In this post I want to take the following attitude. Although there are several promising approaches to solving EDP, I am going to concentrate just on the representation-of-diagonals idea and pretend that that is the problem. That is, I want to pretend that the main problem we are trying to solve is not a problem about discrepancy of sequences in HAPs but the following question instead.
Problem. Is it true that for every positive constant there exists a diagonal matrix with trace at least that can be expressed as a linear combination of the form with and each and the characteristic function of a homogeneous arithmetic progression?
There are other equivalent ways of formulating this problem, but I’ll stick with this one for now. Incidentally, can be thought of as notation for the characteristic function of .
In this post I want to try to encourage a certain stepping back. Our general problem is to construct something with some rather delicate properties. We don’t really know how to go about it. In that kind of situation, what is one to do? Does one wait to be struck suddenly by a brilliant idea? Or is there a way of searching systematically? Of course, I very much hope it will be the latter, or at least nearer to the latter. (more…)