Proof of a Conjecture of Kleinberg-Sawin-Speyer

Abstract

In Ellenberg and Gijswijt's groundbreaking work, 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 $\lambda_p<p$ such that any subset of $\mathbb{Z}_p^{n}$ with no arithmetic progression of length 3 must be of size at most $\lambda_p^n$. Blasiak et al showed that the same bounds apply to tri-coloured sum-free sets, which are triples $\{(a_i,b_i,c_i):a_i,b_i,c_i\in\mathbb{Z}_p^{n}\}$ with $a_i+b_j+c_k=0$ if and only if $i=j=k$. Building on this work, Kleinberg, Sawin and Speyer gave a description of a value $\mu_p$ such that no tri-coloured sum-free sets of size $e^{\mu_p n}$ exist in $\mathbb{Z}_p^{n}$, but for any $\epsilon>0$, 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.