New bounds on cap sets

Abstract

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