Green’s Conjecture and Testing Linear Invariant Properties

Abstract

A system of l linear equations in p unknowns M x = b is said to have the removal property if every set S ⊆ {1, . . . , n} which contains o(np-l) solutions of M x = b can be turned into a set S′ containing no solution of M x = b, by the removal of o(n) elements. Green [GAFA 2005] proved that a single homogenous linear equation always has the removal property, and conjectured that every set of homogenous linear equations has the removal property. In this paper we confirm Green's conjecture by showing that every set of linear equations (even non-homogenous) has the removal property. We also discuss some applications of our result in theoretical computer science, and in particular, use it to resolve a conjecture of Bhattacharyya, Chen, Sudan and Xie [4] related to algorithms for testing properties of boolean functions.