Meshulam bound for cap sets ($|A|=O(3^n/n)$)

A cap set in ${\mathbb{F}}_3^n$ has subset density $O(1/n)$. A Fourier analysis on a finite abelian group proof uses a hyperplane density increment for cap sets repeatedly: absence of nondiagonal zero-sum triples forces a large finite abelian Fourier coefficient, and hence a denser slice. This is a weaker bound than the Ellenberg–Gijswijt cap-set bound, but illustrates the density increment method.