Worked solution: Maynard's multidimensional sieve: bounded gaps between primes (2013)

Maynard's 2013 paper Small gaps between primes (arXiv:1311.4600, Annals of Mathematics 2015) opens with the notion of an admissible set: a finite set of nonnegative integers $\mathcal{H}=\{h_1,\dots,h_k\}$ is admissible if, for every prime $p$, some residue $a_p$ is missed by every shift, i.e. $a_p\not\equiv h_i \pmod{p}$ for all $i$ (Maynard 2013, §1). The prime $k$-tuples conjecture predicts that for every admissible $\mathcal{H}$, infinitely many $n$ make all of $n+h_1,\dots,n+h_k$ simultaneously prime; it is completely open for $k>1$.

Goldston, Pintz and Yıldırım (GPY) had shown in 2005 that a sieve-theoretic sum controls this question, proving unconditionally that $\liminf_n (p_{n+1}-p_n)/\log p_n=0$: infinitely often the gap is a vanishing fraction of the average gap $\log p_n$, though not literally bounded. Zhang's breakthrough pushed the GPY machinery past a technical barrier concerning how evenly primes spread across arithmetic progressions, using new estimates, and obtained the first genuinely bounded gap (Maynard 2013, §1, eq. 1.2).

Maynard's paper takes a different route: rather than matching Zhang's hard new estimate, it refines the sieve itself so that the older, classical Bombieri–Vinogradov theorem already suffices — and gives a far better bound. The remaining steps walk through that refinement, ending with $\liminf_n(p_{n+1}-p_n)\le 600$ from purely classical input (Maynard 2013, Theorem 1.3).