I wasn’t expecting to have to write another post quite this soon, so this is another one where I don’t have much to say. Here are a few bits and bobs from the last lot of comments, but there’s plenty more in the comments themselves, and actually I think that it’s not that hard to browse through them now that we have depth-1 threading and quite a lot of the comments are very short.
Johan de Jong came up with a very interesting variant of the problem, in which is replaced by the space of polynomials over . I confess to being a little sad when the problem was solved negatively soon afterwards, as it had looked as though it might be a rather good model problem. However, the solution was nice.
One of the general aims at the moment is to try to show that a multiplicative function of bounded discrepancy must have some kind of character-like behaviour. Terence Tao has come up with an intriguing argument that shows not that but at least something in the right ball park.
Work has continued on a human proof that completely multiplicative sequences must have discrepancy greater than 2. It looks as though the proof will be complete before too long, and not too painful. Some nice tricks of Jason Dyer have come in helpful in reducing the amount of case analysis that is needed.