A linear bound on the dimension in Green-Ruzsa's theorem
type: Item
instance of: scholarly article
title: A linear bound on the dimension in Green-Ruzsa's theorem
zbMATH Open document ID: 1304.11122
author: Karol Cwalina
author: Tomasz Schoen
published in: Journal of Number Theory
publication date: 2013-02-21T00:00:00Z
review text: Let \(G\) be any abelian group. If \(H\) is a subgroup of \(G\) and \[ P=\{x_0+l_1x_1+\cdots+l_dx_d:\;l_i\in\{-L_i,\ldots,0,\ldots,L_i\}\;\text{for}\;i=1,\ldots,l\} \] is a generalized arithmetic progression of dimension \(d\) and size \(\prod_{i=1}^d(2L_i+1)\), then we call \(P+H=\{x+h:\;x\in P\;\text{and}\;h\in H\}\) a coset progression. Let \(A\) be a nonempty subset of \(G\) with \(|A+A|\leq K|A|\), where \(K\) is a positive constant. \textit{B. Green} and \textit{I. Z. Ruzsa} [J. Lond. Math. Soc., II. Ser. 75, No. 1, 163--175 (2007; Zbl 1133.11058)] showed that \(A\) is contained in a coset progression \(P+H\) of dimension \(d(P+H)\leq CK^4\log(K+2)\) and \(\text{size}(P+H)\leq\text{exp}(CK^4\log^2(K+2))|A|\), where \(C\) is a positive constant. Motivated by \textit{M.-C. Chang}'s work [Duke Math. J. 113, No. 3, 399--419 (2002; Zbl 1035.11048)] on a bound in Freiman's theorem, the authors proved that either there is a proper coset progression \(P+H\) of dimension \(d(P+H)\leq 2|K|\) and \(\text{size}(P+H)\leq \text{exp}(CK^4\log^2(K+2))|A|\) such that \(A\subseteq (P+H)\), or \(A\) is fully contained in at most \(CK^3\log^2K\) cosets (of a subgroup of \(G\)), whose total cardinality is bounded by \(\text{exp}(CK^4\log^2(K+2))|A|\).
Mathematics Subject Classification ID: 11P70
Mathematics Subject Classification ID: 20K01
zbMATH DE Number: 6138268
zbMATH Keywords: Freiman's theorem
zbMATH Keywords: Green-Ruzsa theorem
zbMATH Keywords: additive combinatorics
zbMATH Keywords: sumset
reviewed by: Zhi-Wei Sun
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.09.012
OpenAlex ID: W1963988947
cites work: A polynomial bound in Freiman's theorem.
cites work: Q3215325
cites work: Freiman's theorem in an arbitrary abelian group
cites work: Generalized arithmetical progressions and sumsets
cites work: Appendix to `Roth's theorem on progressions revisited' by J. Bourgain
cites work: On the Bogolyubov-Ruzsa lemma
cites work: Near optimal bounds in Freiman's theorem
cites work: Q5393666
DOI: 10.1016/J.JNT.2012.09.012
Recommended article: The Greene-Krantz conjecture in dimension two
Recommended article: Extensions of the linear bound in the Füredi-Hajnal conjecture
Recommended article: Green's conjecture and testing linear-invariant properties
Recommended article: Green’s Conjecture and Testing Linear Invariant Properties
Recommended article: A criterion for Greenberg's conjecture
Recommended article: Some algebraic consequences of Green's hyperplane restriction theorems
Recommended article: On Greenberg's generalized conjecture
Recommended article: Upper Bound for the Dvoretzky Dimension in Milman-Schechtman Theorem
Recommended article: A generalization of Greenberg's L-invariant
Recommended article: A linear bound for \(\rho(n)\)
description: scientific article; zbMATH DE number 6138268