On the Removal Lemma for Linear Systems over Abelian Groups

Abstract

In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the $k$--determinantal of an integer $(k\times m)$ matrix $A$ is coprime with the order $n$ of a group $G$ and the number of solutions of the system $Ax=b$ with $x_1\in X_1,...,x_m\in X_m$ is $o(n^{m-k})$, then we can eliminate $o(n)$ elements in each set to remove all these solutions. This is a follow-up of our former paper 'A Removal Lemma for Systems of Linear Equations over Finite Fields' arXiv:0809.1846v1, which dealt with the case of finite fields.

18 pages. A slightly more general version where the quantifiers for the main result are independent of the entries of the input matrix (only depend on its dimensions). Explanations concerning the condition on the k-determinantal in the main result are included

References in corpus (2)

• Limits of Hypergraphs, Removal and Regularity Lemmas. A Non-standard Approach
• A simple regularization of graphs

Cited by in corpus (1)

• On the number of monochromatic solutions of integer linear systems on Abelian groups