A connection between matchings and removal in abelian groups

In a finite abelian group $G$, define an additive matching to be a collection of triples $(x_i, y_i, z_i)$ such that $x_i + y_j + z_k = 0$ if and only if $i = j = k$. In the case that $G = \mathbb{F}_2^n$, Kleinberg, building on work of Croot-Lev-Pach and Ellenberg-Gijswijt, proved a polynomial upper bound on the size of an additive matching. Fox and Lov\'{a}sz used this to deduce polynomial bounds on Green's arithmetic removal lemma in $\mathbb{F}_2^n$. If $G$ is taken to be an arbitrary finite abelian group, the questions of bounding the size of an additive matching and giving bounds for Green's arithmetic removal lemma are much less well understood. In this note, we adapt the methods of Fox and Lov\'{a}sz to prove that, provided we can assume a sufficiently strong bound on the size of an additive matching in cyclic groups, a similar bound should hold in the case of removal.