An isoperimetric method in additive theory (Q1906481)

This paper deals with subsets in groups \(G\) showing that certain conditions above all on their cardinality control their structure. The main theorem reads as follows: Let \(G=\langle B\rangle\), with finite \(B\) containing \(1\). If every element of \(G\setminus\{1\}\) has order \(\geq |B|\), then either (1) \(|XB|\geq\min(|G|-1,|X|+|B|)\) for all finite, at least 2-element subsets \(X\); or (2) there are \(r\neq 1\) and \(j\in\mathbb{Z}\) such that \(B=\{r^i;\) \(j\leq i\leq j+|B|-1\}\). Reformulated as a theorem on critical pairs of subsets it includes earlier results in the literature, among others of A. G. Vosper for prime \(|G|\).