I think this will be a bit long for a comment, so I’ll make it a post instead. I want to try to say as clearly as I can (which is not 100% clearly) what we know about a certain way of constructing a decomposition of the identity on Recall from the last post or two that what we want to do is this. Define a square in to be a set of the form where by I mean the set of all positive integers such that Let us identify sets with their characteristic functions. We are trying to find, for any constant a collection of squares and some coefficients with the following properties.
- where and is the number of points in the interval that defines or, more relevantly, the number of points in the intersection of with the main diagonal of
- Let Then for any pair of coprime positive integers we have
The second condition tells us that the off-diagonal elements of the matrix you get when you convert the decomposition into a matrix indexed by are all zero, and the first condition tells us that we have an efficient decomposition in the sense that we care about. In my previous post I showed why obtaining a collection of squares for a constant implies that the discrepancy of an arbitrary sequence is at least In this post I want to discuss some ideas for constructing such a system of squares and coefficients. I’ll look partly at ideas that don’t work, so that we can get a sense of what constraints are operating, and partly at ideas that might have a chance of working. I do not guarantee that the latter class of ideas will withstand even five minutes of serious thought: I have already found many approaches promising, only to dismiss them for almost trivial reasons. [Added later: the attempt to write up even the half promising ideas seems to have killed them off. So in the end this post consists entirely of half-baked ideas that I’m pretty sure don’t work. I hope this will lead either to some new and better ideas or to a convincing argument that the approach I am trying to use to create a decomposition cannot work.]