It looks like it may be time to roll over the search for narrow admissible sets to a new blog post, as we’re approaching 100 comments on the original thread.
In the meantime, an official polymath8 project has started. The wiki page is a good place to get started. Work to understand and improve the bounds in Zhang’s result on prime gaps has split into three main areas.
1) A reading seminar on Zhang’s Theorem 2.
2) A discussion on sieve theory, bridging the gap begin Zhang’s Theorem 2 and the parameter k_0 (see also the follow-up post).
3) Efforts to find narrow admissible sets of a given cardinality k_0 — the width of the narrowest set we find gives the current best bound on prime gaps.
We started on 3) in the previous blog post, and now will continue here. I’ll try to summarize the situation.
Just recently there’s been a significant improvement in , the desired cardinality of the admissible set, and we’re now looking at . Hopefully there’s going to be a whole new round of techniques, made possible by the significantly smaller problem size.
As I write this, the narrowest admissible set of size 34,429 found so far, due to Andrew Sutherland, has width 388,118.
This was found using the “greedy-greedy” algorithm. This starts with some chosen interval of integers, in this case [-185662,202456], and then sieves as follows. First discard 1 mod 2, and then 0 mod p for , for some parameter b. (I’m not actually sure of the value of this parameter in Andrew’s best set.) After that, for each prime we pick a minimally occupied residue class, and sieve that out. Assuming we picked a sufficiently wide interval to begin with, when we’re done the resulting admissible set with still have at least elements.
More generally, there are several directions worth pursuing
1. sharpening bounds on , the maximal cardinality of an admissible set of width at most ,
2. finding new constructions of admissible sets of a given size (and also ‘almost-admissible’ sequences)
3. developing algorithms or search techniques to find narrow admissible sets, perhaps starting from a wider or smaller admissible set, or starting from an ‘almost-admissible’ set.
(If these questions carry us in different directions, there’s always more room on the internet!)
For sufficiently small sizes (at most 372), everything is completely understood due to exhaustive computer searches described at http://www.opertech.com/primes/k-tuples.html. At least for now, we need to look at much larger sizes, so obtaining bounds and finding probabilistic methods is probably the right approach.
I’m writing this on a bus, beginning 30 hours of travel. (To be followed by a short sleep then an intense 3 day conference!) So my apologies if I missed something important!