New Bounds on cap sets

TL;DR

This paper improves the upper bounds on the size of cap sets in F_3^N by analyzing their additive combinatorial properties, showing they are smaller than previously known by a polynomial factor.

Contribution

It introduces a new bound on cap set sizes in F_3^N, surpassing Meshulam's bound, using advanced additive combinatorics techniques.

Findings

Cap sets in F_3^N are at most C * 3^N / N^{1+ε} in size.

Large spectrum of cap sets exhibits strong additive combinatorial properties.

Provides universal constants ε > 0 and C > 0 for the bounds.

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