Archive for January 18th, 2011

More on the cap-set problem

January 18, 2011

Update: Nets Katz and Michael Bateman have posted a preprint to the arxiv that gives an n^\epsilon improvement to the bounds for the cap-set problem. More on this later.

I have been extremely pleased with the response to my first post on the cap-set problem, in that various people have very generously shared their ideas about it. Indeed, Nets Katz has even posted a comment that may give an improvement to the bound by a factor of n^\epsilon for some very small \epsilon. (For reasons I have already explained, this would be very interesting if it worked.) This followed a previous comment in which he outlined some arguments that he had found with Michael Bateman that go a lot further than the ones that appeared in the section entitled “Sets with very weak additive structure” in my previous post.

Also, Ernie Croot, in addition to sketching a different very interesting approach to the problem, suggested that I should get in touch with Seva Lev, which I did. Seva has an example of a cap set that is interestingly different from the usual examples. I will discuss it in a moment.

My plan, by the way, is to try to understand the ideas in Ernie’s and Nets’s comments and then explain them in my own words in a future post. But I’m not yet ready to do that, so in this post I want to discuss some other aspects of the problem, including Seva’s example.