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…)