Let  p  be a fixed prime. A triangle in &#120125;  n    p   is an ordered triple ( x, y, z ) of points satisfying  x  +  y  +  z  = 0. Let  N  =  p   n   = |&#120125;  n    p  |. 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  ⊂ &#120125;  n    p   and the number of triangles in  X  ×  Y  ×  Z  is at most  ΔN  2 , 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 &#120125;  n    p  . 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  Δ  = ( ϵ /3)  C   p   , and we must have  Δ  ≤  ϵ   C   p   − 0(1) . In particular,  C  2  = 1 + 1/(5/3 − log 2  3) ≈ 13.239, and  C  3  = 1 + 1/ c  3  with [EQUATION], and [EQUATION], which gives  C  3  ≈ 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.
['Jacob Fox', 'László Miklós Lovász']
A tight bound for green's arithmetic triangle removal lemma in vector spaces