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.

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 $P_2.$