A proof of Green's conjecture regarding the removal properties of sets of linear equations

A system of ℓ linear equations in p unknowns Mx = b is said to have the removal property if every set S ⊆ {1, …, n} that contains o(np−ℓ) solutions of Mx = b can be turned into a set S′ containing no solution of Mx = b by the removal of o(n) elements. Green (Geom. Funct. Anal. 15 (2005) 340–376) 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 related to algorithms for testing properties of boolean functions.

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