A tight bound for Green's arithmetic triangle removal lemma in vector spaces

TL;DR

This paper establishes a nearly optimal polynomial bound for Green's arithmetic triangle removal lemma in vector spaces over finite fields, resolving a long-standing open problem and improving previous exponential bounds.

Contribution

The authors prove a tight polynomial bound for Green's arithmetic triangle removal lemma in finite fields, determining the exact exponent and confirming polynomial bounds are possible.

Findings

Polynomial bound for the triangle removal lemma in _p^n with explicit exponents

The best possible exponent C_p is explicitly computed for p=2 and p=3

Uses advanced bounds on multicolored sum-free sets and recent breakthroughs in the cap set problem

Abstract

Let $p$ be a fixed prime. A triangle in $\mathbb{F}_p^n$ is an ordered triple $(x,y,z)$ of points satisfying $x+y+z=0$. Let $N=p^n=|\mathbb{F}_p^n|$. Green proved an arithmetic triangle removal lemma which says that for every $\epsilon>0$ and prime $p$, there is a $\delta>0$ such that if $X,Y,Z \subset \mathbb{F}_p^n$ and the number of triangles in $X \times Y \times Z$ is at most $\delta N^2$, then we can delete $\epsilon N$ elements from $X$, $Y$, and $Z$ and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that $1/\delta$ can be taken to be an exponential tower of twos of height logarithmic in $1/\epsilon$. We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in $\mathbb{F}_p^n$. We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is a computable number $C_p$ such that we may take $\delta = (\epsilon/3)^{C_p}$, and we must have $\delta \leq \epsilon^{C_p-o(1)}$. In particular, $C_2=1+1/(5/3 - \log_2 3) \approx 13.239$, and $C_3=1+1/c_3$ with $c_3=1-\frac{\log b}{\log 3}$, $b=a^{-2/3}+a^{1/3}+a^{4/3}$, and $a=\frac{\sqrt{33}-1}{8}$, which gives $C_3 \approx 13.901$. The proof uses Kleinberg, Sawin, and Speyer's essentially sharp bound on multicolored sum-free sets, which builds on the recent breakthrough on the cap set problem by Croot-Lev-Pach, and the subsequent work by Ellenberg-Gijswijt, Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans, and Alon.