Archive for February 19th, 2010

EDP8 — what next?

February 19, 2010

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 {}f is a quasirandom \pm 1-valued function defined on \{1,2,\dots,N\} for some large N, in the sense that all its Fourier coefficients are small. Must there be some HAP along which the sum has absolute value at least C? If so, how quasirandom does {}f 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 \alpha such that \sum_{n=1}^n e(\alpha n) 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. (more…)