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On weighted sums in abelian groups. (English) Zbl 0872.11016

Sei \(G\) eine abelsche Gruppe der Ordnung \(n\) und \(k\in\mathbb{N}\). Sei weiter \(x_0,x_1,\dots,x_{n+k-1}\) eine Folge von Elementen aus \(G\) mit der Eigenschaft, daß \(x_0\) der am häuftigsten auftretende Wert in der Folge ist. Schließlich werden Gewichte \(w_i\) eingeführt als eine Menge \(\{w_i\mid 1\leq i\leq k\}\) ganzer Zahlen teilerfremd zu \(n\). Dann sagt ein Hauptresultat der Arbeit (Theorem 2.1): Es gibt eine Permutation \(\alpha\) von \([1,n+k-1]\), so daß gilt \[ \sum_{1\leq i\leq k}w_ix_{\alpha(i)}= \Biggl(\sum_{1\leq i\leq k}w_i\Biggr)x_0. \] In einem zweiten Teil der Arbeit werden konstante Gewichte \(w_i\) betrachtet.
Reviewer: E.Härtter (Mainz)

MSC:

11B83 Special sequences and polynomials
20K01 Finite abelian groups
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