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 $C {3^N \over N^{1+ε}}$. We do this by obtaining quite strong information about the additive combinatorial properties of the large spectrum.

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