A removal lemma for systems of linear equations over finite fields

The final publication is available at link.springer.comWe 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 and rank k; if the linear system Ax = b has o(q m-k ) solutions with xi ¿ Xi , then we can destroy 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.Supported as project 1M0545 by Czech Ministry of Education. Supported by the Catalan Research Council under project 2005SGR0258. Supported by the Spanish Research Council under project MTM2005-08990-C01-CPeer ReviewedPostprint (author's final draft