I’m having to write posts at a ridiculous rate now, but until the official launch (at which point I expect a significant theoretical component to add to the experimental one — one decision we should make is whether to have separate theoretical and experimental threads or whether it is better to have a unified discussion) I’ll keep them short and mainly aimed at summarizing what is going on for the benefit of people who want to keep up with the discussion without reading hundreds of comments.
Most of the comments in the last post have concerned sequences with low discrepancy and additional constraints. If we define the map to be the one that takes the sequence to the sequence , then it is notable that the long sequences that we have found satisfy, not quite all the time but certainly most of the time, constraints such as or more generally . This suggests searching for sequences that satisfy these constraints exactly. One reason for doing so is that there are many fewer such sequences, so searching for low-discrepancy examples should be much quicker, and we have some reason to expect them to exist. And indeed, although we have not matched the length 1124, we have got sequences of length well over 500 that satisfy additional constraints.
The best way to keep up with what has been done is probably to check out the experimental results page on the Polymath5 wiki. And the wiki itself is still growing fairly quickly, so if you read through that then you will have a digest of most of what has been discussed in the comments.
A phenomenon that has recently been observed is that sometimes pairs of HAP-subsequences of good sequences do not agree all that well to start with, but after a while “lock in” to each other. We have not yet properly thought about this, but it seems pretty interesting.