\(t\)-sum generators of finite Abelian groups
Given a finite Abelian group \(A\) and an integer \(t\), \(1\leq t\leq| A|-1\), a subset \(S\) of \(A\) is called a \(t\)-sum generator of \(A\) if every element of \(A\) can be written as the sum of exactly \(t\) distinct elements of \(S\). The authors investigate the minimal integer \(M(t,A)\) such that every subset of \(A\) of size greater than \(M(t,A)\) is a \(t\)-sum generator of \(A\). (When \(A\) has no \(t\)-sum generator, \(M(t,A)=| A|.)\) For Abelian finite of even order, the authors get the following result: Let \(A\) be a finite Abelian group of even order \(\geq 12\). Then, for \(3\leq t\leq| A|/2-2\), \(M(t,A)=| A|/2\), except when \(A\in\{\mathbb{Z}^ h_ 2,\mathbb{Z}_ 4\times\mathbb{Z}_ 2^{h-1}\}\) and \(t\in\{3,| A|/2-2\}\), in which case \(M(t,A)=| A|/2+1\). For a group \(A\) of odd order, the problem of determining \(M(t,A)\) remains open. The case of odd-order groups seems to be much more complicated than that of even-order groups. The authors present some bounds on \(M(t,\mathbb{Z}^ h_ p)\), \(p\) an odd prime.
scientific article; zbMATH DE number 69891