Open question: best bounds for cap sets

Earlier this month, in the previous incarnation of this page, I posed a question which I thought was unsolved, and obtained the answer (in fact, it was solved 25 years ago) within a week. Now that this new version of the page has better feedback capability, I am now tempted to try again, since I have a large number of such questions which I would like to publicise. (Actually, I even have a secret web page full of these somewhere near my home page, though it will take a non-trivial amount of effort to find it!)

Perhaps my favourite open question is the problem on the maximal size of a cap set – a subset of ( being the finite field of three elements) which contains no lines, or equivalently no non-trivial arithmetic progressions of length three. As an upper bound, one can easily modify the proof of Roth’s theorem to show that cap sets must have size (see e.g. this paper of Meshulam). This of course is better than the trivial bound of once n is large. In the converse direction, the trivial example shows that cap sets can be as large as ; the current world record is , held by Edel. The gap between these two bounds is rather enormous; I would be very interested in either an improvement of the upper bound to , or an improvement of the lower bound to . (I believe both improvements are true, though a good friend of mine disagrees about the improvement to the lower bound.)

Uchiyama’s constructive proof of the Fefferman-Stein decomposition

I’ve just uploaded the short story “Uchiyama’s constructive proof of the Fefferman-Stein decomposition“. In 1982, Uchiyama gave a new proof of the celebrated Fefferman-Stein theorem that expressed any BMO function as the sum of a bounded function, and Riesz transforms of bounded functions. Unlike the original proof (which relied, among other things, on the Hahn-Banach theorem), Uchiyama’s proof was very explicit, constructing the decomposition by building the bounded functions one Littlewood-Paley frequency band at a time while keeping the functions taking values on or near a sphere, and then iterating away the error. Here I have written some notes on how the proof goes. The notes are a little condensed, in that a number of standard computations involving estimations of Schwartz tails, Carleson measures, etc. have been omitted, but hopefully the gist of the argument is still clear.