A weighted generalization of Gao's n+D-1 Theorem

Abstract

Let $G$ denotes a finite abelian group of order $n$ and Davenport constant $D$, and put $m= n+D-1$. Let $x=(x_1, ..., x_m)\in G^m$ be a sequence with a maximal repetition $\ell$ attained by $x_m$ and put $r=\min(D,\ell)$. Let $w=(w_1, ..., w_{m-r})\in \Z^{m-r}.$ Then there are an $n$-subset $I\subset [1,m-r]$ and an injection $f: I\mapsto [1,m]$, such that $m\in f(I)$ and $$\sum_{i\in I}w_{i}x_{f({i})}=(\sum_{i\in I}w_{i})x_{m}.$$