Archive for December, 2009

The next Polymath project on this blog: some polls

December 28, 2009

For reasons that I have already gone into, I do not think that the origin-of-life project is suitable as the next Polymath project on this blog, though there seems to be enough enthusiasm for it that I am quite serious about giving it a try at some point in the not too distant future. The complexity-related project also no longer seems such a great idea for now. That leaves three candidates from amongst the problems that I have posted about recently: the project related to the polynomial-DHJ problem, the project related to the Littlewood problem, and the project to solve Erdős’s discrepancy problem. My impression is that for each of these three projects there is already a small group of highly interested people, and certainly my level of enthusiasm for each of these three problems is enough for me to be ready to devote plenty of time to it. (A theory I want to test is that if I post regularly and am not completely stuck, then that will be enough to keep the project feeling active and attracting contributions from other people as well, even if a proof does not appear to be round the corner.)

To help me decide which of the above three problems to go for, here are four polls. (more…)

Wiles meets his match

December 20, 2009

A brief return to the theme of mathematics in literature: I can’t resist sharing what is, by a long way, the silliest piece of fictional mathematics I have ever come across. It comes in “The Girl Who Played With Fire,” by the late Stieg Larsson, translated (not very well) by someone called Reg Keeling. Here is a little piece of advice for any author who wants to incorporate mathematics into a novel. If you don’t want what you write to be risibly unrealistic, it is not enough to read popular science books: you must also run what you write past a mathematician.

And here is the passage in question. (more…)

Erdős’s discrepancy problem

December 17, 2009

I’ve been pretty busy over the last month or so — hence my blog silence — and continue to be busy. But here is one more discussion of a problem that was on my earlier list of possible future Polymath projects. The problem in question is sometimes known as Erdős’s discrepancy problem. This will be a shortish post, because I don’t have much of a programme for solving the problem: if it were adopted, then we would be starting almost from scratch, but that could be an interesting experiment.

Here is the question.

Problem. Is it possible to find a \pm 1-valued sequence x_1,x_2,\dots and a constant C such that |\sum_{i=1}^nx_{id}|\leq C for every n and every d?

With the help of a little terminology we can ask this question in a slightly different way. If \mathcal{A} is a collection of subsets of a set X and f:X\rightarrow\{-1,1\}, define the discrepancy of f to be the maximum value of |\sum_{a\in A}f(a)| over all A\in\mathcal{A}. In our case, \mathcal{A} is the collection of all arithmetic progressions of the special form \{d,2d,3d,\dots,nd\}, and the question is whether there is any function that has bounded discrepancy for this \mathcal{A}. I say “bounded” rather than “finite” because one can define the function \delta(N,f) to be the discrepancy of f with respect to all those sets in \mathcal{A} that are subsets of \{1,2,\dots,N\}. Then the question is equivalent to asking whether there is any function f for which \delta(N,f) is a bounded function of N. (more…)