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.
Categories: math.NT
