This paper is a contribution to the theory of inverse theorems in additive combinatorics. Let \((G,+)\) be a finite group with exponent \(\exp(G)\) (recall that \(\exp(G)\) is the mininum non-negative integer \(n\) such that \(g+\stackrel{n}{\dots}g=n\cdot g=0\)). For a subset \(S\subseteq G\), let \(\sum(S)\) be the subset of \(G\) obtained by taking all possible non-empty sums of elements in \(S\). Finally, we say that \(S\) is zero-sum free if \(0\) is not an element in \(\sum(G)\).
The main general question addressed in this paper is the following: what can we say about the structure of \(S\) if we have information concerning \(|S|\) and \(|\sum(S)|\)? In this context, the authors determine the structure of \(S\) when \(|S|=\exp(G)+k\) (with \(k\in\{0,1\}\)) and \(|\sum(S)|=(k+2)\exp(G)-1\), which is the main result of the paper (the explicit structure, with the notation of the paper can be found in the main result of the publication, Theorem 4.3). The proof of the theorem is based on a (long and delicate) case analysis.