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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A Disappearing Number on in London

September 10, 2010

I said in my post about the fourth day of the ICM that if you got the chance to see Simon McBurney’s play A Disappearing Number then you should. Well, I have just learned that it has a short run in London coming up — from today to the 25th of this month. If you open their West End Leaflet you will find some information about the play, and tickets can be booked at this page.