Almost Primes in Almost All Short Intervals

Let $E_k$ be the set of positive integers having exactly $k$ prime factors. We show that almost all intervals $[x,x+{\log }^{1+\epsilon }x]$ contain $E_3$ numbers, and almost all intervals $[x,x+{\log }^{3.51}x]$ contain $E_2$ numbers. By this we mean that there are only $o(X)$ integers $1\le x\le X$ for which the mentioned intervals do not contain such numbers. The result for $E_3$ numbers is optimal up to the $\epsilon$ in the exponent. The theorem on $E_2$ numbers improves a result of Harman, which had the exponent $7+\epsilon$ in place of $3.51$. We will also consider general $E_k$ numbers, and find them on intervals whose lengths approach $\log x$ as $k\to \infty$.