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, lim inf(n)(p(n)+(m) - p(n)) < infinity for every integer m. We also show that lim inf(p(n)+(l) - pn) <= 600 and, if we assume the Elliott-Halberstam conjecture, that lim inf(n) (p(n)+(1) - p(n)) <= 12 and lim inf(n) (p(n)+(2) - p(n)) <= 600.