Again, this is just a brief report on what is going on. Plenty of work is still being put into making sure that the main results, questions, ideas, etc. are appearing on the Polymath5 wiki and especially (for the moment) the experimental results page, so looking at that is still a good way to keep up.
Alec came up with a sequence of length 1112 that is fairly different from the sequences of length 1124. See this comment and succeeding ones for information about it. I am particularly intrigued by the diagram showing partial sums of pointwise products of different HAP-subsequences. There is data here that demands to be explained, but the focus of the discussion is elsewhere for now.
Kevin has suggested thinking about how the number of sequences of length n compares with the number of sequences of length m, where by “sequences” I mean sequences with relevant properties such as having discrepancy 2 and perhaps being multiplicative as well. There is some interesting data here too.
The logarithm of the number of multiplicative sequences of discrepancy 2 and length n behaves in a suggestive way. Yet more to chew over.
I myself have started thinking about an algorithm for generating long low-discrepancy multiplicative sequences in the hope of proving that the discrepancy of such sequences can grow more slowly than logarithmically.
Gil has suggested looking at randomized modifications of the problem with a view to obtaining insights about the problem itself.
Alec has looked at the behaviour of Dirichlet series built out of some of the good examples we have.
Finally, I am hoping for good news from Klas in the not too distant future.