A set of linear forms \(\mathcal L = \{ L_1(x), \dots, L_k(x) \} \subset \mathbb Z[x]\) is called admissible if the product \(\prod_{i=1}^k L_i(n)\) has no fixed prime divisor for \(n\in \mathbb N\). The prime \(k\)-tuples conjecture states that if \(\mathcal L\) is admissible, then there exist infinitely many \(n \in \mathbb N\) such that the integers \(L_i(n)\), \(1 \leq i \leq k\), are simultaneously prime. While this conjecture appears to be out of reach at present, remarkable progress was achieved recently by the author [Ann. Math. (2) 181, No. 1, 383–413 (2015; Zbl. 1306.11073)] and T. Tao (unpublished). Maynard and Tao proved independently that for any \(m \geq 1\) and any admissible \(k\)-tuple of forms \(\mathcal L\), with \(k \geq k_0(m)\), there are infinitely many \(n \in \mathbb N\) such that at least \(m\) of the integers \(L_i(n)\), \(1 \leq i \leq k\), are prime.
The main idea in the work of Maynard and Tao is of a rather general nature and has since been applied, with great success, to a number of other problems. In the present paper, the author explores the limits of the inherent flexibility of that method. Let \(\mathcal A\) and \(\mathcal P\) be “interesting” subsets of \(\mathbb N\) and of the primes, respectively. Roughly speaking, the main result of the paper establishes that if \(\mathcal A\) and \(\mathcal P \cap L_i(\mathcal A)\), \(1 \leq i \leq k\), are well-distributed in arithmetic progressions, then there are infinitely many \(n \in \mathcal A\) such that several of the integers \(L_i(n)\), \(1 \leq i \leq k\), are primes from \(\mathcal P\). The well-distribution hypothesis is too technical to include in this review, but it suffices to say that it amounts to assuming average bounds of a Bombieri-Vinogradov type for the number of elements of \(\mathcal A\) and \(\mathcal P \cap L(\mathcal A)\) in arithmetic progressions, plus a hypothesis that \(\mathcal A\) is not concentrated in any particular arithmetic progression of small modulus. The main theorem provides a quantitative lower bound for the number of \(n \in \mathcal A\) with the desired properties. Moreover, in the special case when \(\mathcal P\) is the set of all primes, the author is able to ensure that – after a reasonable tightening of some hypotheses – the primes counted in the theorem can be chosen to be consecutive (in the sequence of all primes).
The paper also includes several interesting consequences of the main theorem. For example, the author proves that if \(\mathcal L\) is admissible, with \(k \geq k_0(m)\), \(x\) is large and \(x^{7/12+\varepsilon} \leq y \leq x\) for some fixed \(\varepsilon > 0\), then the interval \([x, x+y]\) contains many integers \(n\) such that at least \(m\) of the integers \(L_i(n)\), \(1 \leq i \leq k\), are prime. Another interesting corollary is the result (Theorem 3.2 in the paper) that for any \(x,y \geq 2\), the interval \([x,2x]\) contains many integers \(z\) such that \(\pi(z+y) - \pi(z) \gg \log y\) (here, as usual, \(\pi(z)\) is the prime counting function).