Result: ON A PROPERTY OF MINIMAL ZERO-SUM SEQUENCES AND RESTRICTED SUMSETS: On a property of minimal zero-sum sequences and restricted sumsets

Let \(G\) be an additively written abelian group, and let \(S\) be a sequence in \(G\backslash \{0\}\) with length \(| S | \geq 4\). Suppose that \(S\) is a product of two subsequences, say \(S=BC\) such that the element \(g+h\) occurs in the sequence \(S\) whenever \(g \cdot h\) is a subsequence of \(B\) or \(C\). It is proved that \(S\) contains a proper zero-sum subsequence, apart from some explicitly characterized cases. Essentially, this means that minimal zero-sum sequences are not additively closed. An important corollary is: Let \(G\) be an abelian group, \(S \subset G \backslash\{0\}\) a finite subset with \(| S | \geq 7\). Let \(S\) be a partitioned into two sets \(B\) and \(C\) with \(\min (| B | , | C |)\geq 2\). Let \((B\dot+B) \cup (C\dot+C)\subset S\), where \(\dot+\) denotes restricted sumsets. Then there exists a proper subset of \(S\) which sums to zero. The proof makes use of Kneser's theorem.