New lower bounds for cap sets

One of the best known problems in additive combinatorics, the cap set problem, asks how large a subset of 𝐹3𝑛 can be if it contains no non-trivial solutions to the equation 𝑥+𝑦+𝑧=0. (A trivial solution is one where 𝑥=𝑦=𝑧.) Such sets are known as cap sets. For a long time, the best known upper bound was due to Meshulam, who adapted the proof of Roth’s theorem on sets of integers without arithmetic progressions of length 3 to this context, obtaining a bound of 𝑂(𝑛−13𝑛). In the other direction, the set {0,1}𝑛 gives a lower bound of 2𝑛, which had been improved to 𝑐𝑛 for a constant 𝑐 that is a little bigger than 2. In 2011 there was a breakthrough when the upper bound was improved by Bateman and Katz to one of the form 𝑂(𝑛−(1+𝑐)3𝑛), but this still left a big gap. Then in 2016, Croot, Lev and Pach posted a paper to arXiv obtaining an exponential improvement to the upper bound for an analogous problem in 𝐹4𝑛, which was followed swiftly by a paper by Ellenberg and Gijswijt that obtained an upper bound of the form 𝐶𝑛 for a constant 𝐶<3, showing that the best known lower bound was at least of the right form.