New Bounds on cap sets

Abstract

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

New version: The most substantive change is the correction of our overstatement of the efficacy of the asymmetric Balog-Szemeredi-Gowers lemma. (Thanks to Izabella Laba and Olof Sisask for pointing this out.) 38 pages