Small gaps between almost-twin primes

Abstract

Let $m\in \mathbb{N}$ be large. We show that there exist infinitely many primes $q_1<\cdot \cdot \cdot <q_{m+1}$ such that \[ q_{m+1}-q_{1}=O(e^{7.63m}) \] and $q_j+2$ has at most \[ \frac{7.36m}{\log 2} + \frac{4\log m}{\log 2} + 21 \] prime factors for each $1\le j\le m+1$. This improves the previous result of Li and Pan, replacing $m^4{e}^{8m}$ by $e^{7.63m}$ and $\frac{16m}{\log 2}+\frac{5\log m}{\log 2}+37$ by $\frac{7.36m}{\log 2}+\frac{4\log m}{\log 2}+21$. The main inputs are the Maynard-Tao sieve, a minorant for the indicator function of the primes constructed by Baker and Irving, for which a stronger equidistribution theorem in arithmetic progressions to smooth moduli is applicable, and Tao's approach previously used to estimate ${\sum }_{x\le n<2x}{1}_{\mathbb{P}}(n)1_{\mathbb{P}}(n+12)\ddot{I}{‰}_n$, where $1_{\mathbb{P}}$ stands for the characteristic function of the primes and $\ddot{I}{‰}_n$ are multidimensional sieve weights.

21 pages. Revised version, accepted for publication in Forum Mathematicum