TITLE: Large gaps between P2s
QUESTION [5 upvotes]: Gaps between consecutive primes are $O(n^{\theta+\varepsilon})$ for $\theta=0.525$ and any $\varepsilon>0.$ I was wondering if a better result is known for gaps between numbers with at most two prime factors. (It seems that this would be easier because it removes the parity obstacle.)
I am interested in both unconditional results and those conditional on standard hypotheses.
Related result: Harman (1981) showed that almost all intervals of length $(\log x)^{7+\varepsilon}$ contains a $P_2.$

REPLY [10 votes]: Here is a result, due to Halberstam, Heath-Brown, and Richert that seems to be of the flavor you are looking for (Almost-primes in short intervals. Recent progress in analytic number theory, Vol. 1 (Durham, 1979), pp. 69–101, Academic Press, London-New York, 1981.)
Theorem. For all sufficiently large $x$ the interval $(x−x^{0.455},x]$, contains at least $\frac{1}{121}\frac{x^{0.455}}{\log x}$ integers that are either primes or products of two primes.
Reference chasing on MathSciNet, it seems that Iwaniec & Laborde improved the exponent from $0.455$ to $0.45$ and Wenzhi Luo later improved this slightly. I am not sure if Luo's result is still the "best" exponent.