A measure-theoretic approach to the theory of dense hypergraphs

Elek, Gábor; Szegedy, Balázs

doi:10.1016/j.aim.2012.06.022

Cited by 78 publications

(191 citation statements)

References 16 publications

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“…. , t k+2 is a k + 1-th order convolution of functions in F k and thus it is orthogonal to g. Then the non-standard version of Fubini's theorem [3] finishes the proof.…”

Section: By Theorem 15 the Statement Is Equivalent With The Fact Thatmentioning

confidence: 65%

“…One is to show that our regularity lemma is well behaved with respect to counting and the second goal is to show that function sequences in which the density of every fixed configuration converges have a nice limit object which is a measurable function on a nilspace. This fits well into the recently developed graph and hypergraph limit theories [17], [2], [18], [19], [3].…”

Section: Then For Functions On Groups In a Theorem 2 Holds With A-nilmentioning

confidence: 87%

“…It is an important fact (see [3]) that every bounded measurable function on X is almost everywhere equal to some ultra limit function f = lim ω f i .…”

Section: Measurable Functionsmentioning

confidence: 99%

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Higher Order Fourier Analysis

Tao

2012

Graduate Studies in Mathematics

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We develop a theory of higher order structures in compact abelian groups. In the frame of this theory we prove general inverse theorems and regularity lemmas for Gowers's uniformity norms.We put forward an algebraic interpretation of the notion "higher order Fourier analysis" in terms of continuous morphisms between structures called compact k-step nilspaces. As a byproduct of our results we obtain a new type of limit theory for functions on abelian groups in the spirit of the socalled graph limit theory. Our proofs are based on an exact (non-approximative) version of higher order Fourier analysis which appears on ultra product groups.

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“…. , t k+2 is a k + 1-th order convolution of functions in F k and thus it is orthogonal to g. Then the non-standard version of Fubini's theorem [3] finishes the proof.…”

Section: By Theorem 15 the Statement Is Equivalent With The Fact Thatmentioning

confidence: 65%

Section: Then For Functions On Groups In a Theorem 2 Holds With A-nilmentioning

confidence: 87%

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Higher Order Fourier Analysis

Tao

2012

Graduate Studies in Mathematics

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“…Let me mention that later Elek and Szegedy [7] very surprisingly used logic to prove the removal lemma as an ultraproduct. But no combinatorial proof has been given for Theorem 1.6 so far.…”

Section: Ergodic Approachmentioning

confidence: 97%

Arithmetic Progressions, Different Regularity Lemmas and Removal Lemmas

Szemerédi

2015

Commun. Math. Stat.

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This lecture note is mainly about arithmetic progressions, different regularity lemmas and removal lemmas. We will be very brief most of the time, trying to avoid technical details, even definitions. For most technical details, we refer the reader to references. Apart from arithmetic progressions, we also discuss property testing and extremal graph theory. Arithmetic ProgressionLet N be a positive integer. Let r k (N ) denote the cardinality of the largest subset of [N ] = {1, 2, . . . , N } containing no k-term arithmetic progression with distinct terms. The earliest results on r k (N ) are the following results of Roth and Szemerédi. Theorem 1.1 (Roth [23]) r 3 (N ) ≤ N log log N . Theorem 1.2 (Szemerédi [29]) r 4 (N ) = o(N ). Theorem 1.3 (Szemerédi [30]) r k (N ) = o(N ).

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“…We refer to [CKT-D11] for background on ultraproducts of measure preserving actions and also [ES08] for background on ultraproducts of measure spaces. Our notation has some changes from that of [CKT-D11] and is as follows.…”

mentioning

confidence: 99%

Weak equivalence and non-classifiability of measure preserving actions

Tucker-Drob

2013

Ergod. Th. Dynam. Sys.

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Abstract. Abért-Weiss have shown that the Bernoulli shift s Γ of a countably infinite group Γ is weakly contained in any free measure preserving action a of Γ. Proving a conjecture of Ioana we establish a strong version of this result by showing that s Γ × a is weakly equivalent to a. Using random Bernoulli shifts introduced by Abért-Glasner-Virag we generalized this to non-free actions, replacing s Γ with a random Bernoulli shift associated to an invariant random subgroup, and replacing the product action with a relatively independent joining. The result for free actions is used along with the theory of Borel reducibility and Hjorth's theory of turbulence to show that the equivalence relations of isomorphism, weak isomorphism, and unitary equivalence on the weak equivalence class of a free measure preserving action do not admit classification by countable structures. This in particular shows that there are no free weakly rigid actions, i.e., actions whose weak equivalence class and isomorphism class coincide, answering negatively a question of Abért and Elek.We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. A countably infinite residually finite group Γ is said to have property EMD * if the action p Γ of Γ on its profinite completion weakly contains all ergodic measure preserving actions of Γ, and Γ is said to have property MD if ι × p Γ weakly contains all measure preserving actions of Γ, where ι denotes the identity action on a standard non-atomic probability space. Kechris shows that EMD * implies MD and asks if the two properties are actually equivalent. We provide a positive answer to this question by studying the relationship between convexity and weak containment in the space of measure preserving actions.

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A measure-theoretic approach to the theory of dense hypergraphs

Cited by 78 publications

References 16 publications

Higher Order Fourier Analysis

Higher Order Fourier Analysis

Arithmetic Progressions, Different Regularity Lemmas and Removal Lemmas

Weak equivalence and non-classifiability of measure preserving actions

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