A Removal Lemma for Systems of Linear Equations over Finite Fields
نویسندگان
چکیده
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 and rank k; if the linear system Ax = b has o(q) 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.
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