Weighted sums in finite cyclic groups
Let \(C_n\) be the cyclic group of \(n\) elements, \(p\) be a prime, and \(k\in\mathbb{N}\). Let \(\{w_1,\dots,w_k\}\) be a sequence of \(k\) integers such that \(w_1+\cdots+ w_k \equiv 0\pmod{p^2}\). Then, for every sequence \(a_1,a_2,\dots\), of \(p^2+1\) elements in \(C_{p^2}\), there are \(k\) distinct indices \(i_1,i_2,\dots,i_k\) such that \(w_1a_{i_1}+ \cdots+ a_{i_k}=0\). This generalizes a result of \textit{Y. O. Hamidoune} [Discrete Math. 162, No. 1--3, 127--132 (1996; Zbl 0872.11016).
scientific article; zbMATH DE number 2083735