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

Abstract

Let p be a fixed prime. A triangle in Fn p is an ordered triple (x; y; z) of points satisfying x+y+z = 0. Let N = pn = jFn p j. Green proved an arithmetic triangle removal lemma which says that for every 0 and prime p, there is a 0 such that if X; Y;Z Fn p and the number of triangles in X , Y,Z is at most N2, then we can delete 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= can be taken to be an exponential tower of twos of height logarithmic in 1= We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in Fn p . We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is a computable number Cp such that we may take = (=3)Cp, and we must have < Cp-o(1). In particular, C2 = 1 + 1=(5=3 - log2 3) 13:239, and C3 = 1 + 1=c3 with c3 = 1 - log b log 3 , b = a 2=3 + a1=3 + a4=3, and a = p 331 8 , which gives C3 - 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.