It’s taken noticeably longer than usual for the number of comments on the previous EDP post to reach 100, so this is perhaps a good moment to think strategically about what we should do. Individual researchers continually have a choice — whether to take a break from the problem and work on other, possibly more fruitful, projects or to tackle that blank page head on and push towards a new level of understanding — and I see no difference with a Polymath project.
I would be interested in the views of others, but my own feeling is that there is still plenty to think about here. There has been a certain amount of talk about Fourier analysis, and that still feels like an insufficiently explored avenue. A good preliminary question, it seems to me, is this. Suppose that is a quasirandom -valued function defined on for some large , in the sense that all its Fourier coefficients are small. Must there be some HAP along which the sum has absolute value at least ? If so, how quasirandom does need to be? What I like about this question is that I think it should be substantially easier than EDP itself. It could be that a simple calculation would solve it: my attempts so far have failed, but not catastrophically enough to rule out the possibility that they could succeed next time. It also seems a pertinent question, because the functions we know of with very low discrepancy have some very high Fourier coefficients. (I don’t really mean Fourier coefficients, so much as real numbers such that has very large absolute value.) Therefore, proving that low discrepancy implies a high Fourier coefficient would be a result in the direction of proving that these examples are essentially the only ones, which would solve the problem.
Even if we knew in advance that there was no hope of solving EDP, there are still some questions that deserve more attention. As far as I can tell, nobody has even attempted to explain why some of our greedy algorithms appear to give a growth rate of for the partial sums. I would love to see just an informal argument for this.
I am also interested in (without yet having thought much about) a class of examples introduced by Kevin O’Bryant. These are sequences of the form where Kevin also considers the case where is infinite but the tend to zero so that the sum is always finite. Sune has pointed out that every sequence is of this form, but also makes the point that this may still be a good way to think about the problem. At the time of writing, I don’t have a lower bound of even in the case when . If that is not just me being stupid, then it raises the possibility that with an infinite one might be able to get a growth rate of in an interestingly new way. (A first step would be to get with a constant that tends to zero as tends to infinity.)
Another line of enquiry that Sune and Gil have thought about is to come up with toy problems that may be easier than EDP. One of Sune’s suggestions is interesting in that he has variants for which there appear to be counterexamples. This sheds light on EDP itself, because it shows that any proof of EDP would have to distinguish between the real problem and the variants.
I think it might be a good idea if we made a conscious decision that we would all focus on one aspect of the problem. For instance, we could choose a sub-problem with a nice statement and temporarily think of that as the “real” problem that we’re working on. But it might also be a bad idea: perhaps what we’ve done up to now, just letting people comment and others follow up on the comments if they find them sufficiently interesting, is the most efficient process. Incidentally, if you think you had a pretty good idea that got forgotten about, it’s not a bad idea to repeat it from time to time (as various people have done already, both in this project and in the DHJ project).
Anyhow, if anybody has thoughts about how we should proceed, then I’d be very pleased to hear them. In the absence of any suggestions to the contrary, I’ll think about the Fourier question and about trying to come up with interesting new multiplicative functions with low discrepancy.
While I’m at it, here’s a thought about such functions, which could turn out to be a special case of Kevin’s thought. (It might even be the basic idea that led him to his thought.) Our best examples so far (not including finite ones found by brute force) have been character-like functions, which are based on some prime number . Could one have something that is a bit like a character-like function but based on an irrational number? The idea would be to take a Sturmian sequence of some kind but to throw in a few zeros. (One way of doing this would be to have two fairly large intervals mod 1 and one small one, and an irrational number , and make 1 if lies in one of the large intervals, -1 if it lies in the other, and 0 if it lies in the small interval. Then one could do something about the zeros that was somehow based on the original sequence. It might end up being quasimultiplicative rather than multiplicative. I haven’t even remotely checked whether that was a non-crazy idea, but perhaps something vaguely similar could work, and perhaps it might do better than our existing character-like examples. The experimental evidence (by which I mean the 1124 sequences) certainly suggests that there ought to be some theoretical construction that does better than . Perhaps a useful exercise would be to write a character-like function in O’Bryant form and then try to generalize from that.
Another little thought, this time in the direction of toy problems, is this. (It may be a special case of one of Sune’s suggestions — I haven’t checked.) Let us say that an infinite discrete set of positive real numbers is a set of pseudointegers if it is closed under multiplication, and if the growth rate is roughly . I don’t know how rough it would have to be. One could then ask EDP for : a HAP would be defined as a set of the form , where is the set of the first elements of and is some element of . It’s just conceivable that one might be able to find an for which EDP is false, in which case it would tell us that a proof of EDP in would have to distinguish between and .
I believe that something like this has been done for the Riemann hypothesis but I can’t remember the details. Terry, if you’re reading this, you were the one who told me about the RH example — it would be great if you or anyone else could remind me what the precise statement is and give me a reference so that I can add them to this post. [It has now been provided: see the first comment below.]