Almost primes in almost all short intervals II

We show that, for almost all x, the interval (x, x + (log x)^2.1] contains products of exactly two primes. This improves on a work of the second author that had 3.51 in place of 2.1. To obtain this improvement, we prove a new type II estimate. One of the new innovations is to use Heath-Brown's mean value theorem for sparse Dirichlet polynomials.