Small gaps between primes

We introduce a refinement of the GPY sieve method for studying prime $k$-tuples and small gaps between primes. This refinement avoids previous limitations of the method, and allows us to show that for each $k$, the prime $k$-tuples conjecture holds for a positive proportion of admissible $k$-tuples. In particular, ${\mathrm{lim\hspace{0.17em}inf}}_n(p_{n+m}-p_n)<\mathrm{\infty }$ for any integer $m$. We also show that $\mathrm{lim\hspace{0.17em}inf}(p_{n+1}-p_n)\le 600$, and, if we assume the Elliott-Halberstam conjecture, that ${\mathrm{lim\hspace{0.17em}inf}}_n(p_{n+1}-p_n)\le 12$ and ${\mathrm{lim\hspace{0.17em}inf}}_n(p_{n+2}-p_n)\le 600$.