On finite difference sets (Q1271964)

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.