Small gaps between almost-twin primes

Abstract Let m ∈ N m\in\mathbb{N} be large. We show that there exist infinitely many primes q 1 < ⋯ < q m + 1 q_{1}<\cdots<q_{m+1} such that q m + 1 − q 1 = O ⁢ ( e 7.63 ⁢ m ) q_{m+1}-q_{1}=O(e^{7.63m}) and q j + 2 q_{j}+2 has at most 7.36 ⁢ m log ⁡ 2 + 4 ⁢ log ⁡ m log ⁡ 2 + 21 \frac{7.36m}{\log 2}+\frac{4\log m}{\log 2}+21 prime factors for each 1 ≤ j ≤ m + 1 1\leq j\leq m+1 . This improves the previous result of Li and Pan, replacing m 4 ⁢ e 8 ⁢ m m^{4}e^{8m} by e 7.63 ⁢ m e^{7.63m} and 16 ⁢ m log ⁡ 2 + 5 ⁢ log ⁡ m log ⁡ 2 + 37 \frac{16m}{\log 2}+\frac{5\log m}{\log 2}+37 by 7.36 ⁢ m log ⁡ 2 + 4 ⁢ log ⁡ m log ⁡ 2 + 21 \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 ∑ x ≤ n < 2 ⁢ x 1 P ⁢ ( n ) ⁢ 1 P ⁢ ( n + 12 ) ⁢ ω n \sum_{x\leq n<2x}\mathbf{1}_{\mathbb{P}}(n)\mathbf{1}_{\mathbb{P}}(n+12)\omega_{n} , where 1 P \mathbf{1}_{\mathbb{P}} stands for the characteristic function of the primes and ω n \omega_{n} are multidimensional sieve weights.