On a theorem of Plünnecke concerning the sum of a basis and a set of positive density
type: Item
instance of: scholarly article
title: On a theorem of Plünnecke concerning the sum of a basis and a set of positive density
zbMATH Open document ID: 0838.11006
author: Janice L. Malouf
published in: Journal of Number Theory
publication date: 1996-06-03T00:00:00Z
review text: In 1970 \textit{H. Plünnecke} [J. Reine Angew. Math. 243, 171-183 (1970; Zbl 0199.36701)] developed a graph-theoretic method in additive number theory and used it to prove the inequality \((*)\) \(\sigma (A + B) \geq \sigma (A)^{1 -1/h}\), where \(\sigma\) denotes Schnirelmann density, \(A\) is any set and \(B\) is any basis of order \(h\). In spite of its sharpness and the wide applicability of the method, this remained relatively unknown for a long time. The reviewer [Sci., Ser. A 3, 97-109 (1989; Zbl 0743.05052)] simplified this proof and gave further applications. The present paper gives a very clear exposition of a complete proof of \((*)\). The graph-theoretic part follows the reviewer's paper with certain changes, among others, a definitely better terminology. The derivation of \((*)\) from graph theory simplifies Plünnecke's proof.
Mathematics Subject Classification ID: 11B13
zbMATH DE Number: 809104
zbMATH Keywords: sumsets
zbMATH Keywords: positive density
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
full work available at URL: jnth.1995.1098
OpenAlex ID: W2010258078
DOI: 10.1006/JNTH.1995.1098
description: scientific article; zbMATH DE number 809104