On finite difference sets
type: Item
instance of: scholarly article
title: On finite difference sets
zbMATH Open document ID: 0902.05009
DOI: 10.1023/A:1006513923148
published in: Acta Mathematica Hungarica
publication date: 1998-11-22T00:00:00Z
review text: Let \(A\) denote a finite subset of \({\mathbb{R}}^n\). The difference set of \(A\) is given by \(A-A=\{a-b:a,b\in A\}\). The affine dimension of \(A\), denoted by \(d=\dim A\), is defined as the dimension of the smallest affine subspace containing \(A\). \textit{G. A. Freiman}, \textit{A. Heppes}, and \textit{B. Uhrin} derived the general lower bound \(| A-A|\geq(d+1)| A| +{1\over 2}d(d+1)\) which is sharp for \(d=1,2\) [Proc. Conf. Number Theory, Budapest 1987, Colloq. Math. Soc. János Bolyai 51, 125-139 (1990; Zbl 0707.11011)]. They also conjectured that \(\dim A=3\) implies \(| A-A|\geq 4.5| A|-9\). In the paper under review, this estimate is proved, and it is also shown that it is best possible. Moreover, the precise structure of those finite sets \(A\) with difference set of lowest power is described: If \(\dim A=2\), then \(A=P+\{0,v\}\) where \(P\) denotes an arithmetic progression with difference \(\delta\) and \(\{\delta,v\}\) is linearly independent. If \(\dim A=3\), then \(A=P+\{0,v\}+\{0,w\}\) where the set \(\{\delta,v,w\}\) is linearly independent.
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 11B30
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11H06
zbMATH DE Number: 1225598
zbMATH Keywords: difference sets
zbMATH Keywords: lower bounds for cardinality
zbMATH Keywords: best possible lower bounds
zbMATH Keywords: arithmetic progression
author: Y. V. Stanchescu
MaRDI profile type: Publication
full work available at URL: a:1006513923148
OpenAlex ID: W1864933544
description: scientific article; zbMATH DE number 1225598