Right at the moment I’d say that Polymath5 has quietened down a little bit but is nevertheless continuing to make progress. There is quite a lot I could write about, but I thought it made more sense to put it on the wiki, so here I shall just briefly draw attention to some recent additions to the wiki (not just by me).
On the experimental side, I am losing track of how new everything is, but a list of the main data can be found at the start of the experimental page on the wiki.
On the theoretical side, Terence Tao has put proofs of some nice facts. One is a very clean proof of Mathias’s result that if the HAP sums of a sequence never go below -1 then they must be unbounded above. The cleanness of the argument comes partly from working with the positive rationals rather than the positive integers and partly from making use of results from topological dynamics. In particular, the latter allows one to talk about a “random rational number ” and give a sensible meaning to quantities such as the probability that .
Terry has also proved, by a fairly intricate combinatorial argument, that there must exist a HAP such that the difference between the maximum and minimum partial sums along that HAP is at least 3. (In the terminology we have been using, the drift is at least 3.)
I have created some pages about general proof strategies. These are linked to from a section of the main Polymath5 page. One is a proof strategy that I have already discussed in some detail in recent posts: showing that a counterexample must have multiplicative structure, tidying up that multiplicative structure as much as possible, and eventually showing that there cannot be any examples with multiplicative structure. A newer idea is to define a different parameter, a bit like drift but measured on a number of different distance scales and averaged. More details about this idea and the motivation for it can be found here.
Incidentally, it is easy to find out what the most recent changes to the wiki have been: on the left of each page is an internal link called Recent Changes.