An inverse theorem for sums of sets of lattice points
type: Item
instance of: scholarly article
title: An inverse theorem for sums of sets of lattice points
zbMATH Open document ID: 0802.11008
author: Melvyn B. Nathanson
published in: Journal of Number Theory
publication date: 1994-12-15T00:00:00Z
review text: Let \(A\) be a finite subset of an \(n\)-dimensional Euclidean space, \(1<c< 2^ n\), and assume that \(| 2A| \leq c| A|\), where \(2A\) denotes the set of all vectors in the form \(a_ 1+ a_ 2\), \(a_ 1, a_ 2\in A\). It is shown that under this assumption there exists a hyperplane \(H\) such that \(| A\cap H|\geq \varepsilon | A|\), with \(\varepsilon= \varepsilon(n,c) >0\). This important result appears, for sets of lattice points, in \textit{G. A. Freiman's} book [Foundations of a structural theory of set addition (1973; Zbl 0271.10044)], with a somewhat sketchy proof. A different complete proof was given by \textit{P. Fishburn} [J. Number Theory 35, 325-334 (1990; Zbl 0703.11008)] for \(n=2\). Here a complete proof is given of the theorem in complete generality and of several geometrical results applied.
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11H99
zbMATH DE Number: 556371
zbMATH Keywords: sumsets
zbMATH Keywords: sets of lattice points
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
full work available at URL: jnth.1994.1003
OpenAlex ID: W2083485879
DOI: 10.1006/JNTH.1994.1003
description: scientific article; zbMATH DE number 556371