Almost primes in almost all very short intervals

Abstract

We show that as soon as h ->infinity$h\rightarrow \infty$ with X ->infinity$X \rightarrow \infty$, almost all intervals (x-hlogX,x]$(x-h\log X, x]$ with x is an element of(X/2,X]$x \in (X/2, X]$ 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.