Citation:Bateman, M., & Katz, N. H. (2012). New bounds on cap sets. Journal of the American Mathematical Society, 25(2), 585-613. http://dx.doi.org/10.1090/S0894-0347-2011-00725-X
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.
