On a theorem of J. E. Olson and an application. (Vanishing sums in finite abelian \(p\)-groups)
Let \(p\) be a prime number and let \(P\) be the direct product of cyclic groups of order \(p^{n_1}, \cdots, p^{n_s}\). Let \(S_1, \cdots, S_k\) be sets of integers noncongruent modulo \(p\). Assume \(| S_1|+ \cdots+ | S_k| \geq p^{n_1}+ \cdots+ p^{n_s}-s+1\). The authors show that for every sequence \(g_1, \cdots, g_k\) of elements of \(A\), there are \(s_i\in (S_i\cup \{0\})\) such that \(s_1g_1+\cdots +s_kg_k=0\) and \(| s_1| +\cdots +| s_k| >0\).
scientific article; zbMATH DE number 1199969