More on the cap-set problem

Update: Nets Katz and Michael Bateman have posted a preprint to the arxiv that gives an 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 for some very small (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.

What is difficult about the cap-set problem?

An observation that has been made many times is that the internet is a perfect place to disseminate mathematical knowledge that is worth disseminating but that is not conventionally publishable. Another observation that has been made many times is that either of the following features can render a piece of mathematics unpublishable, even if it could be useful to other mathematicians.

1. It is not original.

2. It does not lead anywhere conclusive.

A piece of unoriginal mathematical writing can be useful if it explains a known piece of mathematics in a new way (and “new” here does not mean radically original — just a slightly different slant that can potentially appeal to a different audience), or if it explains something that “all the experts know” but nobody has bothered to write down. And an inconclusive observation such as “This proof attempt looks promising at first, but the following construction suggests that it won’t work, though unfortunately I cannot actually prove a rigorous result along those lines,” can be very helpful to somebody else who is thinking about a problem.

I’m mentioning all this because I have recently spent a couple of weeks thinking about the cap-set problem, getting excited about an approach, realizing that it can’t work, and finally connecting these observations with conversations I have had in the past (in particular with Ben Green and Tom Sanders) in order to see that these thoughts are ones that are almost certainly familiar to several people. They ought to have been familiar to me too, but the fact is that they weren’t, so I’m writing this as a kind of time-saving device for anyone who wants to try their hand at the cap-set problem and who isn’t one of the handful of experts who will find everything that I write obvious.

Added a little later: this post has taken far longer to write than I expected, because I went on to realize that the realization that the approach couldn’t work was itself problematic. It is possible that some of what I write below is new after all, though that is not my main concern. (It seems to me that mathematicians sometimes pay too much attention to who was the first to make a relatively simple observation that is made with high probability by anybody sensible who thinks seriously about a given problem. The observations here are of that kind, though if any of them led to a proof then I suppose they might look better with hindsight.) The post has ended up as a general discussion of the problem with several loose ends. I hope that others may see how to tie some of them up and be prepared to share their thoughts. This may sound like the beginning of a Polymath project. I’m not sure whether it is, but discuss the possibility at the end of the post.