What is the current best result on the greatest lower bound on gaps between P2 almost primes where P2 represents a prime or the product of two semi-primes?

Last week, at Harvard, there was a "special seminar" by Yi Tang Zhang of the University of New Hampshire with the abstract "the speaker proves that there are an infinite number of pair of primes whose difference is bounded by 70 million". If this is true, this is surely the best bound for your question as well. Yet it seems too strong to be true. Anyone here who attended the talk?
– JoëlMay 17 '13 at 17:30

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I don't understand the question. According to the twin prime conjecture, gap 2 occurs infinitely often, hence one cannot give a better lower bound than 2. Perhaps you meant: how large are the gaps between P2 numbers, provably?
– GH from MOMay 17 '13 at 18:31

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Joël: According to several experts the result seems to be true. Very astonishing indeed.
– GH from MOMay 17 '13 at 18:34

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I imagine the question being asked is: what is the greatest lower bound on the largest gap between P2s up to $x$, as a function of $x$? In other words, an analogue for P2s of Rankin's result on large gaps between primes.
– Greg MartinMay 17 '13 at 19:49

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@Greg and others. The burden is on the OP to clarify his question. We shouldn't be trying to guess what he means.
– Felipe VolochMay 17 '13 at 19:53

largestgap between P2s up to $x$, as a function of $x$? In other words, an analogue for P2s of Rankin's result on large gaps between primes. – Greg Martin May 17 '13 at 19:49