An Arithmetic Analogue of Fox's Triangle Removal Argument

We give an arithmetic version of the recent proof of the triangle removal lemma by Fox [Fox11], for the group Fn 2 . A triangle in Fn 2 is a triple (x, y, z) such that x + y + z = 0. The triangle removal lemma for Fn 2 states that for every ε > 0 there is a δ > 0, such that if a subset A of Fn 2 requires the removal of at least ε · 2n elements to make it triangle-free, then it must contain at least δ · 22n triangles. This problem was first studied by Green [Gre05] who proved a lower bound on δ using an arithmetic regularity lemma. Regularity based lower bounds for triangle removal in graphs were recently improved by Fox and we give a direct proof of an analogous improvement for triangle removal in Fn 2 . The improved lower bound was already known to follow (for triangleremoval in all groups), using Fox’s removal lemma for directed cycles and a reduction by Král, Serra and Vena [KSV09] (see [Fox11, CF13]). The purpose of this note is to provide a direct Fourier-analytic proof for the group Fn 2 .