressions in the generalized twin primes
Andrew Lott, Nagendar Reddy Ponagandla
TL;DR
The paper proves that there exist x, y and a fixed b≤246 such that the sets {x+P_1(y),…,x+P_t(y)} and {x+P_1(y)+b,…,x+P_t(y)+b} are all prime for polynomials P_i∈ℤ[y] with P_i(0)=0, extending polynomial progressions in primes to generalized twin-prime configurations. It combines Maynard–Polymath improvements with a transference principle and a W-trick-based pseudorandom measure to model sparse prime patterns by a dense surrogate, and establishes a polynomial forms condition that enables a Bergelson–Leibman-type Szemerédi theorem in this setting. The key contributions are the construction of a pseudorandom majorant ν_𝒜, a robust correlation estimate that tolerates bad primes, and a transference argument that yields actual polynomial prime configurations with a bounded shift, thereby broadening the scope of prime patterns in the widening landscape of sparse combinatorial structures. This has implications for understanding structured prime configurations beyond arithmetic progressions and demonstrates the power of transference techniques in handling polynomial configurations in primes.
Abstract
By Maynard's theorem and the subsequent improvements by the Polymath Project, there exists a positive integer $b\leq 246$ such that there are infinitely many primes $p$ such that $p+b$ is also prime. Let $P_1,...,P_t\in \mathbb{Z}[y]$ with $P_1(0)=\cdots=P_t(0)=0$. We use the transference argument of Tao and Ziegler to prove there exist positive integers $x, y,$ and $b \leq 246 $ such that $x+P_1(y),x+P_2(y),...,x+P_t(y)$ and $x+P_1(y)+b,x+P_2(y)+b,...,x+P_t(y)+b$ are all prime. Our work is inspired by Pintz, who proved a similar result for the special case of arithmetic progressions.