The previous talk in the series is here. The next talk in the series is here.
A result that has played a central role in additive combinatorics is the statement that for every positive c there exists n such that every subset of 𝔽3n of density at least c contains three distinct vectors x, y and z such that x + y + z = 0. For a long time, a major open problem was to decide what relationship between c and n needs to hold for this conclusion to follow: in one direction, it was known to be enough if c is at least C/n for an absolute constant C, but the only known constructions of sets without solutions to x + y + z = 0 gave sets of density an for some a < 1: in other words, exponentially small density.
There was a very interesting but small improvement to the C/n bound by Michael Bateman and Nets Katz in 2011, and then in 2016 a remarkable development occurred, when work of Ernie Croot, Seva Lev and Péter Pál Pach on a similar problem led to a solution by Jordan Ellenberg and Dion Gijswijt, who independently proved an exponential upper bound for the density needed: that is, they showed that there is a constant a less than 1 such that every set of density at least an contains distinct x, y and z with x + y + z = 0.
I devote two videos to this solution, or rather to a modification of the solution due to Terence Tao. Tao’s approach uses a notion called slice rank, which I introduce in this video. The solution to the cap-set problem will be given in the next one.
This video was produced by Tim Gowers as part of his Part III course at the University of Cambridge. Printed notes for this course are available here.