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 $P_2.$

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    $\begingroup$ While it does not answer this question exactly, work on Jacobsthal's function can provide some accessible lower bounds. If this were combined with knowledge (which at this time is nonexistent, from my limited perspective) of how the totients of a number are distributed, I feel one could get a handle on the matter. Gerhard "Ask Me About Jacobsthal's Function" Paseman, 2012.09.08 $\endgroup$ Commented Sep 8, 2012 at 19:35

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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.


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