On arithmetic structures in dense sets of integers
type: Item
instance of: scholarly article
title: On arithmetic structures in dense sets of integers
zbMATH Open document ID: 1020.11010
DOI: 10.1215/S0012-7094-02-11422-7
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: The main aim of the paper is to prove the result that if \(A\subset\{1,2,\dots,N\}\) has density at least \((\log\log N)^{-c}\), where \(c\) is an absolute constant, then \(A\) contains a \(3\)-term arithmetic progression whose common difference is positive and of the form \(x^2+y^2\) for some integers \(x,y\) not both zero. The proof method is based on ideas developed by Gowers and Sárközy combined with an application of the Selberg sieve. Using a new variant of Sárközy's proof, the author also shows that there is a constant \(c\) such that, if \(A\subset\{1,2,\dots,N\}\) has density at least \(c(\log\log N)^{-1/11}\), then \(A\) contains two elements \(a,a^\prime\) with \(a-a^\prime\) a nonzero square.
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 05D10
Mathematics Subject Classification ID: 11N36
zbMATH DE Number: 1820923
zbMATH Keywords: Szemerédi theorem
zbMATH Keywords: Sárközy theorem
zbMATH Keywords: arithmetic progression
zbMATH Keywords: Selberg sieve
zbMATH Keywords: square difference
zbMATH Keywords: sum of two squares
reviewed by: Štefan Porubský
MaRDI profile type: Publication
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full work available at URL: s0012-7094-02-11422-7
OpenAlex ID: W1967102966
author: Ben Green
description: scientific article; zbMATH DE number 1820923