A removal lemma for systems of linear equations over finite fields
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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.
KeywordsAbelian Group Arithmetic Progression Colored Version Combinatorial Proof Regularity Lemma
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