On arithmetic progressions in \(A+B+C\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1362.11014
author: Kevin Henriot
publication date: 2014-11-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1211.4917
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11B30
Mathematics Subject Classification ID: 11B13
zbMATH DE Number: 6365887
zbMATH Keywords: arithmetic progressions
zbMATH Keywords: sumsets
zbMATH Keywords: density-increment strategy
zbMATH Keywords: Bohr sets
zbMATH Keywords: convolutions
MaRDI profile type: Publication
OpenAlex ID: W2963089191
arXiv ID: 1211.4917
DOI: 10.1093/IMRN/RNT121
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published in: IMRN. International Mathematics Research Notices
title: On arithmetic progressions in \(A+B+C\)
review text: In this paper the author investigates the additive structure of the three terms sumsets namely, long arithmetic progressions in the sumset \(A+B+C\), where \(A\), \(B\) and \(C\) are subsets of \(\mathbb{Z}_{N}\). He proves that If \(0 < \varepsilon < 1\) and \(A\), \(B\) and \(C\) are subsets of \(\mathbb{Z}_{N}\) with densities \(\alpha \geq (\log N)^{-2+\varepsilon}\) and \(\beta\), \(\gamma \geq e^{-c(\log N)^{c}}\) respectively, then there exists an absolute positive constant \(c\) such that the sumset \(A+B+C\) contains an arithmetic progression of length at least \(e^{c(\log N)^{c}}\). As a corollary he obtains that if \(A\) is a subset of primes less than \(N\) with cardinality \(\alpha N/\log N\) where \(\alpha \geq (\log N)^{-1}(\lg\log N)^{14}\), then there exists an absolute constant \(c > 0\) such that \(A+A+A\) contains an arithmetic progression of length at least \(e^{c(\alpha\log N)^{1/4}(\log\log N)^{-7/2}}\).NEWLINENEWLINEThe proofs are based on Fourier analytic techniques. The key tool is the well known density-increment argument. To use this he needs some properties of Bohr sets, the Katz-Koester transform established by \textit{T. Sanders} [Ann. Math. (2) 174, No. 1, 619--636 (2011; Zbl 1264.11004)] and the Croot-Sisask lemma [\textit{E. Croot} and \textit{O. Sisask}, Geom. Funct. Anal. 20, No. 6, 1367--1396 (2010; Zbl 1234.11013)] about sets of almost-periods of convolutions.
reviewed by: Sándor Kiss
description: scientific article; zbMATH DE number 6365887