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

Shapira, A.

doi:10.1112/jlms/jdp076

Cited by 47 publications

(72 citation statements)

References 40 publications

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“…This however is not a straightforward task and we devote a forthcoming paper [8] to this issue in the context of finite fields. Similar extensions have been recently addressed by Shapira [14], Candela [3] and Szegedy [16].…”

Section: Corollary 3 Let G Be a Finite Group Of Odd Order N And A A supporting

confidence: 64%

A combinatorial proof of the Removal Lemma for Groups

Kráľ

Serra

Vena

2009

Journal of Combinatorial Theory, Series A

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Green [B. Green, A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005) 340-376] established a version of the Szemerédi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his Removal Lemma that allows us to extend its statement to all finite groups. We also discuss possible extensions of the Removal Lemma to systems of equations.

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Section: Corollary 3 Let G Be a Finite Group Of Odd Order N And A A supporting

confidence: 64%

A combinatorial proof of the Removal Lemma for Groups

Kráľ

Serra

Vena

2009

Journal of Combinatorial Theory, Series A

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“…Independently of us, Conjecture 9.4 from [3] was proved by Shapira [10] whose method also yields a different proof of Theorem 1. Shapira's proof also uses the colored version of the hypergraph Removal Lemma (Theorem 3) as our proof does.…”

Section: Our Main Results Is the Followingmentioning

confidence: 99%

A removal lemma for systems of linear equations over finite fields

2012

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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 and rank k; if the linear system Ax = b has o(q m−k ) solutions with x i ∈ X i , then we can destroy 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.

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“…We note that affine invariant properties as studied here are somewhat different from general (nonlinear) linear-invariant properties studied in [BCSX09,Sha10,KSV08,BGS10]. In their setting they consider properties whose domain is a large vector space over a constant sized field, and the property is invariant under linear-transformations of this domain, but they do not require the property to form a vector space.…”

Section: Motivationmentioning

confidence: 98%

On Sums of Locally Testable Affine Invariant Properties

Ben–Sasson

Grigorescu

Maatouk

et al. 2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Affine-invariant properties are an abstract class of properties that generalize some central algebraic ones, such as linearity and low-degree-ness, that have been studied extensively in the context of property testing. Affine invariant properties consider functions mapping a big field F q n to the subfield F q and include all properties that form an F q -vector space and are invariant under affine transformations of the domain. Almost all the known locally testable affine-invariant properties have so-called "single-orbit characterizations" -namely they are specified by a single local constraint on the property, and the "orbit" of this constraint, i.e., translations of this constraint induced by affine-invariance. Single-orbit characterizations by a local constraint are also known to imply local testability. Despite this prominent role in local testing for affine-invariant properties, single-orbit characterizations are not well-understood.In this work we show that properties with single-orbit characterizations are closed under "summation". Such a closure does not follow easily from definitions, and our proof uses some of the rich developing theory of affine-invariant properties. To complement this result, we also show that the property of being an n-variate low-degree polynomial over F q has a singleorbit characterization (even when the domain is viewed as F q n and so has very few affine transformations). This allows us to exploit known results on the single-orbit characterizability of "sparse" affine-invariant properties to show the following: The sum of any sparse affineinvariant property (properties satisfied by q O(n) -functions) with the set of degree d multivariate polynomials over F q has a single-orbit characterization (and is hence locally testable) when q is prime. Our result leads to the broadest known family of locally testable affine-invariant properties and gives rise to some intriguing questions/conjectures attempting to classify all locally testable affine-invariant properties.

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

Cited by 47 publications

References 40 publications

A combinatorial proof of the Removal Lemma for Groups

A combinatorial proof of the Removal Lemma for Groups

A removal lemma for systems of linear equations over finite fields

On Sums of Locally Testable Affine Invariant Properties

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