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 × m) matrix with coefficients in F _q ; if the linear system Ax = b has o(q ^m−k ) solutions with x _i ∈ X _i , then we can eliminate 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.