Small gaps between products of two primes

Let q(n) denote the nth number that is a product of exactly two distinct primes. We prove that q(n+1) - q(n) <= 6 infinitely often. This sharpens an earlier result of the authors, which had 26 in place of 6. More generally, we prove that if nu is any positive integer, then (q(n+nu) - q(n)) <= nu e(nu - gamma) (1+o(1)) infinitely often. We also prove several other related results on the representation of numbers with exactly two prime factors by linear forms.