Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5778304 | Advances in Mathematics | 2017 | 11 Pages |
Abstract
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 δâ¤ÏµCpâo(1). In particular, C2=1+1/(5/3âlog2â¡3)â13.239, and C3=1+1/c3 with c3=1âlogâ¡blogâ¡3, b=aâ2/3+a1/3+a4/3, and a=33â18, which gives C3â13.901. 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.
Related Topics
Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Jacob Fox, László Miklós Lovász,