An Arithmetic Analogue of Fox's Improved Triangle Removal Lemma

We give an arithmetic version of the recent proof of the improved triangle removal lemma by Fox [Fox11], for the group $F_2^n$.
A triangle in $F_2^n$ is a tuple (x,y,z) such that $x+y+z = 0$. The triangle removal lemma for $F_2^n$ states that for every \eps > 0, there is a \delta >0, such that if a subset A of F_2^n requires the removal of at least eps 2^n elements to make it triangle-free, then it must contain at least \delta 2^{2n} triangles. We give a direct proof which gives an improved lower bound for \delta (as a function of \eps), analogous to the one obtained by Fox for triangle removal in graphs.
This result was previously known via a reduction from the improved removal lemma for directed cycles [Fox11,Král-Serra-Vena 09]. However, we believe our proof in this simplified setting is more transparent, and defines fourier-analytic notions that may be of independent interest.