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 X1, …, Xm be subsets of the finite field Fq and let A be a (k × m) matrix with coefficients in Fq; if the linear system Ax = b has o(qm−k) solutions with xi ∈ Xi, then we can eliminate all these solutions by deleting o(q) elements from each Xi. 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.