  We show that as soon as $h\to \infty$ with $X \to \infty$, almost all
intervals $(x-h\log X, x]$ with $x \in (X/2, X]$ contain a product of at most
two primes. In the proof we use Richert's weighted sieve, with the arithmetic
information eventually coming from results of Deshouillers and Iwaniec on
averages of Kloosterman sums.
