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 $P\_2.$



Large gaps between P2s
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.


