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 $w=(w_1,...,w_{m-r})\in {\mathbb{Z}}^{m-r}.$ 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 $$\sum _{i\in I}{w}_i{x}_{f(i)}=(\sum _{i\in I}{w}_i)x_m.$$

5 pages