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

Опубликовано 12 июля 2016, 18:17

Fix a prime power $q$. A triangle in $\mathbb{F}_q^n$ is a triple $x,y,z$ of elements with $x+y=z$. Green's triangle removal lemma in $\mathbb{F}_q^n$ states that for each $\epsilon>0$ there is a $\delta>0$ such that every subset of $\mathbb{F}_q^n$ which has few triangles (less than $\delta N^2$, where $N=q^n$ is the total number of elements), can be made triangle-free by removing few elements (less than $\epsilon N$). Previously, the best known bound was of tower type. Using a result of Kleinberg-Speyer, which builds on the breakthrough work of Croot-Lev-Pach on the cap set problem, and subsequent work by Ellenberg-Gijswijt and Blasiak-Church-Cohn-Grochow-Umans, we prove a polynomial bound for Green's triangle removal lemma, which we can prove is tight for small q, and conjecture it is tight for all q. Joint work with Jacob Fox.