Weighted sums in finite abelian groups

In this note we prove the following weighted generalization of Bollob'as and Leader theorem (J. Number Theory { \bf{78}} (1999), no. 1, 27--35): Let $G$ be an abelian group of order $n$ and $k$ a positive integer. Let $(w_1,w_2, \dots, w_k)$ be a sequence of integers where each $w_i$ is co-prime to $n$. Then, given a sequence $(x_1,x_2, \dots, x_{k+r})$ of elements of $G$, where $1 \le r \le n-1$, if $0$ is the most repeated element in the sequence, and $ \sum_1^kw_ix_{ \sigma(i)} \neq 0$, for all permutations$ \sigma$ of ${1, 2, \dots, k+r }$, we have $$ |{\sum_1^kw_ix_{\sigma(i)}: \sigma \textrm{is a permutation of} {1,2, \dots, k+r \}}| \ge r+1 .$$