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

Abstract

Let p be a fixed prime. A triangle in Fpn is an ordered triple (x,y,z) of points satisfying x+y+z=0. Let N=pn=|Fpn|. Green proved an arithmetic triangle removal lemma which says that for every ϵ>0 and prime p, there is a δ>0 such that if X,Y,Z⊂Fpn and the number of triangles in X×Y×Z is at most δN2, then we can delete ϵN 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/δ can be taken to be an exponential tower of twos of height logarithmic in 1/ϵ.We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in Fpn. We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is an explicit number Cp such that we may take δ=(ϵ/3)Cp, and we must have δ≤ϵCp−o(1). In particular, C2=1+1/(5/3−log2⁡3)≈13.239, and C3=1+1/c3 with c3=1−log⁡blog⁡3, b=a−2/3+a1/3+a4/3, and a=33−18, which gives C3≈13.901. The proof uses the essentially sharp bound on multicolored sum-free sets due to work of Kleinberg–Sawin–Speyer, Norin, and Pebody, 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.