The discussion in the last thread has noticeably moved on to new topics. In particular, multiplicative functions have been much less in the spotlight. Some progress has been made on the question of whether the Fourier transform of a sequence of bounded discrepancy must be very large somewhere, though the question is far from answered, and it is not even clear that the answer is yes. (One might suggest that the answer is trivially yes if EDP is true, but that is to misunderstand the question. An advantage of this question is that there could in theory be a positive answer not just for -valued functions but also for -valued functions with norm at least , say.)
Another question that has been investigated, mostly by Sune, is the question about what happens if one takes another structure (consisting of “pseudointegers”) for which EDP makes sense. The motivation for this is either to find a more general statement that seems to be true or to find a more general statement that seems to be false. In the first case, one would see that certain features of were not crucial to the problem, which would decrease the size of the “proof space” in which one was searching (since now one would try to find proofs that did not use these incidental features of ). In the second case, one would see that certain features of were crucial to the problem (since without them the answer would be negative), which would again decrease the size of the proof space. Perhaps the least satisfactory outcome of these investigations would be an example of a system that was very similar to where it was possible to prove EDP. For example, perhaps one could find a system of real numbers that was closed under multiplication and had a counting function very similar to that of , but that was very far from closed under addition. That might mean that there were no troublesome additive examples, and one might even be able to prove a more general result (that applied, e.g., to -valued functions). This would be interesting, but the proof, if it worked, would be succeeding by getting rid of the difficulties rather than dealing with them. However, even this would have some bearing on EDP itself, I think, as it would be a strong indication that it was indeed necessary to prove EDP by showing that counterexamples had to have certain properties (such as additive periodicity) and then pressing on from there to a contradiction.
A question I have become interested in is understanding the behaviour of the quadratic form with matrix . The derivation of this matrix (as something to be interested in in connection with EDP) starts with this comment and is completed in this comment. I wondered what the positive eigenvector would look like, and Ian Martin obliged with some very nice plots of it. Here is a link to where these plots start. It seems to be a function with a number-theoretic formula (that is, with a value at that strongly depends on the prime factorization of — as one would of course expect), but we have not yet managed to guess what that formula is.
I now want to try to understand this quadratic form in Fourier space. That is, for any pair of real numbers I want to calculate , and I would then like to try to understand the shape of the kernel . Now looking back at this comment, one can see that
Since the bilinear form is determined by the quadratic form I’ll concentrate on the latter (which in any case is what interests me). So substituting into the above formula gives me
The infinite sum is a geometric progression, so this simplifies to
Note that for each the integrand is bounded unless is a multiple of , and more generally is small unless is close to a multiple of and is close to 0. So we do at least have the condition of being close to a rational with small denominator making an appearance here. (Why small denominator? Because then there will be more such that is a multiple of .)
I plan to investigate the sequence 1, -1, 0, 1, -1, 0, … from this perspective. It takes the value at . I shall attempt to understand from the Fourier side why this gives a sequence with such small discrepancy.
Before I finish this post, let me also mention a nice question of Alec’s, or perhaps it is better to call it a class of questions. It’s a little bit like the “entropy” question that I asked about EDP, but it’s about multiplicative functions. The question is this: you play a game with an adversary in which you take turns assigning values to primes. You want the resulting completely multiplicative function to have as small discrepancy as you can, whereas your adversary wants the discrepancy (that is, growth of partial sums) to be large. How well can you do? One can ask many variants, such as what happens if your adversary is forced to choose certain primes (for instance, every other prime), or if your adversary’s choices are revealed to you in advance (so now the question is what you can do if you are trying to make a low-discrepancy function but someone else has filled in half the values already and done so as badly as possible), or if you choose your values randomly, etc. etc. So far there don’t seem to be any concrete results, and yet it feels as though it ought to be possible to prove at least something non-trivial here.
One other question I’d like to highlight before I finish this post. It seems that we do not know whether EDP is true even if you insist that the HAPs have common differences that are either primes or powers of 2. The powers of 2 rule out all periodic sequences, but for a strange parity reason: for instance, if you have a sequence that’s periodic with period 72, then along the HAP with common difference 8 it is periodic with period 9, which means that the sum along each block of 9 is non-zero (because it is an odd number) and therefore the sums along that HAP grow linearly. Sune points out that the sequence is a simple counterexample over , but it’s not clear what message we can take from that, given that periodic sequences don’t work in the case. I like this question, because finding a counterexample should be easier if there is one, and if there isn’t, then proving the stronger theorem might be easier because HAPs with prime common differences are “independent” in a nice way.
Update: I intended, but forgot, to mention also some interesting ideas put forward by Gil in this comment. He has a proposal for trying to use probabilistic methods, and in particular methods that are suited to proving that rare events have non-zero probability when there is sufficient independence around, to show that there are many sequences with slow-growing discrepancy. It is not clear at this stage whether such an argument can be made to work, but it seems very likely that thinking about the problems that arise when one attempts to make it work will be fruitful.