REAL

On Linear Configurations in Subsets of Compact Abelian Groups, and Invariant Measurable Hypergraphs

Candela, P. and Szegedy, Balázs and Vena, L. (2016) On Linear Configurations in Subsets of Compact Abelian Groups, and Invariant Measurable Hypergraphs. ANNALS OF COMBINATORICS, 20 (3). pp. 487-524. ISSN 0218-0006

[img] Text
art3A10.10072Fs00026_016_0313_1_u.pdf - Published Version
Restricted to Registered users only

Download (1720Kb) | Request a copy
[img]
Preview
Text
44224.pdf - Submitted Version

Download (470Kb) | Preview

Abstract

We prove an arithmetic removal result for all compact abelian groups, generalizing a finitary removal result of Král’, Serra, and the third author. To this end, we consider infinite measurable hypergraphs that are invariant under certain group actions, and for these hypergraphs we prove a symmetry-preserving removal lemma, which extends a finitary result of the same name by the second author. We deduce our arithmetic removal result by applying this lemma to a specific type of invariant measurable hypergraph. As a direct consequence of our removal result, we obtain the following generalization of Szemerédi’s theorem: for any compact abelian group G, any measurable set A⊆ G with Haar probability μ(A) ≥ α> 0 satisfies ∫G∫G1A(x)1A(x+r)..1A(x+(k-1)r)dμ(x)dμ(r)≥c, where the constant c= c(α, k) > 0 is valid uniformly for all G. This result is shown to hold more generally for any translationinvariant system of r linear equations given by an integer matrix with coprime r× r minors. © 2016, Springer International Publishing.

Item Type: Article
Uncontrolled Keywords: removal results; linear configurations; hypergraphs; Compact Abelian groups
Subjects: Q Science / természettudomány > QA Mathematics / matematika
Q Science / természettudomány > QA Mathematics / matematika > QA166-QA166.245 Graphs theory / gráfelmélet
SWORD Depositor: MTMT SWORD
Depositing User: MTMT SWORD
Date Deposited: 03 Jan 2017 07:53
Last Modified: 09 Jan 2017 08:11
URI: http://real.mtak.hu/id/eprint/44224

Actions (login required)

View Item View Item