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

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