On subsets of finite Abelian groups with no 3-term arithmetic progressions
type: Item
instance of: scholarly article
title: On subsets of finite Abelian groups with no 3-term arithmetic progressions
zbMATH Open document ID: 0832.11006
DOI: 10.1016/0097-3165(95)90024-1
published in: Journal of Combinatorial Theory. Series A
publication date: 1996-03-12T00:00:00Z
review text: For a finite commutative group \(G\) let \(D(G)\) denote the maximal cardinality of subsets \(A\subset G\) that do not contain a 3-term arithmetic progression. Heath-Brown and Szemerédi's improvement of Roth's theorem can be formulated as \(D(\mathbb{Z}_m) \ll m(\log m)^{-c}\). Here this is extended to all groups in the form \(D(G) \ll|G|(\log |G|)^{-c}\). The key ingredient is the following result. If \(G= \mathbb{Z}_{k_1} \otimes \dots \otimes \mathbb{Z}_{k_n}\) with \(k_1 |\dots|k_n\), then \(D(G)\leq 2|G|/n\). Thus, for instance, for \(G= \mathbb{Z}^n_3\) the exponent is \(c=1\). (On the other hand, no analog of Behrend's construction is known for such groups, hence it is possible that the real order is \(D(G) \ll|G|^{1- \delta}\) with some \(\delta >0\).) The proof uses a discrete version of Roth's method.
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 11B75
zbMATH DE Number: 774653
zbMATH Keywords: subsets of finite Abelian groups
zbMATH Keywords: maximal cardinality
zbMATH Keywords: 3-term arithmetic progression
zbMATH Keywords: Roth's method
author: Roy Meshulam
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
full work available at URL: 0097-3165(95)90024-1
OpenAlex ID: W2072865951
Wikidata QID: Q60568641
cites work: Q4284607
cites work: A lattice point problem and additive number theory
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cites work: On Certain Sets of Integers
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description: scientific article; zbMATH DE number 774653