Weighted sums in finite cyclic groups. (English) Zbl 1052.11014

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 Y. O. Hamidoune [Discrete Math. 162, No. 1–3, 127–132 (1996; Zbl 0872.11016).


11B75 Other combinatorial number theory
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20K01 Finite abelian groups


Zbl 0872.11016
Full Text: DOI


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