A Removal Lemma for Systems of Linear Equations over Finite Fields

We prove a removal lemma for systems of linear equations over finite fields: let $X_1,...,X_m$ be subsets of the finite field $\F_q$ and let $A$ be a $(k\times m)$ matrix with coefficients in $\F_q$ and rank $k$; if the linear system $Ax=b$ has $o(q^{m-k})$ solutions with $x_i\in X_i$, then we can destroy all these solutions by deleting $o(q)$ elements from each $X_i$. This extends a result of Green [Geometric and Functional Analysis 15(2) (2005), 340--376] for a single linear equation in abelian groups to systems of linear equations. In particular, we also obtain an analogous result for systems of equations over integers, a result conjectured by Green. Our proof uses the colored version of the hypergraph Removal Lemma.