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 $1 \leq j \leq m+1$. 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 $\sum_{x \leq n < 2x} \mathbf{1}_{\mathbb{P}}(n)\mathbf{1}_{\mathbb{P}}(n+12)\omega_{n}$, where $\mathbf{1}_{\mathbb{P}}$ stands for the characteristic function of the primes and $\omega_{n}$ are multidimensional sieve weights.