An inverse theorem for sums of sets of lattice points (Q1320508)

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.