Polynomial progressions in the generalized twin primes

By Maynard’s theorem and the subsequent improvements by the Polymath Project, there exists a positive integer $b\le 246$ such that there are infinitely many primes $p$ such that $p+b$ is also prime. Let $P_1,\mathrm{.}\mathrm{.}\mathrm{.},P_t\in \mathbb{Z}[y]$ 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 $b\le 246$ such that $x+P_1(y),x+P_2(y),\mathrm{.}\mathrm{.}\mathrm{.},x+P_t(y)$ and $x+P_1(y)+b,x+P_2(y)+b,\mathrm{.}\mathrm{.}\mathrm{.},x+P_t(y)+b$ are all prime. Our work is inspired by Pintz, who proved a similar result for the special case of arithmetic progressions.