This is the third in a series of posts in which I discuss problems that could perhaps form Polymath projects. Again, I am elaborating on a brief discussion that appeared in an earlier post on possible future projects. [Not for the first time in my experience, WordPress's dates have gone funny and this was posted not on the 17th as it says above but on the 20th.]
An obvious objection to the Littlewood conjecture as a Polymath project is that it is notoriously hard. On its own that might not necessarily be a convincing argument, since part of the point of Polymath is to attempt to succeed where individual mathematicians have failed. However, a second objection is that the best results in the direction of the Littlewood conjecture, due to Einsiedler, Katok and Lindenstrauss, use methods that are far from elementary (and far from understood by me). I envisage this project as an elementary one, at least to begin with, so does that make it completely unrealistic? I shall try to argue in this post that there is plenty that could potentially be done by elementary methods, even if attempting to prove the conjecture itself is probably too ambitious.
Another advantage of tackling the conjecture by elementary means is that if we find ourselves forced to reintroduce the non-elementary methods that have led to the very interesting results of Einsiedler, Katok and Lindenstrauss, we will have a deeper understanding of those methods than if we had just passively learnt about them. I myself prefer to rediscover things than to learn them: it isn’t always practical, but it’s easier if you half bear in mind that they are there and have a vague idea about them. (more…)