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Let $ A $ be a set of odd primes such that between any two consecutive elements thereof there is at least one prime gap that occurs infinitely often, i.e. an even integer $ g $ such that the equality $ p_{n+1}-p_{n}=g $ is true for infinitely many values of $ n $ and let's denote by $ A' $ the set of such $ g $ 's.

Can it be proven with current technology that $ \lim_{x\to\infty}\dfrac{\#\{g\in A',g\leqslant x\}}{x}=\dfrac{1}{2} $? If not what is the best lower bound currently reachable for the considered limit ? I'm mainly interested in unconditional results.

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    $\begingroup$ I don't understand the definition of $A$. Suppose that by a cruel twist of fate it turns out that the set $A'$ is the singleton set $\{6\}$. Then there could be infinitely many different choices for $A$. right? Each having at most 5 elements. Or do I misunderstand? (Of course in the end the definition of $A$ doesn't matter much since the question is about $A'$ not $A$) $\endgroup$ – Vincent Jun 28 '18 at 19:17
  • $\begingroup$ I doubt anything close to that is currently achievable. I suspect the density of actual gaps and actual differences is similarly unknown, except for the fact that they are arbitrarily large, mostly even, and occupy all the small positive even numbers. And this is without the proviso that the gap occurs infinitely often. Gerhard "Still Working On Gap Bounds" Paseman, 2018.06.28. $\endgroup$ – Gerhard Paseman Jun 28 '18 at 19:20
  • $\begingroup$ @Vincent : I want $ A $ to as "dense" as possible, i.e the distance between two consecutive elements thereof should be minimal. Hence in your example it would be $\{5,7\} $. $\endgroup$ – Sylvain JULIEN Jun 28 '18 at 19:34
  • $\begingroup$ Oh, I got your argument. Indeed there is not necessarily only one $ A $. I edited accordingly. $\endgroup$ – Sylvain JULIEN Jun 28 '18 at 19:39
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    $\begingroup$ Then I don't understand how the set $A$ interacts with anything else in the post. $\endgroup$ – Greg Martin Jun 29 '18 at 7:26
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If $p_{n+1}-p_n=g$ for infinitely many values of $n$, then $g$ is called a Polignac number. Motivated by Yitang Zhang's famous work in 2013, Janos Pintz proved that the set of Polignac numbers has a positive lower asymptotic density, see Section 2.2 of Pintz's paper http://arxiv.org/abs/1305.6289.

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