Proof of a conjecture of Kleinberg-Sawin-Speyer

Abstract

In Ellenberg and Gijswijt's groundbreaking work~\cite{EllenbergGijswijt}, the authors show that a subset of $\mathbb{Z}_3^{n}$ with no arithmetic progression of length 3 must be of size at most $2.755^n$ (no prior upper bound was known of $(3-\epsilon)^n)$), and provide for any prime $p$ a value $\lambda_p0$, such sets of size $e^{(\mu_p-\epsilon) n}$ exist for all sufficiently large $n$. The value of $\mu_p$ was left open, but a conjecture was stated which would imply that $e^{\mu_p}=\lambda_p$, i.e. the Ellenberg-Gijswijt bound is correct for the sum-free set problem. The purpose of this note is to close that gap. The conjecture of Kleinberg, Sawin and Speyer is true, and the Ellenberg-Gijswijt bound is the correct exponent for the sum-free set problem.