  We give an arithmetic version of the recent proof of the triangle removal
lemma by Fox [Fox11], for the group $\mathbb{F}_2^n$.
  A triangle in $\mathbb{F}_2^n$ is a triple $(x,y,z)$ such that $x+y+z = 0$.
The triangle removal lemma for $\mathbb{F}_2^n$ states that for every $\epsilon
> 0$ there is a $\delta > 0$, such that if a subset $A$ of $\mathbb{F}_2^n$
requires the removal of at least $\epsilon \cdot 2^n$ elements to make it
triangle-free, then it must contain at least $\delta \cdot 2^{2n}$ triangles.
This problem was first studied by Green [Gre05] who proved a lower bound on
$\delta$ 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
$\mathbb{F}_2^n$.
  The improved lower bound was already known to follow (for triangle-removal in
all groups), using Fox's removal lemma for directed cycles and a reduction by
Kr\'{a}l, Serra and Vena [KSV09] (see [Fox11,CF13]). The purpose of this note
is to provide a direct Fourier-analytic proof for the group $\mathbb{F}_2^n.$
