New Bounds on cap sets

We provide an improvement over Meshulam’s bound on cap sets in $F_3^N$. We show that there exist universal $ϵ>0$ and $C>0$ so that any cap set in $F_3^N$ has size at most $C\frac{3^N}{N^{1+ϵ}}$. We do this by obtaining quite strong information about the additive combinatorial properties of the large spectrum.