Small gaps between primes

We introduce a refinement of the GPY sieve method for studying prime kk-tuples and small gaps between primes. This refinement avoids previous limitations of the method and allows us to show that for each kk, the prime kk-tuples conjecture holds for a positive proportion of admissible kk-tuples. In particular, lim infn(pn+m−pn)<∞lim infn(pn+m−pn)<∞ for every integer mm. We also show that lim inf(pn+1−pn)≤600lim inf(pn+1−pn)≤600 and, if we assume the Elliott-Halberstam conjecture, that lim infn(pn+1−pn)≤12lim infn(pn+1−pn)≤12 and lim infn(pn+2−pn)≤600lim infn(pn+2−pn)≤600