Julian's musings

I recently had the fun of reading Ellenberg and Gijswijt’s paper on the capset problem, where they bound the size of a subset of $\mathbb{F}_q^n$ with no three terms in arithmetic progression by $c^n$ with $c<q$.

The paper is beautifully written, and amazingly needs only relatively elementary undergraduate algebra. (It is generalised to the Galois field $\mathbb{F}_q$, but if we take $q$ to be prime, then even that is unnecessary to understand the argument.)

I was somewhat stuck on two small points at the start of the proof of Proposition 4, and thought I would share my realisation of the argument here for others’ benefit.