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 $[x,x+\log^{1+\varepsilon} x]$ 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 $1\leq x\leq X$ for which
the mentioned intervals do not contain such numbers. The result for $E_3$
numbers is optimal up to the $\varepsilon$ in the exponent. The theorem on
$E_2$ numbers improves a result of Harman, which had the exponent
$7+\varepsilon$ in place of $3.51$. We will also consider general $E_k$
numbers, and find them on intervals whose lengths approach $\log x$ as $k\to
\infty$.