## Archive for September 10th, 2010

### EDP20 — squares and fly traps

September 10, 2010

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 $\mathbb{Q}.$ Recall from the last post or two that what we want to do is this. Define a square in $\mathbb{N}\times\mathbb{N}$ to be a set of the form $[r,s]^2,$ where by $[r,s]$ I mean the set of all positive integers $n$ such that $r\leq n\leq s.$ Let us identify sets with their characteristic functions. We are trying to find, for any constant $C,$ a collection of squares $S_1,\dots,S_k$ and some coefficients $\lambda_1,\dots,\lambda_k$ with the following properties.

• $C\sum_{i=1}^k|\lambda_i|\leq\sum_{i=1}^k\lambda_it_i,$ where $S_i=[r_i,s_i]^2$ and $t_i=(s_i-r_i+1)$ is the number of points in the interval that defines $S_i,$ or, more relevantly, the number of points in the intersection of $S_i$ with the main diagonal of $\mathbb{N}\times\mathbb{N}.$
• Let $f(x,y)=\sum_i\lambda_iS_i(x,y).$ Then for any pair of coprime positive integers $a,b$ we have $\sum_{n=1}^\infty f(na,nb)=0.$

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 $\mathbb{Q}_+$ 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 $C$ implies that the discrepancy of an arbitrary $\pm 1$ sequence is at least $C^{1/2}.$ 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.]
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