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

作者：Fox Jacob^*; Lovasz Laszlo Miklos

来源：Advances in Mathematics, 2017, 321: 287-297.

DOI：10.1016/j.aim.2017.09.037

• Arithmetic removal lemma
• Additive combinatorics
• Polynomial method
• Subspace sampling

Let p be a fixed prime. A triangle in F-p(n) is an ordered triple (x, y, z) of points satisfying x + y + z = 0. Let N = p(n) = vertical bar F-p(n)vertical bar. 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 of F-p(n) and the number of triangles in X x Y x 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 F-p(n). We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is an explicit number C-p such that we may take delta = (epsilon/3(p)(C), and we must have delta <= epsilon(Cp-o(1)). In particular, C-2 = 1 + 1/(5/3- log(2) 3) approximate to 13.239, and C-3 = 1 + 1/c(3) with c(3) = 1 - log b/log 3, b = a(-2/3) + a(1/3) +a(4/3), and a = root 33-1/8, which gives C-3 approximate to 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.

• 出版日期2017-12-1
• 单位MIT