Subsets with small sums in abelian groups. I: The Vosper property (Q1362996)

This paper characterizes the finite subsets \(B\) in an abelian group \(G\) such that, for any finite subset \(A\) having at least two elements, \(|A + B|\geq\) min\((|G|-1, |A|+|B|)\). The approach uses graph-theoretic ideas on Cayley graphs, including the concepts of ``fragments'' and ``atoms'' in a graph (as in \textit{H. A. Jung} [Math. Ann. 202, 307-320 (1973; Zbl 0239.05133)]). Applications are given to diagonal forms over finite fields and to characterizing Cayley graphs (or loop networks) of high connectivity.