The symmetry preserving removal lemma

Szegedy, Balázs

doi:10.1090/s0002-9939-09-09860-8

Cited by 16 publications

(28 citation statements)

References 9 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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“…It is natural to ask if a similar removal lemma over groups, or even just abelian groups, also holds for sets of linear equations. See [33] for a related recent result.…”

Section: A Removal Lemma Over Groupsmentioning

confidence: 92%

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

Shapira

2010

Journal of the London Mathematical Society

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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(n p−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. We confirm Green's conjecture by showing that every set of linear equations (even non-homogenous) has the removal property.

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“…It was later observed in [10,13] that Theorem 3.1 can also be deduced directly from the graph removal lemma, bypassing the arithmetic regularity lemma. In this way Theorem 3.1 can also be generalized to deal with general linear equations using hypergraph removal lemmas; see [12] and references therein.…”

Section: By Inspecting the Proof One Notes That The Construction Of Amentioning

confidence: 99%

When the sieve works II

Matomäki

Shao

2018

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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AbstractFor a set of primes {\mathcal{P}}, let {\Psi(x;\mathcal{P})} be the number of positive integers {n\leq x} all of whose prime factors lie in {\mathcal{P}}. In this paper we classify the sets of primes {\mathcal{P}} such that {\Psi(x;\mathcal{P})} is within a constant factor of its expected value. This task was recently initiated by Granville, Koukoulopoulos and Matomäki [A. Granville, D. Koukoulopoulos and K. Matomäki, When the sieve works, Duke Math. J. 164 2015, 10, 1935–1969] and their main conjecture is proved in this paper. In particular, our main theorem implies that, if not too many large primes are sieved out in the sense that\sum_{\begin{subarray}{c}p\in\mathcal{P}\\ x^{1/v}<p\leq x^{1/u}\end{subarray}}\frac{1}{p}\geq\frac{1+\varepsilon}{u},for some {\varepsilon>0} and {v\geq u\geq 1}, then\Psi(x;\mathcal{P})\gg_{\varepsilon,v}x\prod_{\begin{subarray}{c}p\leq x\\ p\notin\mathcal{P}\end{subarray}}\bigg{(}1-\frac{1}{p}\bigg{)}.

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The symmetry preserving removal lemma

Cited by 16 publications

References 9 publications

A combinatorial proof of the Removal Lemma for Groups

A combinatorial proof of the Removal Lemma for Groups

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

When the sieve works II

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